Expandierende und kontrahierende Flächen

Oliver C. Schnürer

FU Berlin, Arnimallee 2-6, 14195 Berlin, Germany

E-mail address: Oliver.Schnuerer@math.fu-berlin.de

2000 Mathematics Subject Classification. 53C44, 35B40, 65K05, 68W20

Abstract. We show that strictly convex surfaces contracting with normal velocity equal to |A|^{2} shrink to a point in finite time. After appropriate rescaling, they converge to spheres.

Strictly convex surfaces that expand by the inverse Gauß curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres.

We describe the algorithm to find our main test functions.

Contents

Part 1. Expandierende und kontrahierende Flächen v

Chapter 1. Einleitung vii 1.1. Problemstellung vii 1.2. Literaturüberblick viii 1.3. Existenz einer Lösung ix 1.4. Resultate und Beweisüberblick x 1.5. Perspektiven xiii 1.6. Gliederung xiii 1.7. Danksagung xiv

Chapter 2. Differential Geometry and Evolution Equations xv 2.1. Differential Geometry and Notation xv 2.2. Evolution Equations xvii

Part 2. Surfaces Contracting with Normal Velocity |A|^{2} 1

Chapter 3. Introduction 3

Chapter 4. Surfaces Flowing by |A|^{2} 5 4.1. Convergence to a Point 5 4.2. A Monotone Quantity 6 4.3. Direct Consequences 9 4.4. Convergence to a Round Point 10 4.5. Convergence Rate 15

Chapter 5. Finding Monotone Quantities 17 5.1. The Algorithm 17 5.2. Motivation and Randomized Tests 17

Chapter 6. Computing Evolution Equations 19

Chapter 7. Other Normal Velocities 21 7.1. H^{3}-Flow 21 7.2. H^{2}-Flow 27 7.3. H^{4}-Flow 28 7.4. |A|^{2} + βH^{2}-Flow 30 7.5. tr A^{3}-Flow 32 7.6. tr A^{α}-Flow 35 7.7. H|A|^{2}-Flow 38 7.8. |A|^{4}-Flow 40 Part 3. Surfaces Expanding by the Inverse Gauß Curvature Flow 43

Chapter 8. Introduction 45

Chapter 9. Inverse Gauß Curvature Flow 47 9.1. A Monotone Quantity 47 9.2. Convergence to Infinity 51 9.3. Convergence to a Sphere 53 9.4. Smooth Convergence to a Sphere 54 9.5. Improved Convergence Rate 55 9.6. Convergence Rate 59

Chapter 10. Finding Monotone Quantities 61 10.1. The Algorithm 61 10.2. Motivation and Randomized Tests 61

Chapter 11. Other Normal Velocities 63 11.1. Homogeneity less than minus one 63 11.2. Homogeneity minus one 67

Bibliography 69

Part 1

Expandierende und kontrahierende Flächen

CHAPTER 1 Einleitung

1.1. Problemstellung

In dieser Arbeit beschäftigen wir uns mit Flächen im dreidimensionalen euklidischen Raum, die sich in Abhängigkeit ihrer Krümmung bewegen. Wir betrachten stets glatte, zusammenhängende, strikt konvexe, geschlossene, eingebettete Flächen, die sich entweder zusammenziehen oder ausdehnen.

Unsere beiden wichtigsten Beispiele sind dabei die Flussgleichung

(1.1)  \frac{d}{dt}X = −|A|^{2}ν

für kontrahierende Flächen, wobei sich eine Fläche mit Normalengeschwindigkeit gleich dem Quadrat der Norm der zweiten Fundamentalform zusammenzieht, und der inverse Gaußkrümmungsfluss

(1.2)  \frac{d}{dt}X = \frac{1}{K} ν

für expandierende Flächen.

Wir erklären nun, wann sich eine Fläche gemäß einer Evolutionsgleichung

(1.3)  \frac{d}{dt}X = −F ν

bewegt. Sei M eine abstrakte, d. h. nicht notwendigerweise eingebettete, zweidimensionale Mannigfaltigkeit. Für unsere Zwecke können wir hier stets die zweidimensionale Sphäre 𝕊^{2} ⊂ ℝ^{3} betrachten. Sei X(·, t) : M → ℝ^{3} eine glatte Familie von regulären Einbettungen in einem Zeitintervall, t ∈ [0, T), T > 0. Dies definiert eine Familie von Flächen M_{t} := X(M, t) in ℝ^{3}. Diese Flächen M_{t} besitzen in jedem Punkt X(x, t) ∈ M_{t} zwei Normalenvektoren der Länge eins. Davon wählen wir einen aus, in unserer Situation konvexer geschlossener Flächen M_{t} stets die äußere Normale, ν(x). Wir sagen nun, dass Flächen M_{t} die Flussgleichung (1.3) lösen, wenn es eine zugehörige Abbildung X wie oben beschrieben gibt, so dass in jedem Punkt (x, t) ∈ M × [0, T) die Gleichung (1.3) für eine gegebene Funktion F erfüllt ist.

Die Einbettungen X(·, t) induzieren auf M in jedem Punkt eine Metrik

(g_{ij})_{1≤i, j≤2}

und eine zweite Fundamentalform

(h_{ij})_{1≤i, j≤2}.

Wir identifizieren M ×{t} mit X(M, t) und erhalten in jedem Punkt x ∈ M_{t} Metrik und zweite Fundamentalform. In dieser Arbeit ist die Normalengeschwindigkeit F stets eine symmetrische Funktion der Eigenwerte (λ_{1}, λ_{2}) von h_{ij} bezüglich g_{ij},

|A|^{2} = λ_{1}^{2} + λ_{2}^{2}, K = λ_{1} · λ_{2}.

Diese Eigenwerte heißen Hauptkrümmungen und sind nur bis auf Permutation wohldefiniert. Daher sind nur symmetrische Funktionen der Hauptkrümmungen wohldefiniert.

Wir werden Flussgleichungen (1.3) mit der Anfangsbedingung, dass M_{0} eine vorgegebene glatte, strikt konvexe, geschlossene Fläche in ℝ^{3} ist, untersuchen. Strikt konvex bedeutet hier, dass die Hauptkrümmungen überall auf M_{0} jeweils strikt positiv sind.

1.2. Literaturüberblick

1.2.1. Kontrahierende Flüsse. Flussgleichungen der Form (1.3) und ihre Analoga für Hyperflächen werden schon seit mehreren Jahrzehnten intensiv untersucht.

Wir betrachten zunächst kontrahierende Flüsse, d. h. solche, für die F positiv ist. William Firey betrachtete den Gaußkrümmungsfluss für Flächen als Modell, das beschreibt, wie konvexe Steine, die in einem Fluss oder am Strand aus allen Richtungen mit gleicher Wahrscheinlichkeit kleine Stöße abbekommen, allmählich schrumpfen. Er vermutete, dass sich die Flächen in endlichen Zeit zu einem Punkt zusammenziehen und dass sie nach geeignetem Reskalieren zu einer runden Sphäre konvergieren. Unter den Annahmen, dass eine Lösung der Flussgleichung existiert und dass die Anfangswerte punktsymmetrisch sind, konnte er dies auch beweisen [24]. Kaising Tso zeigte, dass Lösungen in jeder Dimension existieren [59] und Ben Andrews führte den Beweis für Flächen ohne anfängliche Symmetrieannahme [12]. Die letztgenannte Arbeit ist der vorliegenden in einigen Aspekten ähnlich. Daher werden wir später nochmals hierauf zurückkommen.

Gerhard Huisken bewies [32], dass sich konvexe Hyperflächen unter dem Fluss entlang ihrer mittleren Krümmung in endlicher Zeit zu einem Punkt zusammenziehen und, geeignet reskaliert, zu Sphären konvergieren, d. h. sie konvergieren zu “runden Punkten”. Die mittlere Krümmung ist positiv homogen vom Grade eins in den Hauptkrümmungen,

H = λ_{1} + … + λ_{n}.

Konvergenz zu runden Punkten für (strikt) konvexe Anfangswerte wurde danach noch häufiger für andere Flussgleichungen bewiesen, bei denen die Normalengeschwindigkeit positiv homogen vom Grade eins in den Hauptkrümmungen ist, z. B. von Ben Andrews [5–7] und Bennett Chow [15, 16]. Für Hyperflächen, die sich unter einer Flussgleichung zusammenziehen, bei denen die Normalengeschwindigkeit positiv homogen von einem Grad strikt größer als eins ist, haben Ben Andrews [5] und Felix Schulze [54] gezeigt, dass eine geeignete anfängliche “Pinchingbedingung”, d. h. eine geeignete Schranke für den Quotienten beliebiger Hauptkrümmungen λ_{i}/λ_{j}, in jedem Punkt auf M_{0} ausreicht, um Konvergenz gegen einen runden Punkt nachzuweisen.

Ist die Homogenität der Normalengeschwindigkeit positiv, aber strikt kleiner als eins, so konvergieren im allgemeinen nicht mehr alle strikt konvexen Hyperflächen gegen runde Punkte. Dies erhält Ben Andrews [13], falls die Normalengeschwindigkeit F eine geeignete Potenz der Gaußkrümmung ist. Insbesondere schrumpft jedes Ellipsoid als affines Bild einer Sphäre unter dem affin invarianten Gaußkrümmungsfluss mit F = K^{1/(n+2)} homothetisch zu einem Punkt. Analoge Resultate gibt es auch von Koichi Anada und Masayoshi Tsutsumi für den harmonischen mittleren Krümmungsfluss [2, 3].

Kontrahierende Hyperflächen wurden auch in Mannigfaltigkeiten betrachtet, vgl. z. B. [8, 9, 33], ebenso für anisotrope Krümmungsflüsse, vgl. z. B. [11]. Für kontrahierende Kurven gibt es weitergehende Resultate, z. B. [25, 28]. Hier braucht man keine Konvexität für die Anfangskurve zu fordern. Für den Riccifluss wurde ebenso das Rundwerden von Mannigfaltigkeiten untersucht [29, 30]. Kompliziertere Verhaltensweisen in Singularitäten sind für den mittleren Krümmungsfluss [1, 37–39] und für den Riccifluss [31, 46–48] untersucht worden.

Außer dem Resultat von Ben Andrews ([12] und [13, Theorem 3/4] im nicht strikt konvexen Fall) für den Gaußkrümmungsfluss für Flächen ist uns kein Resultat bekannt, das für einen kontrahierenden Krümmungsfluss mit Normalengeschwindigkeit einer Homogenität strikt größer als eins in den Hauptkrümmungen Konvergenz konvexer Anfangswerte gegen einen runden Punkt ohne anfängliche Pinchingbedingung liefert. In der vorliegenden Schrift zeigen wir dies beispielsweise für strikt konvexe Flächen, die sich mit Normalengeschwindigkeit |A|^{2} zusammenziehen (1.1).

1.2.2. Expandierende Flüsse. Bisher wurden expandierende Flüsse intensiv für Normalengeschwindigkeiten untersucht, die positiv homogen vom Grade minus eins in den Hauptkrümmungen sind. Claus Gerhardt und John Urbas haben jeweils sternförmige Hyperflächen betrachtet, die für t → ∞ exponentiell gegen Unendlich und nach geeignetem Reskalieren gegen Sphären konvergieren [26, 60, 61]. John Urbas erhält hier entsprechende Resultate auch für negative Homogenitäten, die größer als minus eins sind. Wir wollen dieses Langzeitverhalten auch als Konvergenz gegen Sphären im Unendlichen bezeichnen. Knut Smoczyk fand eine Darstellungsformel für den inversen harmonischen mittleren Krümmungsfluss [56]. Gerhard Huisken und Tom Ilmanen haben mittels einer schwachen Formulierung des inversen mittleren Krümmungsflusses die Penroseungleichung bewiesen [34]. Weitergehende Untersuchungen hierzu finden sich in [35, 55].

Inhomogene Flüsse, deren Lösungen für t → ∞ gegen Sphären im Unendlichen konvergieren, finden sich in [14, 19, 20, 42].

Expandierende Krümmungsflüsse einer Homogenität strikt kleiner als minus eins scheinen bisher noch überhaupt nicht auf Konvergenz gegen Sphären nach Reskalieren untersucht worden zu sein. Unter solchen Flüssen expandieren Sphären in endlicher Zeit nach Unendlich. In dieser Arbeit zeigen wir, dass beispielsweise Lösungen des inversen Gaußkrümmungsflusses (1.2) für strikt konvexe Anfangsflächen in endlicher Zeit gegen runde Sphären im Unendlichen konvergieren.

1.3. Existenz einer Lösung

Die Normalengeschwindigkeiten F in (1.1) und (1.2) erfüllen die Bedingung

\frac{∂F}{∂λ_{i}} > 0

genau in den Punkten, in denen M_{t} strikt konvex ist. Daher erhalten wir für glatte strikt konvexe Anfangswerte einen parabolischen Krümmungsfluss, der auf einem maximalen Zeitintervall [0, T), 0 < T ≤ ∞, eine glatte Familie strikt konvexer, kompakter Flächen M_{t} besitzt, die (1.1) bzw. (1.2) erfüllen [36]. Dies bezeichnet man als Kurzzeitexistenzresultat.

Ist für alle Zeiten 0 ≤ t < T und alle x ∈ M die Norm des Einbettungsvektors |X(x, t)| gleichmäßig beschränkt und erfüllen auf M ×[0, T) die Hauptkrümmungen λ_{i} für eine Konstante c > 0 die Ungleichung 0 < \frac{1}{c} ≤ λ_{i} ≤ c, i = 1, 2, so erhalten wir aufgrund der Hölderschranken für zweite Ableitungen von Nicolai Krylov, Mikhail Safonov, Lawrence Evans, Ben Andrews [4] und der Schauderabschätzungen durch Betrachtung der Gleichung für die im Folgenden noch definierte Stützfunktion, dass sich die Familie M_{t} noch in glatter Weise bis zur Zeit t = T fortsetzen lässt. Auf die Fläche M_{T} können wir nun das Kurzzeitexistenzresultat anwenden und erhalten eine glatte Familie strikt konvexer Flächen M_{t} auf einem Zeitintervall [0, T + ε), ε > 0. Dies widerspricht der Annahme, dass T maximal war. Daher existiert eine Lösung auf einem Zeitintervall [0, T) mit T = ∞, was aus anderen Gründen für (1.1) und (1.2) leicht auszuschließen ist, oder 0 < T < ∞ und es gilt

lim sup_{t↑T} sup_{M_{t}} |X(·, t)| =∞,
lim inf_{t↑T} inf_{M_{t}} min_{i=1, 2} λ_{i} =0,

oder

lim sup_{t↑T} sup_{M_{t}} max_{i=1, 2} λ_{i} =∞.

1.4. Resultate und Beweisüberblick

1.4.1. Kontrahierende Flächen. Für kontrahierende Flächen lautet unser wichtigstes Resultat

Theorem 1.1. Sei M ⊂ ℝ^{3} eine glatte, strikt konvexe, geschlossene Fläche. Dann gibt es T, 0 < T < ∞, und eine glatte Familie M_{t}, t ∈ [0, T), von strikt konvexen geschlossenen Flächen in ℝ^{3}, die die Flussgleichung

\frac{d}{dt}X = −|A|^{2}ν

mit M_{0} = M lösen. Für t ↑ T konvergieren die Flächen M_{t} gegen einen Punkt Q ∈ ℝ^{3}. Die reskalierten Flächen (M_{t} − Q) · (6(T − t))^{−1/3} konvergieren in glatter Weise für t ↑ T gegen die Einheitssphäre 𝕊^{2} ⊂ ℝ^{3}.

Das Theorem besagt also, dass zentrierte Sphären, die sich homothetisch zusammenziehen und deren Radius durch (6(T − t))^{−1/3} gegeben ist, das typische Verhalten für diesen Fluss zeigen.

Im ersten Teil des Beweises zeigen wir ganz ähnlich wie Kaising Tso in [59], dass sich die Flächen M_{t} in endlicher Zeit zu einem Punkt zusammenziehen: Schon der Vergleich mit einer anfänglich M umschließenden Sphäre zeigt, dass Lösungen aufgrund des Maximumprinzips nur für endliche Zeit existieren können. Solange die Flächen eine Sphäre mit positivem Radius umschließen, erhalten wir obere Schranken an |A|^{2}. Da positive untere Schranken an die Hauptkrümmungen erhalten bleiben, folgt die Konvergenz zu einem Punkt.

In Beweisen, dass Hyperflächen nach Reskalieren gegen Sphären konvergieren, spielt häufig eine Größe, deren Skalierungsverhalten “besser” als skalierungsinvariant ist, eine Schlüsselrolle. Hier haben wir das folgende Resultat

Theorem 1.2. Unter den Voraussetzungen von Theorem 1.1 ist

max_{M_{t}} \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}}

während des Flusses nichtwachsend.

Aufgrund des Skalierungsverhaltens dieser Testgröße erhalten wir sofort, dass die einzigen glatten, strikt konvexen, geschlossenen Lösungen, die sich homothetisch zusammenziehen, sich homothetisch kontrahierende Sphären sind. Auch solch eine Klassifikation selbstähnlicher Lösungen scheint für die hier betrachteten Flussgleichungen neu zu sein.

Heuristisch erwarten wir, dass Theorem 1.2 sofort Konvergenz gegen Sphären nach Reskalieren liefert, denn wir erwarten, dass die beiden Hauptkrümmungen überall gegen Unendlich konvergieren. Damit wird die Fläche überall fast umbilisch. Da die einzigen umbilischen, strikt konvexen, vollständigen Flächen in ℝ^{3} Sphären sind, sollte also aus obigen Argumenten Konvergenz gegen Sphären folgen.

Im Gegensatz zu Flussgleichungen der Homogenität eins (d. h. die Normalengeschwindigkeit ist positiv homogen vom Grade eins in den Hauptkrümmungen) ist die hier gefundene monotone Größe keine Modifikation einer skalierungsinvarianten Größe wie in [32]. Aufgrund der Gegenbeispiele in [5] erwarten wir hier überhaupt nicht, dass es entsprechende, für den Konvergenzbeweis gegen Sphären geeignete skalierungsinvariante Größen gibt.

Theorem 1.2 folgt unmittelbar aus dem Maximumprinzip. Integralabschätzungen sind hier nicht nötig.

Wir finden auch noch andere Testgrößen, die eine zu Theorem 1.2 analoge Aussage liefern. Durch die Betrachtung komplizierterer Größen können wir bessere Konvergenzraten erhalten.

Eine weitere Folgerung aus Theorem 1.2 ist, dass der Pinchingquotient λ_{1}/λ_{2} (und damit auch λ_{2}/λ_{1}, da λ_{1} und λ_{2} nur bis auf Permutation definiert sind) während der Evolution beschränkt bleibt.

Der Rest des Beweises ähnelt nun dem von Ben Andrews für den Gaußkrümmungsfluss [12]. Basierend auf einer besser als skalierungsinvarianten Abschätzung für |λ_{1} − λ_{2}| aus Theorem 1.2 zeigen wir zunächst, dass die Flächen zwischen je zwei zentrierten Sphären liegen, deren Radien auch nach geeignetem Reskalieren für t ↑ T noch gegeneinander konvergieren. Mittels einer Harnackungleichung erhalten wir dann, dass sich auch die Normalengeschwindigkeit nur wenig von der einer selbstähnlich kontrahierenden Sphäre unterscheidet. Dies gilt dann auch für die Hauptkrümmungen und damit haben wir so gute Pinchingabschätzungen, dass Konvergenz auch nach Reskalieren aus bekannten Resultaten [5] folgt.

Zu Theorem 1.1 und Theorem 1.2 analoge Resultate haben wir auch für weitere Flussgleichungen mit Homogenitäten strikt größer als eins bewiesen.

1.4.2. Algorithmen. Die Testgröße in Theorem 1.2 haben wir mit Hilfe eines Computerprogrammes gefunden, das testet, ob eine symmetrische, als Quotient zweier Polynome gegebene Funktion, geeignet ist, um Konvergenz zu Sphären zu liefern, d. h. mögliche Testgrößen sollen insbesondere in umbilischen Punkten (λ_{1} = λ_{2}) ein globales Minimum haben und ein geeignetes Skalierungsverhalten aufweisen. Wir rechnen dann in Normalkoordinaten für solche Kandidaten w die Evolutionsgleichung, d. h. \frac{d}{dt}w − F^{ij}w_{; ij}, in der Notation von Kapitel 2, in einem kritischen Punkt von w aus und werten diese an Zufallszahlen für die Hauptkrümmungen und die Ableitungen der zweiten Fundamentalform aus, die so gewählt sind, dass w_{; i} = 0 erfüllt ist, d. h. wir benützen auch, dass wir in einem kritischen Punkt von w sind. Dies wiederholen wir einige Male. Erhalten wir stets ein Vorzeichen, das die Anwendbarkeit des Maximumprinzips nicht ausschließt, untersuchen wir diesen Kandidaten von Hand genauer. Bisher waren alle so gewonnenen Kandidaten geeignet, um Konvergenz gegen runde Punkte zu zeigen. Wir können auf diese Weise viele mögliche Testgrößen untersuchen.

Die weiteren Betrachtungen hängen jedoch nicht von der Richtigkeit des Programmes ab. Wir benutzen es lediglich, um geeignete Testgrößen zu “raten”.

Im Gegensatz zu den analogen Betrachtungen für expandierende Flüsse scheint es für kontrahierende Flächen zu den meisten Flussgleichungen monotone Größen zu geben.

Der angegebene Algorithmus läßt sich auch auf Hyperflächen anwenden. Für dreidimensionale Hyperflächen lieferte er uns jedoch keine monotonen Größen.

Wir wollen noch bemerken, dass es in den Evolutionsgleichungen für Flächen auch mit Hilfe des Sturmschen Algorithmusses anstatt mit Zufallszahlen möglich wäre, das Vorzeichen der in der Evolutionsgleichung auftretenden Terme algebraisch zu untersuchen.

Die hier betrachteten Testgrößen lassen sich leicht als Funktionen der Elemente einer algebraischen Basis, H und |A|^{2} für Flächen, darstellen. Die zugehörigen Evolutionsgleichungen erhält man daraus mit Hilfe einfacher algebraischer Formeln. Auf diese Weise können wir sie leicht mit Hilfe eines Computeralgebraprogrammes ausrechnen lassen. Für unsere beiden Modellgleichungen, (1.1) und (1.2), haben wir jedoch ausführliche und ohne Computer nachvollziehbare Berechnungen der jeweiligen Evolutionsgleichungen in dieser Arbeit.

1.4.3. Expandierende Flächen. In Teil 3 der Arbeit erweitern wir unsere Betrachtungen auf expandierende Flächen. Dabei interessieren wir uns insbesondere für solche Flussgleichungen, die eine Homogenität kleiner oder gleich minus zwei haben. Für Homogenitäten strikt kleiner als minus eins konvergieren Sphären in endlicher Zeit homothetisch nach Unendlich.

Unser Modelltheorem für expandierende Flächen lautet

Theorem 1.3. Sei M ⊂ ℝ^{3} eine glatte, strikt konvexe, geschlossene Fläche. Dann gibt es T, 0 < T < ∞, und eine glatte Familie M_{t}, t ∈ [0, T), von strikt konvexen geschlossenen Flächen in ℝ^{3}, die die Flussgleichung

\frac{d}{dt}X = \frac{1}{K} ν

mit M_{0} = M lösen. Für t ↑ T konvergieren die Flächen M_{t} gegen Unendlich. Die reskalierten Flächen (M_{t} −Q)·(T −t) konvergieren in glatter Weise für t ↑ T gegen die Einheitssphäre 𝕊^{2} ⊂ ℝ^{3}.

Auch hier weisen wir also nach, dass das Verhalten von Sphären unter dieser Evolutionsgleichung das typische ist.

Hier benötigen wir schon ganz zu Anfang des Beweises das folgende

Theorem 1.4. Unter den Voraussetzungen von Theorem 1.3 ist

max_{M_{t}} \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}}

während des Flusses nichtwachsend.

Der Beweis davon benutzt wiederum nur das Maximumprinzip.

Auch diese monotone Größe liefert eine Klassifikation selbstähnlicher Lösungen wie im Falle kontrahierender Flächen, da die Größe ein entsprechendes Skalierungsverhalten besitzt.

Im expandierenden Fall benötigen wir diese Größe schon um zu zeigen, dass die Flächen während des Flusses gepincht bleiben, d. h. dass \frac{λ_{1}}{λ_{2}} + \frac{λ_{2}}{λ_{1}} beschränkt bleibt. Wir erhalten positive untere Schranken an die Geschwindigkeit, solange die betrachteten Flächen in einem Ball enthalten sind. Dies sichert, dass die Evolutionsgleichung so lange eine Lösung besitzt, bis ein Teil der Fläche nach Unendlich konvergiert. Durch Vergleich mit Sphären erhalten wir, dass dies schon in endlicher Zeit passiert. Nach einem Spiegelungsargument von Aleksandrov muss dann aber auch die gesamte Fläche gegen Unendlich konvergieren und die zugehörigen Oszillationsabschätzungen für die Stützfunktion liefern bereits Konvergenz gegen runde Sphären im Unendlichen.

Durch Betrachtung komplizierterer Testgrößen können wir auch hier die Konvergenzraten verbessern. Dafür argumentieren wir dann wieder ähnlich wie in Teil 2.

Auch hier erhalten wir ähnliche monotone Größen für andere Flussgleichungen, es handelt sich dabei um solche einer Homogenität kleiner oder gleich minus zwei.

Der zum Auffinden der monotonen Größen verwendete Algorithmus ist nahezu identisch zu dem im Falle kontrahierender Flächen oben bereits beschriebenen.

1.5. Perspektiven

In der geometrischen Analysis ist es wichtig, geeignete Testfunktionen zu finden. Häufig weiß man dabei, von welchen Größen die Testfunktionen abhängen dürfen und welche Eigenschaften sie idealerweise zu erfüllen haben. In solchen Situationen kann man dann wie hier den Computer zum Einsatz bringen um herauszufinden, ob eine Vermutung richtig sein kann, ähnlich wie hier beim Auffinden der monotonen Größen, oder einfach nur, um kompliziertere Rechnungen durchzuführen.

Durch das Testen mit Zufallszahlen haben wir uns schon früher plausibel gemacht, dass Lemma 4.1 in [50] gelten sollte, aber auch, dass es nicht sinnvoll ist, die I-Methode, vgl. [21–23], weiter zu iterieren.

Im Abschnitt 4.5 untersuchen wir eine Testgröße, deren Evolutionsgleichung keine “konstanten Terme” mehr enthält. Für alle anderen Evolutionsgleichungen treten solche Terme aber noch auf. Wir erwarten, dass man bei aufwändigerer Suche auch in diesen Fällen monotone Größen finden könnte, die eine bessere Konvergenzrate liefern und in deren Evolutionsgleichung keine konstanten Terme mehr auftreten. Möglicherweise reicht es hierfür nicht mehr aus, lediglich Quotienten von Polynomen in den Hauptkrümmungen zu betrachten.

Für den mittleren Krümmungsfluss sind nach punktförmigen Singularitäten [32] auch zylinderförmige untersucht worden [39]. Vielleicht lassen sich ähnliche Singularitäten auch für Flussgleichungen höherer Homogenität untersuchen.

Die Arbeiten [40] und [45] zeigen die Verbindungen zwischen Blätterungen und physikalischen Größen auf. So etwas könnte auch für die durch den inversen mittleren Krümmungsfluss gegebene Blätterung möglich sein. Zumindest im euklidischen Fall ist es möglich, mit Hilfe der hier untersuchten Größen das Verhalten solche Blätterungen genauer zu studieren.

1.6. Gliederung

Der Rest der Arbeit gliedert sich wie folgt. In Kapitel 2 erklären wir unsere differentialgeometrischen Notationen und leiten einige Evolutionsgleichungen her.

Im zweiten Teil betrachten wir Flächen, die sich mit Normalengeschwindigkeit |A|^{2} oder ähnlichen Geschwindigkeiten zusammenziehen. In Kapitel 4 beweisen wir das Theorem 1.1 unter Benutzung der monotonen Größe aus Theorem 1.2. Dann beschreiben wir in Kapitel 5 unseren Algorithmus und in Kapitel 6 erläutern wir, wie man Evolutionsgleichungen mit Computeralgebra ausrechnen kann. Am Ende dieses Teiles betrachten wir noch einige weitere kontrahierende Flüsse und beweisen auch hier Konvergenz gegen runde Punkte.

Im dritten Teil betrachten wir den inversen Gaußkrümmungsfluss. In Kapitel 9 beweisen wir Theorem 1.3 mit Hilfe der monotonen Größe aus Theorem 1.4. Unser Algorithmus zum Auffinden monotoner Größen bei expandierenden Flächen unterscheidet sich etwas von dem für den kontrahierenden Fall. Wir beschreiben ihn in Kapitel 10. In Kapitel 11 betrachten wir Flüsse, deren Lösungen wie beim inversen Gaußkrümmungsfluss in endlicher Zeit gegen runde Sphären im Unendlichen konvergieren.

1.7. Danksagung

Diese Arbeit entstand als der Autor an der Freien Universität in Berlin in der Arbeitsgruppe von Klaus Ecker war. Wir möchten uns recht herzlich für die Unterstützung während dieser Zeit bedanken.

Wir beschäftigten uns schon früher mit ähnlichen Problemen: Während unseres Aufenthaltes als Feodor-Lynen Stipendiat der Alexander von Humboldt-Stiftung bei Shing-Tung Yau in Harvard, Cambridge, Massachussetts, U.S.A., und während unseres Stipendiums am Max-Planck-Institut für Mathematik in den Naturwissenschaften bei Jürgen Jost in Leipzig. Wir möchten auch hier für die Unterstützung danken.

Speziell möchten wir uns noch bedanken bei

• Klaus Ecker, der uns auf die parabolische Version der Aleksandrovreflexion aufmerksam machte,

• Jörg Härterich, Gerhard Huisken und Felix Otto für Diskussionen über Verbesserungen der Konvergenzrate,

• Kashif Rasul, dessen C-Compiler Optionen unser Programm schneller laufen ließen,

• Olaf Schnürer, der uns an gute algebraische Basen erinnerte,

• Felix Schulze für Diskussionen über den Konvergenzbeweis in [12],

• John Stalker, der uns auf den Sturmschen Algorithmus aufmerksam machte.

CHAPTER 2 Differential Geometry and Evolution Equations

In this chapter, we recall standard equations from differential geometry and evolution equations, see e. g. [32, 36, 50, 51].

2.1. Differential Geometry and Notation

We use X = X(x, t) = (X^{α})_{1≤α≤3} to denote the embedding vector of a manifold M into ℝ^{3} and \frac{d}{dt}X = Ẋ for its total time derivative. Set M_{t} := X(M, t) ⊂ ℝ^{3}. We will assume that M_{t} is strictly convex, closed, smooth, and connected. We choose ν = ν(x) = (ν^{α})_{1≤α≤3} to be the outer unit normal vector to M_{t} at x ∈ M_{t}. The embedding X(·, t) induces at each point on M_{t} a metric (g_{ij})_{1≤i, j≤2} and a second fundamental form (h_{ij})_{1≤i, j≤2} on M_{t}. These tensors are symmetric. The principal curvatures λ_{1}, λ_{2} are the eigenvalues of the second fundamental form with respect to that metric. A surface is called strictly convex, if all principal curvatures are strictly positive. We will assume this throughout the paper. The inverse of the second fundamental form is denoted by \binom{}{h̃^{ij}}_{1≤i, j≤2}.

Latin indices range from 1 to 2 and refer to geometric quantities on the surface, Greek indices range from 1 to 3 and refer to components in the ambient space ℝ^{3}. In ℝ^{3}, we will always choose Euclidean coordinates. We use the Einstein summation convention. Latin indices are raised and lowered with respect to the induced metric or its inverse (g^{ij}), for Greek indices we use the flat metric of ℝ^{3}.

Denoting by ⟨·, ·⟩ the Euclidean scalar product in ℝ^{3}, we have

g_{ij} = ⟨X_{, i}, X_{, j}⟩ = X_{, i}^{α}δ_{αβ}X_{, j}^{β},

where we used indices, preceded by commas, to denote partial derivatives. We write indices, preceded by semi-colons, e. g. h_{ij; k}, to indicate covariant differentiation with respect to the induced metric, e. g.

X_{; ij}^{α} = X_{, ij}^{α} − Γ_{ij}^{k}X_{, k}^{α},

where

Γ_{ij}^{k} = \frac{1}{2}g^{kl}(g_{il, j} + g_{jl, i} − g_{ij, l})

are the Christoffel symbols of the metric (g_{ij}).

We can use the Gauß formula (2.1) or the Weingarten equation (2.2) to compute the second fundamental form.

The Gauß formula relates covariant derivatives of the position vector to the second fundamental form and the normal vector

(2.1)  X_{; ij}^{α} = −h_{ij}ν^{α}.

The Weingarten equation allows to compute derivatives of the normal vector

(2.2)  ν_{; i}^{α} = h_{i}^{k}X_{; k}^{α}.

Symmetric functions of the principal curvatures are well-defined, we will use the mean curvature H = λ_{1} + λ_{2}, the square of the norm of the second fundamental form |A|^{2} = λ_{1}^{2} + λ_{2}^{2}, tr A^{k} = λ_{1}^{k} + λ_{2}^{k}, and the Gauß curvature K = λ_{1}λ_{2}. It is often convenient to choose coordinate systems such that, at a fixed point, the metric tensor equals the Kronecker delta, g_{ij} = δ_{ij}, and (h_{ij}) is diagonal, (h_{ij}) = diag(λ_{1}, λ_{2}), e. g.

∑ λ_{k}h_{ij; k}^{2} = ∑_{i, j, k=1}^{2} λ_{k}h_{ij; k}^{2} = h^{kl}h_{j; k}^{i}h_{i; l}^{j} = h_{rs}h_{ij; k}h_{ab; l}g^{ia}g^{jb}g^{rk}g^{sl}.

Whenever we use this notation, we will also assume that we have fixed such a coordinate system.

A normal velocity F can be considered as a function of (λ_{1}, λ_{2}) or (h_{ij}, g_{ij}). We set F^{ij} = \frac{∂F}{∂h_{ij}}, F^{ij, kl} = \frac{∂^{2}F}{∂h_{ij}∂h_{kl}}. Note that in coordinate systems with diagonal h_{ij} and g_{ij} = δ_{ij} as mentioned above, F^{ij} is diagonal. For F = |A|^{2}, we have F^{ij} = 2h^{ij} = 2λ_{i}g^{ij}, and for F = −K^{−1}, we have F^{ij} = K^{−1}h̃^{ij} = K^{−1}λ_{i}^{−1}g^{ij}.

The Gauß equation expresses the Riemannian curvature tensor of the surface in terms of the second fundamental form

(2.3)  R_{ijkl} = h_{ik}h_{jl} − h_{il}h_{jk}.

As we use only Euclidean coordinate systems in ℝ^{3}, h_{ij; k} is symmetric according to the Codazzi equations.

The Ricci identity allows to interchange covariant derivatives. We will use it for the second fundamental form

(2.4)  h_{ik; lj} = h_{ik; jl} + h_{k}^{a}R_{ailj} + h_{i}^{a}R_{aklj}.

2.1.1. The Support Function. Let M be a smooth, strictly convex, compact surface in ℝ^{3} and X its embedding vector. Then ν is a diffeomorphism from M to 𝕊^{2}. Assume that M encloses the origin. We define the support function u : 𝕊^{2} → ℝ_{+} of M as

u(z) := ⟨X(ν^{−1}(z)), t⟩.

The support functions u(·, t) of a family of surfaces flowing according to

\frac{d}{dt}X = −F ν

fulfill the evolution equation (see [10])

u̇ = −F.

In order to write F in terms of u, we note that the eigenvalues of u_{; ij} + uσ_{ij} with respect to the standard metric (σ_{ij}) are given by \frac{1}{λ_{1}} ≡ κ_{1} and \frac{1}{λ_{2}} ≡ κ_{2}. Thus κ_{i} are the radii of curvature. Here we used u_{; ij} to denote covariant derivatives of u with respect to the standard metric on the sphere. So we obtain for a family of surfaces flowing according to \frac{d}{dt}X = \frac{1}{K} ν the following evolution equation for the support function

u̇ = \frac{det(u_{; ij} + uσ_{ij})}{det(σ_{ij})}.

and the evolution equation

u̇ = −\frac{κ_{1}^{2} + κ_{2}^{2}}{κ_{1}^{2}κ_{2}^{2}} = − (σ^{ij}(u_{;jk} + uσ_{jk})σ^{kl}(u_{;li} + uσ_{li})) \frac{det(σ_{ij})^{2}}{det(u_{;ij} + uσ_{ij})^{2}}

for the support function corresponds to \frac{d}{dt}X = −|A|^{2}ν, where (σ^{ij}) denotes the inverse of (σ_{ij}).

We wish to remark that the calculations in [10] use support functions. This motivates the definition of the dual function in (4.8).

For tensors A and B, A_{ij} ≥ B_{ij} means that (A_{ij} − B_{ij}) is positive definite. Finally, we use c to denote universal, estimated constants.

2.2. Evolution Equations

For a family M_{t} solving the evolution equation

\frac{d}{dt}X = −F ν

as described in Section 1.1, we have the following evolution equations

(2.5)  \frac{d}{dt}g_{ij} = − 2F h_{ij},
(2.6)  \frac{d}{dt}h_{ij} =F_{; ij} − F h_{i}^{k}h_{kj},
(2.7)  \frac{d}{dt}ν^{α} =g^{ij}F_{; i}X_{; j}^{α}.

As a direct consequence of these results, we obtain the following evolution equations for a family M_{t} of surfaces (or hypersurfaces) flowing according to

\frac{d}{dt}X = −F ν.

Lemma 2.1. We have

\frac{d}{dt}F − F^{ij}F_{; ij} =F F^{ij}h_{i}^{k}h_{kj},
\frac{d}{dt}h_{ij} − F^{kl}h_{ij; kl} =F^{kl}h_{k}^{a}h_{al} · h_{ij} − F^{kl}h_{kl} · h_{i}^{a}h_{aj}
− F h_{i}^{k}h_{kj} + F^{kl, rs}h_{kl; i}h_{rs; j},
\frac{d}{dt}X^{α} − F^{ij}X_{; ij}^{α} = (F^{ij}h_{ij} − F ) ν^{α},
\frac{d}{dt}|X|^{2} − F^{ij}(|X|^{2})_{; ij} =2 (F^{ij}h_{ij} − F ) ⟨X, ν⟩ − 2F^{ij}g_{ij},
\frac{d}{dt}ν^{α} − F^{ij}(ν^{α})_{; ij} =F^{ij}h_{i}^{k}h_{kj} · ν^{α},

and

\frac{d}{dt}⟨X, ν⟩ − F^{ij}⟨X, ν⟩_{; ij} = − F^{ij}h_{ij} − F + F^{ij}h_{i}^{k}h_{kj}⟨X, ν⟩.

Proof. We have, see [49, Lemma 5.4],

\frac{∂F}{∂g_{kl}} = −F^{il}h_{i}^{k}

and compute the evolution equation of the normal velocity F

\frac{d}{dt}F − F^{ij}F_{; ij} = − F^{il}h_{i}^{k} \frac{d}{dt}g_{kl} + F^{ij} \frac{d}{dt}h_{ij} − F^{ij}F_{; ij}

=F F^{ij}h_{i}^{k}h_{kj},

where we used (2.5) and (2.6) and the fact that g_{ij; k} ≡ 0.

The computations for the evolution equation for h_{ij} are more involved as we have to interchange covariant derivatives.

\frac{d}{dt}h_{ij} − F^{ij}h_{ij; kl} =F^{kl}h_{kl; ij} + F^{kl, rs}h_{kl; i}h_{rs; j}

− F h_{i}^{k}h_{kj} − F^{ij}h_{ij; kl}

=F^{kl}h_{ik; lj} + F^{kl, rs}h_{kl; i}h_{rs; j}
− F h_{i}^{k}h_{kj} − F^{ij}h_{ik; jl}  by Codazzi
=F^{kl} (h_{k}^{a}R_{ailj} + h_{i}^{a}R_{aklj}) − F h_{i}^{k}h_{kj}
+ F^{kl, rs}h_{kl; i}h_{rs; j}  by (2.4)
=F^{kl}h_{k}^{a}h_{al}h_{ij} − F^{kl}h_{k}^{a}h_{aj}h_{il}

+ F^{kl}h_{i}^{a}h_{al}h_{kj} − F^{kl}h_{i}^{a}h_{aj}h_{kl}

− F h_{i}^{k}h_{kj} + F^{kl, rs}h_{kl; i}h_{rs; j}  by (2.3)
=F^{kl}h_{k}^{a}h_{al}h_{ij} − F^{kl}h_{i}^{a}h_{aj}h_{kl}

− F h_{i}^{k}h_{kj} + F^{kl, rs}h_{kl; i}h_{rs; j}.

Combining (1.3) and (2.1) implies the evolution equation for X^{α}. Then, we get

\frac{d}{dt}|X|^{2} − F^{ij}(|X|^{2})_{; ij} =2X^{α}δ_{αβ} (\frac{d}{dt}X^{β} − F^{ij}X_{; ij}^{β}) − 2F^{ij}X_{; i}^{α}δ_{αβ}X_{; j}^{β}
=2X^{α}δ_{αβ}ν^{β} (F^{ij}h_{ij} − F ) − 2F^{ij}g_{ij}.

We now come to the evolution equation for the unit normal ν^{α}.

\frac{d}{dt}ν^{α} − F^{ij}(ν^{α})_{; ij} =g^{ij}F_{; i}X_{; j}^{α} − F^{ij} (h_{i}^{k}X_{; k}^{α})_{; j}  by (2.7) and (2.2)
=g^{ij}F^{kl}h_{kl; i}X_{; j}^{α} − F^{ij}h_{i; j}^{k}X_{; k}^{α} − F^{ij}h_{i}^{k}X_{; kj}^{α}
=F^{ij}h_{i}^{k}h_{kj}ν^{α}  by (2.1).

Finally, we have

\frac{d}{dt}⟨X, ν⟩ − F^{ij}⟨X, ν⟩_{; ij} =X^{α}δ_{αβ} (\frac{d}{dt}ν^{β} − F^{ij}(ν^{β})_{; ij})

+ (\frac{d}{dt}X^{α} − F^{ij}X_{; ij}^{α}) δ_{αβ}ν^{β}

− 2F^{ij}X_{; i}^{α}δ_{αβ}ν_{; j}^{β}

=F^{ij}h_{i}^{k}h_{kj}⟨X, ν⟩ + (F^{ij}h_{ij} − F ) ⟨ν, ν⟩
− 2F^{ij}X_{; i}^{α}δ_{αβ}h_{j}^{k}X_{; k}^{β}  by (2.2)
=F^{ij}h_{i}^{k}h_{kj}⟨X, ν⟩ − F^{ij}h_{ij} − F.
□

Part 2

Surfaces Contracting with Normal Velocity |A|^{2}

CHAPTER 3 Introduction

We consider compact strictly convex surfaces M_{t} in ℝ^{3} that contract with normal velocity equal to the square of the norm of the second fundamental form

(3.1)  \frac{d}{dt}X = −|A|^{2}ν.

This is a parabolic flow equation. We obtain a solution on a maximal time interval [0, T), 0 < T < ∞. For t ↑ T, the surfaces converge to a point. After appropriate rescaling, they converge to a round sphere. We say that the surfaces M_{t} converge to a “round point”. The key step in the proof, Theorem 4.3, is to show that

(3.2)  max_{M_{t}} (\frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} )

is non-increasing in time.

Our main theorem in this part is

Theorem 3.1. For any smooth strictly convex closed surface M in ℝ^{3}, there exists a smooth family of surfaces M_{t}, t ∈ [0, T), solving (3.1) with M_{0} = M. For t ↑ T, M_{t} converges to a point Q. The rescaled surfaces (M_{t} − Q) · (6(T − t))^{−1/3} converge smoothly to the unit sphere 𝕊^{2}.

Note that we do not have to impose any pinching assumption on the initial surface and that the normal velocity is homogeneous of degree two, i. e. especially of some degree larger than one.

We will also consider other normal velocities for which similar results hold. Therefore, we have to find monotone quantities like in (3.2).

In Table 1, we have collected some normal velocities F (1^{st} column) and quantities w (2^{nd} column) such that max_{M_{t}} w is non-increasing in time for surfaces contracting with normal velocity F. In each case, we obtain convergence to round points for smooth compact strictly convex initial surfaces M_{0}.

\begin{matrix}|A|^{2} & \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} \\ K [12] & (λ_{1} − λ_{2})^{2} \\ H^{2} & \frac{(λ_{1} + λ_{2})^{3}(λ_{1} − λ_{2})^{2}}{(λ_{1}^{2} + λ_{2}^{2}) λ_{1}λ_{2}} \\ H^{3} & \frac{(λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1} + λ_{2})^{2}(λ_{1} − λ_{2})^{2}}{(λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2}) λ_{1}λ_{2}} \\ H^{4} & \frac{(λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1} + λ_{2})^{6}(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}} \\ {|A|^{2} + βH^{2}, \atop 0 ≤ β ≤ 5} & \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} \\ tr A^{3} & \frac{(3λ_{1}^{2} + 2λ_{1}λ_{2} + 3λ_{2}^{2}) (λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} \\ {tr A^{α}, \atop α = 2, 4, 5, 6} & \frac{(λ_{1}^{α−2} + λ_{2}^{α−2}) (λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} \\ H|A|^{2} & \frac{(λ_{1} + λ_{2})^{2}(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} \\ |A|^{4} & \frac{(λ_{1}^{4} + 2λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} + 2λ_{1}λ_{2}^{3} + λ_{2}^{4}) (λ_{1} − λ_{2})^{2}}{(λ_{1} + λ_{2})λ_{1}λ_{2}} \end{matrix}

CHAPTER 4 Surfaces Flowing by |A|^{2}

4.1. Convergence to a Point

It is known, that (3.1) is a parabolic evolution equation for strictly convex initial data and that is has a solution on a maximal time interval.

We show that M_{t} stays uniformly strictly convex. The following lemma is similar to results in [7].

Lemma 4.1. For a smooth compact strictly convex surface M in ℝ^{3}, flowing according to Ẋ = −|A|^{2}ν, the minimum of the principal curvatures is non-decreasing.

Proof. Consider M_{ij} = h_{ij} − εg_{ij} with ε > 0 so small that M_{ij} is positive semi-definite for some time t_{0}. We wish to show that M_{ij} is positive semi-definite for t > t_{0}. Lemma 2.1 yields

\frac{d}{dt}h_{ij} − F^{kl}h_{ij; kl} = 2 tr A^{3}h_{ij} − 3|A|^{2}h_{i}^{k}h_{kj} + 2g^{kr}g^{ls}h_{kl; i}h_{rs; j}.

In the evolution equation for M_{ij}, we drop the positive definite terms involving derivatives of the second fundamental form

\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl} ≥ 2 tr A^{3}h_{ij} − 3|A|^{2}h_{i}^{k}h_{kj} + 2ε|A|^{2}h_{ij}.

Let ξ be a zero eigenvalue of M_{ij} with |ξ| = 1, M_{ij}ξ^{j} = h_{ij}ξ^{j} − εg_{ij}ξ^{j} = 0. So we obtain in a point with M_{ij} ≥ 0

(2 tr A^{3}h_{ij} − 3|A|^{2}h_{i}^{k}h_{kj} + 2ε|A|^{2}h_{ij}) ξ^{i}ξ^{j} =2ε tr A^{3} − 3ε^{2}|A|^{2} + 2ε^{2}|A|^{2}
=2ε tr A^{3} − ε^{2}|A|^{2}
≥2ε^{2}|A|^{2} − ε^{2}|A|^{2} > 0

and the maximum principle for tensors [18, 29, 30] gives the result. □

The next result shows that |A|^{2} stays uniformly bounded as long as M_{t} encloses a ball of fixed positive radius. A similar estimate is used in [59].

Lemma 4.2. For a strictly convex solution of (3.1), |A|^{2} is uniformly bounded in terms of the radius R of an enclosed sphere B_{R}(x_{0}), max_{M_{0}} \frac{|A|^{2}}{⟨X−x_{0}, ν⟩− \frac{1}{2} R}, and max_{M_{0}} |X − x_{0}|. More precisely, we have

(4.1)  sup_{t} max_{M_{t}} |A|^{2} ≤ max \{max_{M_{0}} |X − x_{0}| · max_{M_{0}} \frac{|A|^{2}}{⟨X − x_{0}, ν⟩ − \frac{1}{2}R}, \frac{18}{R^{2}} \} .

Proof. We may assume that x_{0} = 0. Let α = \frac{1}{2}R. Then α is a positive lower bound for ⟨X, ν⟩ − α. Lemma 2.1 yields the evolution equations

\frac{d}{dt}X^{β} − F^{ij}X_{; ij}^{β} =|A|^{2}ν^{β},

6 4. SURFACES FLOWING BY |A|^{2}

\frac{d}{dt}ν^{β} − F^{ij}ν_{; ij}^{β} =2 tr A^{3}ν^{β},
\frac{d}{dt}⟨X, ν⟩ − F^{ij}⟨X, ν⟩_{; ij} = − 3|A|^{2} + 2 tr A^{3}⟨X, ν⟩,
\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =2|A|^{2} tr A^{3}.

In a critical point of \frac{|A|^{2}}{|X, ν|−α}, we obtain

\frac{d}{dt} log \frac{|A|^{2}}{⟨X, ν⟩ − α} − F^{ij} (log \frac{|A|^{2}}{⟨X, ν⟩ − α})_{; ij} = \frac{1}{⟨X, ν⟩ − α} (3|A|^{2} − 2 tr A^{3}α) .

Note that ⟨X, ν⟩−α ≤ max_{M_{0}} |X| as a sphere of radius max_{M_{0}} |X|, centered at the origin, will enclose any M_{t}. We only have to prove that we preserve the bound in Equation (4.1), when max_{M_{t}} \frac{|A|^{2}}{⟨X, ν⟩−α} increases. Then we have 0 ≤ 3|A|^{2} − 2 tr A^{3}α

at a point, where max_{M_{t}} \frac{|A|^{2}}{⟨X, ν⟩−α} is attained. This inequality and elementary calculations for convex surfaces give

|A|^{2} ≤ 2^{1/3} · (tr A^{3})^{2/3} ≤ 2\begin{matrix}(tr A^{3}) & 2 \\ (|A|^{2}) & 2 \end{matrix} ≤ \frac{9}{2α^{2}}

at such a maximum point and the Lemma follows. □

We obtain that the second fundamental form of the surface stays bounded as long as M_{t} encloses some ball. The estimates of Krylov, Safonov, Evans (see also [4]), and Schauder imply that the solution stays smooth. Then, similarly as in [59], the positive lower bound on the minimum principal curvature implies that the surfaces converge to a point in finite time.

4.2. A Monotone Quantity

Theorem 4.3. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3} flowing according to Ẋ = −|A|^{2}ν,

(4.2)  max_{M(t)} \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{2λ_{1}λ2} = max_{M(t)} \frac{H · (2|A|^{2} − H^{2})}{H^{2} − |A|^{2}} ≡ max_{M(t)} w

is non-increasing in time.

An immediate consequence of this theorem is

Corollary 4.4. The only homothetically shrinking smooth compact strictly convex surfaces M_{t}, solving the flow equation Ẋ = −|A|^{2}ν in ℝ^{3}, are spheres.

Proof. The quantity \frac{(λ_{1}+λ_{2})(λ_{1}−λ_{2})^{2}}{λ_{1}λ_{2}} is positive homogeneous of degree one in the principal curvatures and non-negative. If M is homothetically shrinking, Theorem 4.3 implies that (λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2} = 0 everywhere. Thus M(t) is umbilic and [57, Lemma 7.1] implies that M(t) is a sphere. □

Proof of Theorem 4.3. We apply Lemma 2.1 and get the following general evolution equation

\frac{d}{dt} tr A^{α} − F^{kl} (tr A^{α})_{; kl} =α ∑_{i} F^{ii}λ_{i}^{2} tr A^{α} + α (F − ∑_{i} F^{ii}λ_{i}) tr A^{α+1}

− α ∑_{r=0}^{α−2} ∑_{i, j, k} F^{kk}λ_{i}^{α−2−r}λ_{j}^{r}h_{ij; k}^{2}

+ α ∑_{k, l, r, s, i} F^{kl, rs}h_{kl; i}h_{rs; i}λ_{i}^{α−1}

(4.3)

for solutions to the flow equation

\frac{d}{dt}X = −F ν.

Using (4.3) for F = |A|^{2} yields

\frac{d}{dt}H − F^{ij}H_{; ij} = − (|A|^{2})^{2} + 2H tr A^{3} + 2 ∑ h_{ij; k}^{2} (4.4)

and

\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =2|A|^{2} tr A^{3}. (4.5)

For the rest of the proof, we consider a critical point of w|_{M_{t}} for some t > 0, where w > 0. It suffices to show that w̃ := log w is non-increasing in such a point. Then our theorem follows.

We rewrite w̃

w̃ = log H + log (2|A|^{2} − H^{2}) − log(H^{2} − |A|^{2})
≡ log A + log B − log C.

In a critical point of w̃, we obtain

\begin{aligned}\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} = & \frac{1}{A} (\frac{d}{dt}A − F^{ij}A_{; ij}) + \frac{1}{B} (\frac{d}{dt}B − F^{ij}B_{; ij}) \\ & − \frac{1}{C} (\frac{d}{dt}C − F^{ij}C_{; ij}) − \frac{1}{AB} F^{ij}(A_{; i}B_{; j} + A_{; j}B_{; i}) \end{aligned}

and

\begin{matrix}0 = & \frac{1}{H} H_{; k} + \frac{1}{2|A|^{2} − H^{2}} (2|A|^{2} − H^{2})_{; k} − \frac{1}{H^{2} − |A|^{2}} (H^{2} − |A|^{2})_{; k} \\ = & \frac{2λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}}{λ_{1} (λ_{1}^{2} − λ_{2}^{2})} h_{11; k} + \frac{2λ_{2}^{2} + λ_{1}λ_{2} + λ_{1}^{2}}{λ_{2} (λ_{2}^{2} − λ_{1}^{2})} h_{22; k}. \end{matrix}

So we deduce that

h_{22; 1} = \frac{λ_{2}}{λ_{1}} \frac{2λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}}{2λ_{2}^{2} + λ_{1}λ_{2} + λ_{1}^{2}} h_{11; 1} ≡ a_{1}h_{11; 1}

and a similar formula holds for h_{11; 2}, h_{11; 2} = a_{2} · h_{22; 2}. We now combine all these results and obtain in a straightforward calculation

\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} = (\frac{1}{H} − \frac{2H}{2|A|^{2} − H^{2}} − \frac{2H}{H^{2} − |A|^{2}} ) · (\frac{d}{dt}H − F^{ij}H_{; ij})

+ (\frac{2}{2|A|^{2} − H^{2}} + \frac{1}{H^{2} − |A|^{2}} ) · (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij})

+ (\frac{6}{2|A|^{2} − H^{2}} + \frac{2}{H^{2} − |A|^{2}} ) F^{ij}H_{; i}H_{; j}

− \frac{2}{H · (2|A|^{2} − H^{2})}F^{ij} ((|A|^{2})_{; i} H_{; j} + (|A|^{2})_{; j} H_{; i})

= − \frac{λ_{1}^{4} + λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} + λ_{1}λ_{2}^{3} + λ_{2}^{4}}{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}} ·

· (− (|A|^{2})^{2} + 2H tr A^{3})

+ \frac{(λ_{1} + λ_{2})^{2}}{2(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}} · 2|A|^{2} tr A^{3}

− 2\frac{λ_{1}^{4} + λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} + λ_{1}λ_{2}^{3} + λ_{2}^{4}}{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}} ∑ h_{ij; k}^{2}

+ 2\frac{λ_{1}^{2} + 4λ_{1}λ_{2} + λ_{2}^{2}}{(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}} ∑ λ_{k}h_{ii; k}h_{jj; k}

− \frac{8}{(λ_{1} − λ_{2})^{2}(λ_{1} + λ_{2})} ∑ λ_{k}(λ_{i} + λ_{j})h_{ii; k}h_{jj; k}

= − \frac{λ_{1}^{8} + 3λ_{1}^{7}λ_{2} + 4λ_{1}^{6}λ_{2}^{2} + 9λ_{1}^{5}λ_{2}^{3} − 2λ_{1}^{4}λ_{2}^{4}}{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}}

− \frac{9λ_{1}^{3}λ_{2}^{5} + 4λ_{1}^{2}λ_{2}^{6} + 3λ_{1}λ_{2}^{7} + λ_{2}^{8}}{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}}

+ \frac{λ_{1}^{7} + 2λ_{1}^{6}λ_{2} + 2λ_{1}^{5}λ_{2}^{2} + 3λ_{1}^{4}λ_{2}^{3} + 3λ_{1}^{3}λ_{2}^{4} + 2λ_{1}^{2}λ_{2}^{5} + 2λ_{1}λ_{2}^{6} + λ_{2}^{7}}{(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}}

− 2\frac{λ_{1}^{4} + λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} + λ_{1}λ_{2}^{3} + λ_{2}^{4}}{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}} ·

· ((1 + 3a_{1}^{2}) · h_{11; 1}^{2} + (1 + 3a_{2}^{2}) · h_{22; 2}^{2})

+ 2\frac{λ_{1}^{2} + 4λ_{1}λ_{2} + λ_{2}^{2}}{(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}} · (λ_{1}(1 + a_{1})^{2} · h_{11; 1}^{2} + λ_{2}(1 + a_{2})^{2} · h_{22; 2}^{2})

− \frac{16}{(λ_{1} − λ_{2})^{2}(λ_{1} + λ_{2})} · (λ_{1}(λ_{1} + λ_{2}a_{1})(1 + a_{1}) · h_{11; 1}^{2}

+λ_{2}(λ_{2} + λ_{1}a_{2})(1 + a_{2}) · h_{22; 2}^{2})

= − 4\frac{K^{2}}{H}

− 2\frac{(5λ_{1}^{8} − 4λ_{1}^{7}λ_{2} + 46λ_{1}^{6}λ_{2}^{2} + 48λ_{1}^{5}λ_{2}^{3} + 72λ_{1}^{4}λ_{2}^{4}) λ_{2}}{(λ_{1} + λ_{2}) (λ_{1} − λ_{2})^{2} (λ_{1}^{2} + λ_{1}λ_{2} + 2λ_{2}^{2})^{2} λ_{1}^{3}} h_{11; 1}^{2}

− 2\frac{(44λ_{1}^{3}λ_{2}^{5} + 34λ_{1}^{2}λ_{2}^{6} + 8λ_{1}λ_{2}^{7} + 3λ_{2}^{8}) λ_{2}}{(λ_{1} + λ_{2}) (λ_{1} − λ_{2})^{2} (λ_{1}^{2} + λ_{1}λ_{2} + 2λ_{2}^{2})^{2} λ_{1}^{3}} h_{11; 1}^{2}

− 2\frac{(5λ_{2}^{8} − 4λ_{2}^{7}λ_{1} + 46λ_{2}^{6}λ_{1}^{2} + 48λ_{2}^{5}λ_{1}^{3} + 72λ_{2}^{4}λ_{1}^{4}) λ_{1}}{(λ_{2} + λ_{1}) (λ_{2} − λ_{1})^{2} (λ_{2}^{2} + λ_{2}λ_{1} + 2λ_{1}^{2})^{2} λ_{2}^{3}} h_{22; 2}^{2}

− 2\frac{(44λ_{2}^{3}λ_{1}^{5} + 34λ_{2}^{2}λ_{1}^{6} + 8λ_{2}λ_{1}^{7} + 3λ_{1}^{8}) λ_{1}}{(λ_{2} + λ_{1}) (λ_{2} − λ_{1})^{2} (λ_{2}^{2} + λ_{2}λ_{1} + 2λ_{1}^{2})^{2} λ_{2}^{3}} h_{22; 2}^{2}

≤0.

We finally apply the maximum principle and our theorem follows. □

4.3. Direct Consequences

We obtain a pinching estimate Corollary 4.5. For a smooth compact strictly convex surface M_{t} in ℝ^{3}, flow-

ing according to Ẋ = −|A|^{2}ν, there exists c = c(M_{0}) such that 0 < \frac{1}{c} ≤ \frac{λ_{1}}{λ_{2}} ≤ c.

Proof. Choose ε > 0 such that λ_{1}, λ_{2} > ε at t = 0. Theorem 4.3 and Lemma 4.1 imply that

2ε\frac{(\frac{λ_{1}}{λ_{2}} − 1)^{2}}{\frac{λ_{1}}{λ_{2}}} = 2ε\frac{(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} ≤ \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} ≤ c.

We obtain the bound on \frac{λ_{1}}{λ_{2}} claimed above. □

Let ρ_{+} be the minimal radius of enclosing spheres and ρ_{−} the maximal radius of enclosed spheres. The quotient of these radii can be estimated as follows

Corollary 4.6. Under the assumptions of Corollary 4.5, ρ_{+}/ρ_{−} is bounded above by a constant depending only on the constant c(M_{0}) in Corollary 4.5.

Proof. Combine Corollary 4.5, [7, Theorem 5.1], and [7, Lemma 5.4]. □ We also obtain a bound for |λ_{1} − λ_{2}| Corollary 4.7. For a smooth compact strictly convex surface M_{t} in ℝ^{3}, flow-

ing according to Ẋ = −|A|^{2}ν, there exists a constant c = c(M_{0}) such that

|λ_{1} − λ_{2}| ≤ c · (|A|^{2})^{1/4} ≤ c · \sqrt{H}.

Proof. This is a direct consequence of Theorem 4.3 and Corollary 4.5. □ As in [12], this estimate on |λ_{1} − λ_{2}| is “better” than scaling invariant. It is

crucial for the rest of the proof of Theorem 3.1.

Let us recall a form of the maximum principle for evolving hypersurfaces. Lemma 4.8. Let M_{t} and M̃_{t} be two smooth compact strictly convex solutions

to (3.1) on some time interval [0, T^{∗}). If M_{0} encloses M̃_{0}, then M_{t} encloses M̃_{t} at any time t ∈ [0, T^{∗}), for which both solutions exist.

Proof. This is a standard consequence of the maximum principle. □ The next result describes the evolution of spheres. Lemma 4.9. Spheres ∂B_{r(t)}(x_{0}) solve (3.1) for t ∈ [0, T) with

r(t) = (6(T − t))^{1/3} and T = \frac{1}{6}r^{3}(0).

Proof. The evolution equation for the radius of a sphere is

ṙ(t) = −\frac{2}{r^{2}(t)}.

□

As a consequence, we can estimate the life span of a solution in terms of inner and outer radii.

Lemma 4.10. Let ρ_{+}(t) and ρ_{−}(t) be the inner and outer radii of M_{t}, respectively. Assume that M_{t} is a smooth compact strictly convex solution of (3.1) on a maximal time interval [0, T). Then we have for t ∈ [0, T)

\frac{1}{6}ρ_{−}^{3}(t) ≤ T − t ≤ \frac{1}{6}ρ_{+}^{3}(t).

Proof. As M_{t} contracts to a point, we deduce from Lemma 4.8 that T − t is bounded below by the life span of ∂B_{ρ_{−}(t)} evolving according to (3.1). So the lower bound follows from Lemma 4.9. The upper bound is obtained similarly. □

4.4. Convergence to a Round Point

We closely follow the corresponding part of [12].

Proposition 4.11. Define q(t) := \frac{1}{4π} ∫_{M_{t}} KX. Then

|⟨X − q, ν⟩ − \frac{1}{8π} ∫_{M_{t}} H| ≤ \frac{1}{4π} · sup_{M_{t}} |λ_{1} − λ_{2}| · H^{2}(M_{t}),

where H^{2}(M_{t}) denotes the area of M_{t}.

Proof. This is [12, Proposition 4]. □

We define r_{+}(t) to be the minimal radius of a sphere, centered at q(t), that encloses M_{t}. Similarly, we define r_{−}(t) to be the maximal radius of a sphere, centered at q(t), that is enclosed by M_{t}.

Lemma 4.12. Under the assumptions of Theorem 3.1, for T − t sufficiently small, r_{+} and r_{−} are estimated as follows

r_{+}(t) ≤(6(T − t))^{1/3} · (1 + c · (T − t)^{1/6}) ,
r_{−}(t) ≥(6(T − t))^{1/3} · (1 − c · (T − t)^{1/6}) ,

and

1 ≤ \frac{r_{+}}{r_{−}} ≤1 + c · (T − t)^{1/6}.

Proof. Denote the bounded component of ℝ^{3} \M_{t} by E_{t}. The transformation formula for integrals implies that

\frac{1}{4π} ∫_{M_{t}} KX = \frac{1}{4π} ∫_{𝕊^{2}} X (ν^{−1}(·)) .

4.4. CONVERGENCE TO A ROUND POINT 11

So we see that q(t) ∈ E_{t}. We have

r_{+} = max_{M_{t}} ⟨X − q(t), ν⟩,  r_{−} = min_{M_{t}} ⟨X − q(t), ν⟩,
ρ_{+} = min_{p∈ℝ^{3}} max_{M_{t}} ⟨X − p, ν⟩, and ρ_{−} = max_{p∈E_{t}} min_{M_{t}} ⟨X − p, ν⟩.

Recall the first variation formula for a vector field Y along M_{t} [41]

∫_{M_{t}} H⟨Y, ν⟩ = ∫_{M_{t}} div_{M_{t}} Y

and get for p ∈ E_{t} such that ρ_{+} = max_{M_{t}}⟨X − p, ν⟩

∫_{M_{t}} H ≥ \frac{1}{ρ_{+}} ∫_{M_{t}} H · ⟨X − p, ν⟩ = \frac{1}{ρ_{+}} ∫_{M_{t}} div_{M_{t}} X = \frac{1}{ρ_{+}} ∫_{M_{t}} 2 = \frac{2}{ρ_{+}} H^{2}(M_{t}).

We employ Proposition 4.11 and deduce that

r_{−} ≥\frac{1}{8π} ∫_{M_{t}} H · \{1 − 2 (∫_{M_{t}} H)^{−1} · sup_{M_{t}} |λ_{1} − λ_{2}| · H^{2}(M_{t})\}
≥\frac{1}{8π} ∫_{M_{t}} H · \{1 − ρ_{+} · sup_{M_{t}} |λ_{1} − λ_{2}|\} .

We estimate as follows

ρ_{+} · sup_{M_{t}} |λ_{1} − λ_{2}| ≤c · ρ_{+} · (|A|^{2})^{1/4}  by Corollary 4.7
≤c · ρ_{+} · (c + \frac{c}{ρ_{−}^{2}} )^{1/4}  by Lemma 4.2
≤c · (T − t)^{1/6}

by Corollary 4.6 and Lemma 4.10 for (T − t) small. So we obtain

(4.6)  r_{−}(t) ≥ \frac{1}{8π} ∫_{M_{t}} H · (1 − c · (T − t)^{1/6}) .

Similar calculations yield

(4.7)  r_{+}(t) ≤ \frac{1}{8π} ∫_{M_{t}} H · (1 + c · (T − t)^{1/6}) .

We employ Lemma 4.10

r_{−} ≤ ρ_{−} ≤ (6(T − t))^{1/3} ≤ ρ_{+} ≤ r_{+}

and obtain for (T − t) small

(6(T − t))^{1/3} · (1 − c · (T − t)^{1/6}) ≤ \frac{1}{8π} ∫_{M_{t}} H ≤ (6(T − t))^{1/3} · (1 + c · (T − t)^{1/6}) .

Using (4.6) and (4.7) gives the claimed estimates on r_{−} and r_{+}, and r_{+}/r_{−} is bounded as stated above. □

Corollary 4.13. Under the assumptions of Theorem 3.1, we have the estimate

|q(t) − Q| ≤ c · (T − t)^{1/3+1/18}.

Therefore, we obtain estimates of the same form as in Lemma 4.12, if we define r_{+} and r_{−} using Q instead of q(t).

Proof. Fix t_{0} ∈ [0, T). A sphere of radius (6(T − t_{0}))^{1/3} (1 + c(T − t_{0})^{1/6}), centered at q(t_{0}) as defined in Proposition 4.11, will enclose M_{t} for all t_{0} ≤ t < T. Thus its radius at time t = T is an upper bound for |q(t_{0}) − Q|. According to Lemma 4.9, the radius of that sphere evolves as follows

r(t) = (6 (\frac{1}{6}r^{3}(0) − (t − t_{0})))^{1/3} .

Therefore, we get

|q(t_{0}) − Q| ≤r(T)
= (6 (\frac{1}{6}6(T − t_{0}) (1 + c(T − t_{0})^{1/6})^{3} − (T − t_{0})))^{1/3}

=(6(T − t_{0}))^{1/3} · ((1 + c(T − t_{0})^{1/6})^{3} − 1)^{1/3}
≤(6(T − t_{0}))^{1/3} · c · (T − t_{0})^{1/18}.
□

Next, we want to check, that we can apply a Harnack inequality [10, Theorem 5.17]. For F = F (λ_{i}), λ_{i} > 0, we define

(4.8)  Φ(κ_{i}) := −F (κ_{i}^{−1}) .

We say that Φ is α-concave, if Φ = sgn α · B^{α} for some B, where B is positive and concave. The function Φ is called the dual function to F.

Lemma 4.14. The dual function to F = |A|^{2} = λ_{1}^{2}+λ_{2}^{2} is α-concave for α ≤ −2.

Proof. We define for λ_{i} > 0

Φ = Φ(λ_{i}) = Φ(h_{ij}, g_{ij}) = − (\frac{1}{λ_{1}^{2}} + \frac{1}{λ_{2}^{2}} ) = \frac{2K − H^{2}}{K^{2}} = \frac{2 det h_{j}^{i} − (tr h_{j}^{i})^{2}}{(det h_{j}^{i})^{2}}

and obtain

Φ^{ij} ≡ \frac{∂Φ}{∂h_{ij}} =\frac{1}{K^{2}} \{[2H^{2} − 2K] h̃^{ij} − 2Hg^{ij}\} ,
Φ^{ij, kl} =\frac{1}{K^{2}} \{[2K − 4H^{2}] h̃^{ij}h̃^{kl} + [2K − 2H^{2}] h̃^{ik}h̃^{jl}
4H [g^{ij}h̃^{kl} + g^{kl}h̃^{ij}] − 2g^{ij}g^{kl}} .

According to [10, (5.4)], it suffices to show that

(4.9)  Φ^{ij, kl}η_{ij}η_{kl} ≤ \frac{α − 1}{αΦ}Φ^{ij}η_{ij}Φ^{kl}η_{kl}

for symmetric matrices (η_{ij}). Terms involving η_{12} clearly have the right sign. The remaining terms are a quadratic form in (η_{11}, η_{22}). Thus (4.9) is fulfilled, if

|A|^{2}· \{[2K − 4H^{2}] (\begin{matrix}\frac{1}{λ_{1}^{2}} & \frac{1}{λ_{1}λ_{2}} \\ \frac{1}{λ_{1}λ_{2}} & \frac{1}{λ_{2}^{2}} \end{matrix} ) + [2K − 2H^{2}] (\begin{matrix}\frac{1}{λ_{1}^{2}} & 0 \\ 0 & \frac{1}{λ_{2}^{2}} \end{matrix} )

+4H (\begin{matrix}\frac{2}{λ_{1}} & \frac{1}{λ_{1}} + \frac{1}{λ_{2}} \\ \frac{1}{λ_{1}} + \frac{1}{λ_{2}} & \frac{2}{λ_{2}^{2}} \end{matrix} ) − 2 (\begin{matrix}1 & 1 \\ 1 & 1 \end{matrix})\} ≤

≤ − \frac{α − 1}{α} \{[2H^{2} − 2K] (\frac{1}{λ_{1}} ; \frac{1}{λ_{2}} ) − 2H (\begin{matrix}1 \\ 1 \end{matrix})\} ⊗ \{[2H^{2} − 2K] (\frac{1}{λ_{1}} ; \frac{1}{λ_{2}} ) − 2H (\begin{matrix}1 \\ 1 \end{matrix})\}

or equivalently

−6 (λ_{1}^{2} + λ_{2}^{2}) (\begin{matrix}\frac{λ_{2}^{2}}{λ_{1}^{2}} & 0 \\ 0 & \frac{λ_{1}^{2}}{λ_{2}^{2}} \end{matrix} ) ≤ −4\frac{α − 1}{α} (\begin{matrix}\frac{λ_{2}^{4}}{λ_{1}^{2}} & λ_{1}λ_{2} \\ λ_{1}λ_{2} & \frac{λ_{1}^{4}}{λ_{2}^{2}} \end{matrix}) .

As \frac{α−1}{α} ≤ \frac{3}{2} for α ≤ −2, we obtain that Φ is α-concave. □

We are now able to improve our velocity bounds.

Lemma 4.15. Under the assumptions of Theorem 3.1, we obtain

2(6(T −t))^{−2/3} ·(1 − c · (T − t)^{1/12}) ≤ |A|^{2} ≤ 2(6(T −t))^{−2/3} ·(1 + c · (T − t)^{1/12})

everywhere on M_{t} for (T − t) sufficiently small.

Proof. We may assume that T − t > 0 is so small that we can use the results obtained before. Parameterize M_{t} by 𝕊^{2} such that the normal image of M_{t} at X(z, t) equals z ∈ 𝕊^{2}. Let us define the support function s of M_{t} as

s(z, t) := ⟨X(z, t), z⟩.

Its evolution equation, see e. g. [10], is

(4.10)  \frac{d}{dt}s(z, t) = −|A|^{2}(z, t).

The α-concavity proved in Lemma 4.14 allows us to use [10, Theorem 5.17]. We obtain for 0 < t_{1} < t_{2} < T and z ∈ 𝕊^{2}, for two points (z, t_{1}) and (z, t_{2}) with the same normal,

(4.11)  \frac{|A|^{2}(z, t_{2})}{|A|^{2}(z, t_{1})} ≥ (\frac{t_{1}}{t_{2}} )^{2/3} .

Let us assume that q(t) is the origin for some fixed time t. As M_{t} lies between ∂B_{r_{+}(t)}(0) and ∂B_{r_{−}(t)}(0), M_{t+τ} lies outside B_{(r_{−}^{3}(t)−6τ)}_{1/3}(0) for any 0 < τ < T −t, so

(4.12)  r_{−}(t) ≤ s(·, t) ≤ r_{+}(t) and (r_{−}^{3} − 6τ)^{1/3} ≤ s(·, t + τ).

Set τ = r_{−}^{5/2}(t)·(r_{+}(t)−r_{−}(t))^{1/2} and observe that t+τ < T, if (r_{+}−r_{−})^{1/2} ≤ \frac{1}{6}r_{−}^{1/2} (by Lemma 4.10), or, if T − t is sufficiently small (by Lemma 4.12). We estimate

|A|^{2}(z, t) ≤ inf_{0≤τ̃≤τ} \{(\frac{t + τ̃}{t})^{2/3} · |A|^{2}(z, t + τ̃)\}  by (4.11)

≤ (\frac{t + τ}{t} )^{2/3} · \frac{1}{τ} · ∫_{t}^{t+τ}|A|^{2}(z, τ̃)dτ̃

≤ (1 + \frac{τ}{t} )^{2/3} · \frac{1}{τ} · (s(z, t) − s(z, t + τ))  by (4.10)
≤ (1 + \frac{τ}{t} ) · \frac{1}{τ} · (r_{+}(t) − (r_{−}^{3}(t) − 6τ)^{1/3})  by (4.12)

=\frac{1 + \frac{τ}{t}}{r_{−}^{2} · (\frac{r_{+}}{r_{−}} − 1)^{1/2}} · (\frac{r_{+}}{r_{−}} − (1 − 6 · (\frac{r_{+}}{r_{−}} − 1)^{1/2})^{1/3}) .

The maximal existence time T is bounded in terms of the radius of a sphere enclosed by M_{0}. So we may also assume that t is bounded below by a positive constant. A very crude estimate gives

τ ≤ r_{−}^{5/2} · r_{+}^{1/2} ≤ r_{+}^{3} ≤ c · (T − t),

so we obtain

\frac{1 + \frac{τ}{t}}{r_{−}^{2}} ≤ (6(T − t))^{−2/3} · (1 + c · (T − t)^{1/6}) .

Observe that for |x| ≤ \frac{1}{2}, we have

−(1 − x)^{1/3} ≤ −1 + \frac{1}{3}x + \frac{1}{3}x^{2} and (1 + x)^{1/3} ≥ 1 + \frac{1}{3}x − \frac{1}{3}x^{2}.

We conclude for small (T − t)

|A|^{2}(z, t) ≤(6(T − t))^{−2/3} · (1 + c · (T − t)^{1/6}) · (2 + 13 · (\frac{r_{+}}{r_{−}} − 1)^{1/2})
≤2 · (6(T − t))^{−2/3} · (1 + c · (T − t)^{1/12}) .

For the lower bound on |A|^{2}, we proceed similarly and use τ = r_{−}^{5/2}(t) · (r_{+}(t) − r_{−}(t))^{1/2},

(r_{−}^{3} + 6τ)^{1/3} ≤ s(z, t − τ) and s(z, t) ≤ r_{+}(t).
□

We have the following estimate for the principal curvatures

Lemma 4.16. Under the assumptions of Theorem 3.1, we obtain

(6(T −t))^{−1/3} ·(1 − c · (T − t)^{1/12}) ≤ λ_{1}, λ_{2} ≤ (6(T −t))^{−1/3} ·(1 + c · (T − t)^{1/12})

on M_{t} for small (T − t).

Proof. As H^{2} = 2|A|^{2} − (λ_{1} − λ_{2})^{2}, we obtain

\begin{matrix}λ_{1} = & \frac{1}{2}(λ_{1} + λ_{2}) + \frac{1}{2}(λ_{1} − λ_{2}) \\ = & \frac{1}{2} \sqrt{2|A|^{2} − (λ_{1} − λ_{2})^{2}} + \frac{1}{2}(λ_{1} − λ_{2}). \end{matrix} (4.13)

Combining Lemmata 4.7 and 4.15, we get |λ_{1} − λ_{2}| ≤ c·(T −t)^{−1/6}. We use Lemma 4.15 and (4.13). The claimed inequality follows. □

Proof of Theorem 3.1: Lemma 4.16 implies, that, everywhere on M_{t}, the quotient λ_{1}/λ_{2} tends to 1 as t ↑ T. Then we can apply known results, see e. g. [5, Theorem 2], to conclude that the rescaled surfaces converge smoothly to the unit sphere 𝕊^{2} ⊂ ℝ^{3}. □

A standard way of rescaling [7] is to consider the embeddings X̃(·, t),

X̃(z, t) := (6(T − t))^{−1/3}(X(z, t) − Q)

with Q as in Theorem 3.1. Define the time function τ(t) := \frac{1}{6} log T − \frac{1}{6} log(T − t). Then we have, using suggestive notation, the following evolution equation

\frac{d}{dτ} X̃ = − |Ã|^{2} ν̃ + 2X̃

and our a priori estimates imply, that, for τ → ∞, M̃_{τ} converges exponentially to 𝕊^{2}.

4.5. Convergence Rate

In order to find out what the optimal convergence rate might be, we consider the evolution equation

\frac{d}{dt}X = −|A|^{2}ν + 2X,

that appropriately rescaled solutions of \frac{d}{dt}X = −|A|^{2}ν fulfill. As in [51, Appendix], we represent the surfaces M_{t} as graphs over the sphere with embeddings

𝕊^{2} ∋ x ↦ x · u(x),

where u : 𝕊^{2} → ℝ_{+}. Let (σ_{ij}) be the standard metric on the sphere and (σ^{ij}) its inverse. Then we get as in [51], using indices to denote covariant derivatives on 𝕊^{2},

g_{ij} =u^{2}(σ_{ij} + φ_{i}φ_{j}), where φ = log u,
h_{ij} =\frac{1}{uw} g_{ij} − \frac{u}{w} φ_{ij}, where w = \sqrt{1 + φ_{i}σ^{ij}φ_{j}},
\frac{∂u}{∂t} = − |A|^{2}w + 2u.

We linearize our equation around the stationary solution u = 1 and take u = 1+εv. Then

\frac{d}{dε}w|_{ε=0} =0,
\frac{d}{dε}g_{ij}|_{ε=0} =2vσ_{ij},
\frac{d}{dε}h_{ij}|_{ε=0} =vσ_{ij} − v_{ij}.

So the linearized equation becomes

\frac{∂v}{∂t} = 2Δv + 6v.

There are spherical harmonics u solving Δu = −l(l + 1)u for l ∈ ℕ [43]. We do not need to consider l = 0 as a corresponding eigenfunction is positive (or negative) everywhere. Thus the resulting surface does not contract to a point for t ↑ T in the unrescaled setting. Similarly, we can exclude l = 1 as the respective eigenfunctions correspond to translations and translated surfaces converge to infinity in the rescaled setting.

Using the ansatz eigenfunction multiplied with e^{−λt}, we get solutions for the linearized equation for λ = 2l(l + 1) − 6. This is positive for l = 2, λ = 6. So we should not expect a convergence rate better than ||X| − 1| ≤ c · e^{−6t} after rescaling, corresponding to estimates like

r_{+} ≤ (6(T − t))^{1/3} · (1 + c · (T − t)).

Here, we have assumed that q(t) = 0.

We can still improve our convergence rate by considering

max_{M_{t}} \frac{(λ_{1}^{2} + λ_{2}^{2})^{2} (λ_{1} − λ_{2})^{2}}{4λ_{1}^{2}λ_{2}^{2}} = max_{M_{t}} \begin{matrix}(2|A|^{2} − H^{2}) & (|A|^{2}) & 2 \\ (H^{2} − |A|^{2}) & 2 \end{matrix} ≡ max_{M_{t}} w.

Under the flow equation \frac{d}{dt}X = −|A|^{2}ν, we obtain, using calculations as above, that w fulfills in a critical point of w

\frac{d}{dt}w − F^{ij}w_{; ij} = − \frac{(λ_{1}^{2} + λ_{2}^{2})^{2}}{(λ_{1}^{3} − λ_{1}λ_{2}^{2} + 2λ_{2}^{3})^{2} λ_{1}^{5}λ_{2}} · (5λ_{1}^{9} − 3λ_{1}^{8}λ_{2} − 6λ_{1}^{7}λ_{2}^{2} + 26λ_{1}^{6}λ_{2}^{3}
−20λ_{1}^{5}λ_{2}^{4} + 8λ_{1}^{4}λ_{2}^{5} + 6λ_{1}^{3}λ_{2}^{6} − 2λ_{1}^{2}λ_{2}^{7} − λ_{1}λ_{2}^{8} + 3λ_{2}^{9}) · h_{11; 1}^{2}
+ (…) · h_{22; 2}^{2}
≤0.

This implies convergence rates like

r_{+} ≤ (6(T − t))^{1/3} · (1 + c · (T − t)^{1/3})

similar to [12], where the author also has a scaling invariant upper bound for |λ_{1} − λ_{2}|. As there are no negative constant terms left for this choice of w, it might be, that there is no monotone quantity as studied in this paper, that allows to further improve this convergence rate.

CHAPTER 5 Finding Monotone Quantities

5.1. The Algorithm

We use a sieve algorithm and start with symmetric rational functions of the principal curvatures as candidates for test functions, e. g.

w = \frac{p_{1}(λ_{1}, λ_{2})}{p_{2}(λ_{1}, λ_{2})} = \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} .

Here, p_{1} ≠ 0 and p_{2} ≠ 0 are homogeneous polynomials.

In the end, we want to find functions w such that W := sup_{M_{t}} w is monotone and ensures convergence to round spheres.

We check, whether these test functions w fulfill the following conditions.

(1) (a) p_{1}(λ_{1}, λ_{2}), p_{2}(λ_{1}, λ_{2}) ≥ 0 for 0 < λ_{1}, λ_{2}, (b) p_{1}(λ_{1}, λ_{2}) = 0 for λ_{1} = λ_{2} > 0.

(2) deg p_{1} > deg p_{2}.

(3) \frac{∂w(1,λ_{2})}{∂λ_{2}} < 0 for 0 < λ_{2} < 1 and \frac{∂w(1,λ_{2})}{∂λ_{2}} > 0 for λ_{2} > 1.

(4) \frac{d}{dt}w − F^{ij}w_{; ij} ≤ 0 (a) for terms without derivatives of (h_{ij}),

(b) for terms involving derivatives of (h_{ij}), if w_{; i} = 0 for i = 1, 2.

5.2. Motivation and Randomized Tests

We restrict our attention to non-negative polynomials p_{i}. For all flow equations considered, spheres contract to points and stay spherical. So we can only find monotone quantities, if deg p_{1} ≤ deg p_{2} or p_{1}(λ, λ) = 0.

If deg p_{1} < deg p_{2}, we obtain that W is decreasing on any self-similarly shrinking surface. So this does not imply convergence to a sphere. The counterexamples in [5] show for normal velocities of homogeneity larger than 1, that the pinching ratio sup_{M_{t}} λ_{2}/λ_{1} (for λ_{2} > λ_{1}) will increase on some time interval during the flow for appropriate initial surfaces. Therefore, we require in step (2), that deg p_{1} > deg p_{2}.

Condition (3) ensures that the quantity decreases, if the eigenvalues approach each other. This excludes especially local zeroes of w(1, λ_{2}) for λ_{2} ≠ 1.

In steps (1), (3), and (4), inequalities are tested by evaluating both sides at random numbers. If an inequality is not violated for sufficiently many tuples of random numbers, we move to the next step and keep the candidate, otherwise we start with another candidate.

The essential part is to test, whether \frac{d}{dt}w −F^{ij}w_{; ij} ≤ 0 holds at a point, where w_{; i} = 0.

We assume in step (4a), that h_{ij; k} = 0. Recall the algebraic fact, that all products of H and |A|^{2} of given homogeneity form a basis for the symmetric homogeneous polynomials of that degree. We represent the polynomials p_{i} in this basis. At random values for λ_{1}, λ_{2}, we compute \frac{d}{dt}H−F^{ij}H_{; ij} and \frac{d}{dt}|A|^{2}−F^{ij}(|A|^{2})_{; ij}. This can be combined according to the rules of differentiation and yields \frac{d}{dt}w − F^{ij}w_{; ij}, evaluated at these numbers.

If not all components of h_{ij; k} vanish (step (4b)), we also have to choose random numbers for h_{ij; k} that fulfill the extremal condition w_{; i} = 0. As above, we evaluate \frac{d}{dt}w − F^{ij}w_{; ij} at the random numbers chosen. Here we can ignore all terms that do not contain derivatives of the second fundamental form. The evaluation of the remaining terms is more involved than in the last step, but can be done similarly according to the various rules of differentiation.

Now we iterate steps (4a) and (4b). If all tests yield \frac{d}{dt}w−F^{ij}w_{; ij} ≤ 0 at critical points of w, it seems likely that we have found an appropriate test quantity. Indeed, if this inequality is fulfilled, the maximum principle implies that W = max_{M_{t}} w is non-increasing in time.

We implemented this algorithm in a C-program and used it to find all the new test functions of that form in this paper.

Obviously, the computing time depends on the number of tests performed. The following computing times are measured for the quantity (4.2) on an Intel Pentium 4, 2.4 GHz, running Linux 2.4.24 and GNU C-compiler 2.95.4. The number of tests per second for step (4a) is 1.6 · 10^{5} and 5.8 · 10^{3} for step (4b). The other steps are comparable to step (4a). In steps (1) to (3), the calculations do not depend on the normal velocity.

It seems worth noting, that, after testing with enough random numbers in an appropriate range, every candidate for a monotone quantity that we checked, turned out to be a useful test quantity. In that sense, algorithm and program seem to be correct.

We are convinced that it is possible to implement this algorithm for surfaces without using random numbers. Evolution equations can be computed algebraically and Sturm’s algorithm can be used to test for non-negativity.

We expect that similar algorithms will be used to find monotone test functions for other (geometric) problems.

CHAPTER 6 Computing Evolution Equations

It is straightforward to use a computer algebra system to obtain evolution equations of test quantities w, evaluated at a critical point of w. More precisely, let M_{t} be a family of surfaces in ℝ^{3}, moving with normal velocity F = F (λ_{1}, λ_{2}), where F > 0 for contracting surfaces. Assume that the test quantity w is a function of H and |A|^{2}. Then w fulfills the evolution equation

\frac{d}{dt}w − F^{ij}(w)_{; ij} = C_{w}(λ_{1}, λ_{2}) + G_{w}(λ_{1}, λ_{2})h_{11; 1}^{2} + G_{w}(λ_{2}, λ_{1})h_{22; 2}^{2}

in a critical point of w. It remains to compute C_{w} (“constant terms”) and G_{w} (“gradient terms”). We use Lemma 2.1 and the following equation of [7, 27]

F^{ij, kl}η_{ij}η_{kl} = ∑ \frac{∂^{2}F}{∂λ_{i}∂λ_{j}} η_{ii}η_{jj} + ∑_{i≠j} \frac{\frac{∂F}{∂λ_{i}} − \frac{∂F}{∂λ_{j}}}{λ_{i} − λ_{j}} (η_{ij})^{2},

for symmetric matrices (η_{ij}) and λ_{1} ≠ λ_{2} or λ_{1} = λ_{2}, when we interpret the last term as a limit. For w = H, we obtain

C_{H} = (\frac{∂F}{∂λ_{1}} λ_{1}^{2} + \frac{∂F}{∂λ_{2}} λ_{2}^{2}) H + (F − \frac{∂F}{∂λ_{1}} λ_{1} − \frac{∂F}{∂λ_{2}} λ_{2}) |A|^{2},

G_{H} =\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} + 2\frac{∂^{2}F}{∂λ_{1}∂λ_{2}} a_{1} + \frac{∂^{2}F}{∂λ_{2}∂λ_{2}} a_{1}^{2} + 2 \frac{\frac{∂F}{∂λ_{1}} − \frac{∂F}{∂λ_{2}}}{λ_{1} − λ_{2}} a_{1}^{2},

where

a_{1} = −\frac{∂w (λ_{1} + λ_{2}, λ_{1}^{2} + λ_{2}^{2})}{∂λ_{1}} (\frac{∂w (λ_{1} + λ_{2}, λ_{1}^{2} + λ_{2}^{2})}{∂λ_{2}})^{−1}

is such that h_{22; 1} = a_{1}h_{11; 1} in a critical point of w. Similarly, we get for w = |A|^{2}

C_{|A|^{2}} =2 (\frac{∂F}{∂λ_{1}} λ_{1}^{2} + \frac{∂F}{∂λ_{2}} λ_{2}^{2}) |A|^{2} + 2 (F − \frac{∂F}{∂λ_{1}} λ_{1} − \frac{∂F}{∂λ_{2}} λ_{2}) tr A^{3},
G_{|A|^{2}} = − 2 (\frac{∂F}{∂λ_{1}} (1 + a_{1}^{2}) + 2\frac{∂F}{∂λ_{2}} a_{1}^{2})

+ 2 (\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} + 2\frac{∂^{2}F}{∂λ_{1}∂λ_{2}} a_{1} + \frac{∂^{2}F}{∂λ_{2}∂λ_{2}} a_{1}^{2}) λ_{1} + 4 \frac{\frac{∂F}{∂λ_{1}} − \frac{∂F}{∂λ_{2}}}{λ_{1} − λ_{2}} a_{1}^{2}λ_{2}.

We also need some mixed terms

−F^{ij}H_{;i}H_{;j} = − \frac{∂F}{∂λ_{1}} (1 + a_{1})^{2},
−F^{ij} (|A|^{2})_{; i} (|A|^{2})_{; j} = − 4\frac{∂F}{∂λ_{1}} (λ_{1} + a_{1}λ_{2})^{2},
−2F^{ij}H_{;i} (|A|^{2})_{; j} = − F^{ij} (H_{;i} (|A|^{2})_{; j} + H_{;j} (|A|^{2})_{; i})

= − 4\frac{∂F}{∂λ_{1}} (1 + a_{1})(λ_{1} + a_{1}λ_{2}).

Combining these expressions yields

C_{w} =\frac{∂w}{∂H} C_{H} + \frac{∂w}{∂|A|^{2}} C_{|A|^{2}},
G_{w} =\frac{∂w}{∂H} G_{H} + \frac{∂w}{∂|A|^{2}} G_{|A|^{2}} − \frac{∂^{2}w}{∂H∂H} \frac{∂F}{∂λ_{1}} (1 + a_{1})^{2}

− 4\frac{∂^{2}w}{∂|A|^{2}∂|A|^{2}} \frac{∂F}{∂λ_{1}} (λ_{1} + a_{1}λ_{2})^{2} − 4\frac{∂^{2}w}{∂H∂|A|^{2}} \frac{∂F}{∂λ_{1}} (1 + a_{1})(λ_{1} + a_{1}λ_{2}).

This formulae allow to easily compute evolution equations in critical points.

CHAPTER 7 Other Normal Velocities

7.1. H^{3}-Flow

In this and the following sections, we will consider strictly convex surfaces contracting according to \frac{d}{dt}X = −F ν for several normal velocities F. We will not repeat parts of the argument that are very similar to the respective parts in the proof for F = |A|^{2}.

As the theorems for these flow equations agree essentially with Theorem 3.1, we will state them in concise form as follows.

Theorem 7.1. A smooth compact strictly convex surface in ℝ^{3}, contracting with normal velocity H^{3}, converges to a round point in finite time.

7.1.1. A Monotone Quantity.

Theorem 7.2. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3}, flowing according to \frac{d}{dt}X = −H^{3}ν,

max_{M_{t}} \frac{(λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1} + λ_{2})^{2}(λ_{1} − λ_{2})^{2}}{2 (λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2}) λ_{1}λ_{2}}

is non-increasing in time.

Proof. We compute

F =H^{3},
F^{ij} =3H^{2}g^{ij},
\frac{d}{dt}H − F^{ij}H_{; ij} =H^{3}|A|^{2} + 6H ∑ h_{ii; k}h_{jj; k},
\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =6H^{2} (|A|^{2})^{2} − 4H^{3} tr A^{3}
− 6H^{2} ∑ h_{ij; k}^{2} + 12H ∑ λ_{k}h_{ii; k}h_{jj; k},
w̃ = log (\frac{(λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1} + λ_{2})^{2}(λ_{1} − λ_{2})^{2}}{2 (λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2}) λ_{1}λ_{2}} )

= log (\frac{−H^{6} + |A|^{2}H^{4} + 2 (|A|^{2})^{2} H^{2}}{−H^{4} + 4|A|^{2}H^{2} − 3 (|A|^{2})^{2}} ) .

In a critical point of w̃, we obtain

h_{22; 1} =\frac{λ_{2}}{λ_{1}} · \frac{3λ_{1}^{6} − 2λ_{1}^{5}λ_{2} + 4λ_{1}^{4}λ_{2}^{2} + 4λ_{1}^{3}λ_{2}^{3} + 4λ_{1}^{2}λ_{2}^{4} − 2λ_{1}λ_{2}^{5} + λ_{2}^{6}}{3λ_{2}^{6} − 2λ_{2}^{5}λ_{1} + 4λ_{2}^{4}λ_{1}^{2} + 4λ_{2}^{3}λ_{1}^{3} + 4λ_{2}^{2}λ_{1}^{4} − 2λ_{2}λ_{1}^{5} + λ_{1}^{6}} · h_{11; 1},

\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} = (\frac{−6H^{5}+4|A|^{2}H^{3}+4(|A|^{2})^{2}H}{−H^{6}+|A|^{2}H^{4}+2(|A|^{2})^{2}H^{2}} − \frac{−4H^{3}+8|A|^{2}H}{−H^{4}+4|A|^{2}H^{2}−3(|A|^{2})^{2}} ) ·

· (|A|^{2}H^{3} + 6H ∑ h_{ii; k}h_{jj; k})

+ (\frac{H^{4}+4|A|^{2}H^{2}}{−H^{6}+|A|^{2}H^{4}+2(|A|^{2})^{2}H^{2}} − \frac{4H^{2}−6|A|^{2}}{−H^{4}+4|A|^{2}H^{2}−3(|A|^{2})^{2}} ) ·

· (6 (|A|^{2})^{2} H^{2} − 4H^{3} tr A^{3}

−6H^{2} ∑ h_{ij; k}^{2} + 12H ∑ λ_{k}h_{ii; k}h_{jj; k})

+ (\frac{−30H^{4}+12|A|^{2}H^{2}+4(|A|^{2})^{2}}{−H^{6}+|A|^{2}H^{4}+2(|A|^{2})^{2}H^{2}} − \frac{−12H^{2}+8|A|^{2}}{−H^{4}+4|A|^{2}H^{2}−3(|A|^{2})^{2}} ) ·

· (−3H^{2} ∑ h_{ii; k}h_{jj; k})

+ (\frac{4H^{2}}{−H^{6}+|A|^{2}H^{4}+2(|A|^{2})^{2}H^{2}} − \frac{−6}{−H^{4}+4|A|^{2}H^{2}−3(|A|^{2})^{2}} ) ·

· (−12H^{2} ∑ λ_{i}λ_{j}h_{ii; k}h_{jj; k})

+ (\frac{4H^{3}+8H|A|^{2}}{−H^{6}+|A|^{2}H^{4}+2(|A|^{2})^{2}H^{2}} − \frac{8H}{−H^{4}+4|A|^{2}H^{2}−3(|A|^{2})^{2}} ) ·

· (−6H^{2} ∑(λ_{i} + λ_{j})h_{ii; k}h_{jj; k})

= − 12\frac{(λ_{1} + λ_{2})^{2}λ_{1}^{3}λ_{2}^{3}}{(λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2})}

− \frac{12(λ_{1} + λ_{2})^{4}λ_{2}}{(λ_{1}^{6} − 2λ_{1}^{5}λ_{2} + 4λ_{1}^{4}λ_{2}^{2} + 4λ_{1}^{3}λ_{2}^{3} + 4λ_{1}^{2}λ_{2}^{4} − 2λ_{1}λ_{2}^{5} + 3λ_{2}^{6})^{2}} ·

· \frac{1}{(λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1} − λ_{2})^{2}λ_{1}^{3}} ·

· (2λ_{1}^{16} − 10λ_{1}^{15}λ_{2} + 10λ_{1}^{14}λ_{2}^{2} + 42λ_{1}^{13}λ_{2}^{3} − 135λ_{1}^{12}λ_{2}^{4}

+274λ_{1}^{11}λ_{2}^{5} − 272λ_{1}^{10}λ_{2}^{6} + 258λ_{1}^{9}λ_{2}^{7} − 144λ_{1}^{8}λ_{2}^{8} + 262λ_{1}^{7}λ_{2}^{9}

−166λ_{1}^{6}λ_{2}^{10} + 98λ_{1}^{5}λ_{2}^{11} + 6λ_{1}^{4}λ_{2}^{12} − 20λ_{1}^{3}λ_{2}^{13} + 14λ_{1}^{2}λ_{2}^{14}

−4λ_{1}λ_{2}^{15} + λ_{2}^{16}) · h_{11; 1}^{2}

− (…) · h_{22; 2}

≤0.

We applied Sturm’s algorithm [58] to obtain the last inequality.

Here and in the rest of the paper, we have sometimes used a computer algebra program for the calculations involving longer polynomials. Moreover, we use (…) in (…) · h_{22; 2}^{2} to abbreviate terms that are, up to interchanging λ_{1} and λ_{2}, equal to the respective factors in front of h_{11; 1}^{2}.

We have applied the following two identities in order to rewrite terms involving derivatives of the second fundamental form

∑ λ_{i}^{α}λ_{j}^{β}λ_{k}^{γ}h_{ij; k}^{2} = (λ_{1}^{α}λ_{1}^{β}λ_{1}^{γ} + λ_{1}^{α}λ_{2}^{β}λ_{2}^{γ}a_{1}^{2} + λ_{2}^{α}λ_{1}^{β}λ_{2}^{γ}a_{1}^{2} + λ_{2}^{α}λ_{2}^{β}λ_{1}^{γ}a_{1}^{2}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2},

∑ λ_{i}^{α}λ_{j}^{β}λ_{k}^{γ}h_{ii; k}h_{jj; k} = (λ_{1}^{α} + λ_{2}^{α}a_{1}) (λ_{1}^{β} + λ_{2}^{β}a_{1}) λ_{1}^{γ}h_{11; 1}^{2} + (…) · h_{22; 2}^{2}.

7.1. H^{3}-FLOW 23

They hold for all α, β, γ ∈ ℝ. □

7.1.2. Velocity Bounds. The following lemma is known, see [53]. Nevertheless, we include it, as we will use some of the calculations of its proof later on.

Lemma 7.3. For a family of compact smooth strictly convex surfaces M_{t} ⊂ ℝ^{3}, 0 ≤ t < T, flowing according to \frac{d}{dt}X = −F ν with F = H^{α}, α > 1, a positive lower bound on the principal curvatures, λ_{1}, λ_{2} ≥ ε > 0, is preserved during the evolution.

Proof. We wish to apply the maximum principle for tensors [18, 29, 30]. So we define

M_{ij} = h_{ij} − εg_{ij}.

We use (2.5) and Lemma 2.1 to obtain

\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl} =F^{kl}h_{k}^{a}h_{al} · h_{ij} − F^{kl}h_{kl}h_{i}^{a}h_{aj} − F h_{i}^{k}h_{kj}
+ F^{kl, rs}h_{kl; i}h_{rs; j} + 2εF h_{ij}.

We now specialize to the normal velocity F = H^{α}. It is easy to see that

F^{kl, rs}h_{kl; i}h_{rs; j} ≥ 0.

We have to test the zero eigenvalue condition. Assume that ξ is a zero eigenvalue of (M_{ij}), h_{ij}ξ^{j} = εg_{ij}ξ^{j}, with g_{ij}ξ^{i}ξ^{j} = 1. We may assume, that in our coordinate system, we have ξ = (1, 0) and (h_{ij}) = ( \begin{matrix}ε & 0 \\ 0 & λ \end{matrix}) with λ ≥ ε > 0. Direct calculations yield

(\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl}) ξ^{i}ξ^{j} ≥αH^{α−1}|A|^{2}ε − αH^{α}ε^{2} − H^{α}ε^{2} + 2εH^{α}ε
=H^{α−1} (αελ(λ − ε) + ε^{3} + ε^{2}λ) > 0.

The lemma follows from the maximum principle. □

The next lemma appears also in [53]. Once again, we will use the following calculations later on.

Lemma 7.4. For a family of compact smooth strictly convex surfaces M_{t} ⊂ ℝ^{3}, 0 ≤ t < T, flowing according to \frac{d}{dt}X = −F ν with F = H^{β}, β > 0, the velocity F is bounded as in Lemma 4.2, in terms of β, the initial data, and the radius R of an enclosed sphere.

Proof. We proceed as above and get

\frac{d}{dt} log (\frac{F}{⟨X, ν⟩ − α})
−F^{ij} (log (\frac{F}{⟨X, ν⟩ − α}))_{; ij} =\frac{1}{⟨X, ν⟩ − α} (F + F^{ij}h_{ij} − αF^{ij}h_{i}^{k}h_{kj}) .

So far, we did not use the fact, that F = H^{β}. In an increasing maximum of \frac{F}{⟨X, ν⟩−α}, we get the inequality F + F^{ij}h_{ij} − αF^{ij}h_{i}^{k}h_{kj} ≥ 0 and deduce there

\frac{1}{α} \frac{1 + β}{β} ≥ \frac{|A|^{2}}{H} ≥ \frac{1}{2}H.

Our Lemma follows. □

7.1.3. Concavity.

Lemma 7.5. The dual function to F = H^{α}, α > 0, defined before Lemma 4.14, is −α-concave.

Proof. We use the same notation as before and obtain

Φ = − H^{α}K^{−α},
Φ^{ij} = − αH^{α−1}K^{−α}g^{ij} + αH^{α}K^{−α}h̃^{ij},
Φ^{ij, kl} = − α(α − 1)H^{α−2}K^{−α}g^{ij}g^{kl}
+ α^{2}H^{α−1}K^{−α}g^{ij}h̃^{kl} + α^{2}H^{α−1}K^{−α}h̃^{ij}g^{kl}
− α^{2}H^{α}K^{−α}h̃^{ij}h̃^{kl} − αH^{α}K^{−α}h̃^{ik}h̃^{jl}.

We have to prove that

Φ^{ij, kl}η_{ij}η_{kl} ≤ \frac{α + 1}{αΦ}Φ^{ij}η_{ij}Φ^{kl}η_{kl}

or

0 ≤ − (α + 1) (\begin{matrix}\frac{λ_{2}^{2}}{λ_{1}^{2}} & 1 \\ 1 & \frac{λ_{1}^{2}}{λ_{2}^{2}} \end{matrix} ) + (α − 1) (\begin{matrix}1 & 1 \\ 1 & 1 \end{matrix})

− αH (\begin{matrix}\frac{2}{λ_{1}} & \frac{1}{λ_{1}} + \frac{1}{λ_{2}} \\ \frac{1}{λ_{1}} + \frac{1}{λ_{2}} & \frac{2}{λ_{2}} \end{matrix} ) + αH^{2} (\begin{matrix}\frac{1}{λ_{1}^{2}} & \frac{1}{λ_{1}λ_{2}} \\ \frac{1}{λ_{1}λ_{2}} & \frac{1}{λ_{2}^{2}} \end{matrix} )

+ H^{2} (\begin{matrix}\frac{1}{λ_{1}^{2}} & 0 \\ 0 & \frac{1}{λ_{2}^{2}} \end{matrix} )

= (\begin{matrix}2\frac{λ_{2}}{λ_{1}} & −2 \\ −2 & 2\frac{λ_{1}}{λ_{2}} \end{matrix} ) .
□

7.1.4. Some Constants. We wish to obtain precise bounds on the normal velocity F near t = T. This proof is almost identical for all our test functions in Table 1 with a factor λ_{1}λ_{2} in the denominator. So it seems appropriate to state this proof only once with appropriate constants depending on the normal velocity and the test function. These constants are

• c_{h}: the homogeneity of F in terms of the principal curvatures.

• c_{1}: the value of F at λ_{1} = λ_{2} = 1.

• c_{α}: a positive constant, such that the dual function to F, defined before Lemma 4.14, is a −c_{α}-concave function. It turns out, that, for all flows considered here, we can choose c_{α} = c_{h}.

• c_{d}: a constant depending on the difference of the degrees of the numerator d_{n} and the denominator d_{d} of the test function w, c_{d} := \frac{1}{2}(2 − d_{n} + d_{d}). This constant is defined such that

(7.1)  |λ_{1} − λ_{2}| ≤ c · H^{c_{d}}

for a pinched surface for which max_{M_{t}} w is non-increasing.

\begin{matrix}F & tr A^{α} & H^{2} & H^{3} & H^{4} & H|A|^{2} & |A|^{4} & |A|^{2} + βH^{2} \\ c_{h} & α & 2 & 3 & 4 & 3 & 4 & 2 \\ c_{1} & 2 & 4 & 8 & 16 & 4 & 4 & 2 + 4β \\ c_{α} & α & 2 & 3 & 4 & 3 & 4 & 2 \\ c_{d} & \frac{1}{2}(3 − α) & \frac{1}{2} & 0 & −2 & 0 & −\frac{1}{2} & \frac{1}{2} \\ \frac{1−c_{d}}{1+c_{h}} & \frac{α−1}{2(α+1)} & \frac{1}{6} & \frac{1}{4} & \frac{3}{5} & \frac{1}{4} & \frac{3}{10} & \frac{1}{6} \end{matrix}

Table 1. Some constants

For the flow equations considered here, these constants are as in Table 1. We assume there that α ≥ 2 and β ≥ 0.

It is important for us that c_{d} < 1 as it implies that Inequality (7.1) is not scaling invariant.

7.1.5. Pinching. We show that our surfaces are pinched during the evolution, i. e. that there exists a constant c > 0, depending on our test quantity, especially on the upper bound for it, and on the positive lower bound for the principal curvatures, ε, such that

0 < \frac{1}{c} ≤ \frac{λ_{i}}{λ_{j}} ≤ c for i, j ∈ \{1, 2\}.

The following proof does not apply directly to the case F = K considered in [12], but the result is also true in that case.

By direct inspection, we see that all our test quantities w are such that w · \frac{λ_{1}λ_{2}}{(λ_{1}−λ_{2})^{2}} is bounded below by a positive constant, depending especially on ε. So we see that

\frac{(\frac{λ_{1}}{λ_{2}} − 1)^{2}}{\frac{λ_{1}}{λ_{2}}} = \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} ≤ c.

Thus \frac{λ_{1}}{λ_{2}} is bounded above and the surface is pinched.

7.1.6. Evolution of Spheres. The radius of contracting spheres fulfills the ordinary differential equation

ṙ(t) = −\frac{c_{1}}{r(t)^{c_{h}}} .

A solution is given by

r(t) = (c_{1} · (1 + c_{h}) · (T − t))^{\frac{1}{1+c_{h}}} ,

so that inner and outer radii are related to the life span T − t of M_{t} as follows

ρ_{−} ≤ (c_{1} · (1 + c_{h}) · (T − t))^{\frac{1}{1+c_{h}}} ≤ ρ_{+}(t).

7.1.7. Bounds for Radii. In order to prove bounds for the radii _{+} and r_{−}, we proceed as in the proof of Lemma 4.12 and use |λ_{1} − λ_{2}| ≤ c · H^{c_{d}}

r_{−} ≥ \frac{1}{8π} ∫_{M_{t}} H · \{1 − c · ρ_{+} · H^{c_{d}}\} .

For each F in Table 1, there exists c_{F} > 0 such that

0 < \frac{1}{c_{F}} H ≤ F^{1/c_{h}} ≤ c_{F} H,

so we can apply the variants of Lemma 4.2 for other curvature functions. For t close to T, we get

r_{−} ≥\frac{1}{8π} ∫_{M_{t}} H · \{1 − c · (T − t)^{\frac{1−c_{d}}{1+c_{h}}} \}

and similarly, we obtain

r_{+} ≤\frac{1}{8π} ∫_{M_{t}} H · \{1 + c · (T − t)^{\frac{1−c_{d}}{1+c_{h}}} \} .

Then we get for small T − t

(c_{1} · (1 + c_{n}) · (T − t))^{\frac{1}{1+c_{h}}} · \{1 − c · (T − t)^{\frac{1−c_{d}}{1+c_{h}}} \} ≤
≤ \frac{1}{8π} ∫_{M_{t}} H ≤(c_{1} · (1 + c_{h}) · (T − t))^{\frac{1}{1+c_{h}}} · \{1 + c · (T − t)^{\frac{1−c_{d}}{1+c_{h}}} \} ,

r_{−} ≥(c_{1} · (1 + c_{h}) · (T − t))^{\frac{1}{1+c_{h}}} · \{1 − c · (T − t)^{\frac{1−c_{d}}{1+c_{h}}} \} ,
r_{+} ≤(c_{1} · (1 + c_{h}) · (T − t))^{\frac{1}{1+c_{h}}} · \{1 + c · (T − t)^{\frac{1−c_{d}}{1+c_{h}}} \} ,

and

1 ≤\frac{r_{+}}{r_{−}} ≤ 1 + c · (T − t)^{\frac{1−c_{d}}{1+c_{h}}} .

7.1.8. Precise Velocity Bounds. We use the notation of Lemma 4.15. The Harnack inequality [10, Theorem 5.17] implies for t_{2} > t_{1} > 0

\frac{F (z, t_{2})}{F (z, t_{1})} ≥ (\frac{t_{1}}{t_{2}} )^{\frac{c_{α}}{1+c_{α}}} .

Note that spheres evolve such that we get

(r_{−}(t)^{1+c_{h}} − c_{1}(1 + c_{h})τ)^{\frac{1}{1+c_{h}}} ≤ s(·, t + τ).

We set

τ := r_{−}(t)^{1+c_{h}} · (\frac{r_{+}(t)}{r_{−}(t)} − 1)^{1/2}

7.2. H^{2}-FLOW 27

and get

F (z, t) ≤\frac{1 + c · (T − t)}{r_{−}(t)^{c_{h}} (\frac{r_{+}(t)}{r_{−}(t)} − 1)^{1/2}} ·

· (\frac{r_{+}(t)}{r_{−}(t)} − (1 − c_{1} · (1 + c_{h}) · (\frac{r_{+}(t)}{r_{−}(t)} − 1)^{1/2})^{\frac{1}{1+c_{h}}} ) .

Use for 0 ≤ x ≤ c(c_{h})

−(1 − x)^{\frac{1}{1+c_{h}}} ≤ −1 + \frac{1}{1 + c_{h}} x + \frac{1}{1 + c_{h}} x^{2}.

We get

(7.2)  F (z, t) ≤\frac{1 + c · (T − t)}{r_{−}(t)^{c_{h}}} · (c_{1} + c (\frac{r_{+}}{r_{−}} − 1)^{1/2})
≤c_{1} · (c_{1} · (1 + c_{h}) · (T − t))^{− \frac{c_{h}}{1+c_{h}}} · (1 + c · (T − t)\frac{1}{2} · \frac{1−c_{d}}{1+c_{h}} )

and a similar lower bound follows.

7.1.9. Convergence of Principal Curvatures. We consider F = H^{α} and obtain

λ_{1} =\frac{1}{2}(λ_{1} + λ_{2}) + \frac{1}{2}(λ_{1} − λ_{2})
=\frac{1}{2}F^{1/α} + \frac{1}{2}(λ_{1} − λ_{2})
≤ (2^{α} · (1 + α) · (T − t))^{− \frac{1}{1+α}} · (1 + c · (T − t)\frac{1}{2} · \frac{1−c_{d}}{1+α} ) .

A similar lower bound is proved analogously. Theorem 7.1 follows.

7.2. H^{2}-Flow Theorem 7.6. A smooth compact strictly convex surface in ℝ^{3}, contracting

with normal velocity H^{2}, converges to a round point in finite time.

Theorem 7.7. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3}, flowing according to \frac{d}{dt}X = −H^{2}ν,

max_{M_{t}} \frac{(λ_{1} + λ_{2})^{3}(λ_{1} − λ_{2})^{2}}{2 (λ_{1}^{2} + λ_{2}^{2}) λ_{1}λ_{2}}

is non-increasing in time.

Proof. We set

w =\frac{(λ_{1} + λ_{2})^{3}(λ_{1} − λ_{2})^{2}}{2 (λ_{1}^{2} + λ_{2}^{2}) λ_{1}λ_{2}} = \frac{H^{3} (2|A|^{2} − H^{2})}{|A|^{2} (H^{2} − |A|^{2})}.

In a critical point of w̃, we get, based on computer algebra calculations,

\frac{d}{dt}w − F^{ij}w_{; ij} = − 2\frac{(λ_{1} + λ_{2})^{4}(λ_{1} − λ_{2})^{2}λ_{1}λ_{2}}{(λ_{1}^{2} + λ_{2}^{2})^{2}}

− \frac{(λ_{1} + λ_{2})^{4}}{(λ_{1}^{2} + λ_{2}^{2})^{2} (λ_{1}^{4} − λ_{1}^{3}λ_{2} + 7λ_{1}^{2}λ_{2}^{2} − λ_{1}λ_{2}^{3} + 2λ_{2}^{4})^{2} λ_{1}^{4}} ·

· (5λ_{1}^{12} − 24λ_{1}^{11}λ_{2} + 112λ_{1}^{10}λ_{2}^{2} − 164λ_{1}^{9}λ_{2}^{3} + 529λ_{1}^{8}λ_{2}^{4}
−448λ_{1}^{7}λ_{2}^{5} + 952λ_{1}^{6}λ_{2}^{6} − 312λ_{1}^{5}λ_{2}^{7} + 391λ_{1}^{4}λ_{2}^{8} − 72λ_{1}^{3}λ_{2}^{9}
+56λ_{1}^{2}λ_{2}^{10} − 4λ_{1}λ_{2}^{11} + 3λ_{2}^{12}) · h_{11; 1}^{2} + (…) · h_{22; 2}^{2}
≤0.

We apply the maximum principle. Our claim follows. □

Theorem 7.6 follows.

7.3. H^{4}-Flow

Theorem 7.8. A smooth compact strictly convex surface in ℝ^{3}, contracting with normal velocity H^{4}, converges to a round point in finite time.

Theorem 7.9. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3}, flowing according to \frac{d}{dt}X = −H^{4}ν,

max_{M_{t}} \frac{(λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1} + λ_{2})^{6}(λ_{1} − λ_{2})^{2}}{2λ_{1}^{2}λ_{2}^{2}}

is non-increasing in time.

Proof. We proceed as above.

F =H^{4},
F^{ij} =4H^{3}g^{ij},
F^{ij, kl} =12H^{2}g^{ij}g^{kl},
w̃ = log (\frac{−H^{10} + H^{8}|A|^{2} + 2H^{6} (|A|^{2})^{2}}{H^{4} − 2H^{2}|A|^{2} + (|A|^{2})^{2}} )

= log (\frac{(λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1} + λ_{2})^{6}(λ_{1} − λ_{2})^{2}}{2λ_{1}^{2}λ_{2}^{2}} ) .

In a critical point of w̃, we get

h_{22; 1} =a_{1}h_{11; 1}
=\frac{λ_{2}}{λ_{1}} \frac{8λ_{1}^{4} + 3λ_{1}^{3}λ_{2} + 2λ_{1}^{2}λ_{2}^{2} − 3λ_{1}λ_{2}^{3} + 2λ_{2}^{4}}{8λ_{2}^{4} + 3λ_{2}^{2}λ_{1} + 2λ_{2}^{2}λ_{1}^{2} − 3λ_{2}λ_{1}^{3} + 2λ_{1}^{4}} h_{11; 1},
\frac{d}{dt}H − F^{ij}H_{; ij} =H^{4}|A|^{2} + 12H^{2}(1 + a_{1})^{2} · h_{11; 1}^{2} + (…) · h_{22; 2}^{2},
\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =8H^{3} (|A|^{2})^{2} − 6H^{4} tr A^{3}
− 8H^{3} (1 + 3a_{1}^{2}) · h_{11; 1}^{2} + 24H^{2}λ_{1}(1 + a_{1})^{2} · h_{11; 1}^{2}
+ (…) · h_{22; 2}^{2}.
−F^{ij}H_{; i}H_{; j} = − 4H^{3}(1 + a_{1})^{2} · h_{11; 1} + (…) · h_{22; 2}^{2},
−F^{ij} ((|A|^{2})_{; i} H_{; j} + (|A|^{2})_{; j} H_{; i}) = − 16H^{3}(λ_{1} + λ_{2}a_{1})(1 + a_{1}) · h_{11; 1}^{2}
+ (…) · h_{22; 2}^{2},

−F^{ij} (|A|^{2})_{; i} (|A|^{2})_{; j} = − 16H^{3}(λ_{1} + λ_{2}a_{1})^{2} · h_{11; 1}^{2} + (…) · h_{22; 2}^{2},

\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} = (\frac{−10H^{9} + 8H^{7}|A|^{2} + 12H^{5} (|A|^{2})^{2}}{−H^{10} + H^{8}|A|^{2} + 2H^{6} (|A|^{2})^{2}}

−\frac{4H^{3} − 4H|A|^{2}}{H^{4} − 2H^{2}|A|^{2} + (|A|^{2})^{2}} ) ·

· (\frac{d}{dt}H − F^{ij}H_{; ij})

+ (\frac{H^{8} + 4H^{6}|A|^{2}}{−H^{10} + H^{8}|A|^{2} + 2H^{6} (|A|^{2})^{2}} − \frac{−2H^{2} + 2|A|^{2}}{H^{4} − 2H^{2}|A|^{2} + (|A|^{2})^{2}} ) ·

· (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij})

+ (\frac{−90H^{8} + 56H^{6}|A|^{2} + 60H^{4} (|A|^{2})^{2}}{−H^{10} + H^{8}|A|^{2} + 2H^{6} (|A|^{2})^{2}}

−\frac{12H^{2} − 4|A|^{2}}{H^{4} − 2H^{2}|A|^{2} + (|A|^{2})^{2}} ) ·

· (−F^{ij}H_{; i}H_{; j})

+ (\frac{8H^{7} + 24H^{5}|A|^{2}}{−H^{10} + H^{8}|A|^{2} + 2H^{6} (|A|^{2})^{2}} − \frac{−4H}{H^{4} − 2H^{2}|A|^{2} + (|A|^{2})^{2}} ) ·

· (−F^{ij} (H_{; i}|A|_{; j}^{2} + H_{; j}|A|_{; i}^{2}))

+ (\frac{4H^{6}}{−H^{10} + H^{8}|A|^{2} + 2H^{6} (|A|^{2})^{2}} − \frac{2}{H^{4} − 2H^{2}|A|^{2} + (|A|^{2})^{2}} ) ·

· (−F^{ij}|A|_{; i}^{2}|A|_{; j}^{2})

=\frac{−3(λ_{1} − λ_{2})^{2}(λ_{1} + λ_{2})^{3}λ_{1}λ_{2}}{λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}}

− \frac{4λ_{2}(λ_{1} + λ_{2})^{5}}{λ_{1}^{3}(λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2})(λ_{1} − λ_{2})^{2}} ·

· \frac{1}{(2λ_{1}^{4} − 3λ_{1}^{3}λ_{2} + 2λ_{1}^{2}λ_{2}^{2} + 3λ_{1}λ_{2}^{2} + 8λ_{2}^{4})^{2}} ·

· (4λ_{1}^{10} + 202λ_{1}^{9}λ_{2} − 447λ_{1}^{8}λ_{2}^{2} + 809λ_{1}^{7}λ_{2}^{3} − 16λ_{1}^{6}λ_{2}^{4} + 696λ_{1}^{5}λ_{2}^{5}

−511λ_{1}^{4}λ_{2}^{6} + 161λ_{1}^{3}λ_{2}^{7} − 78λ_{1}^{2}λ_{2}^{8} + 4λ_{1}λ_{2}^{9} + 40λ_{2}^{10}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2}

≤0.

Here we used once more a computer algebra system and Sturm’s theorem to obtain the last inequality. □

Theorem 7.8 follows.

There is also a monotone quantity (involving polynomials of higher homogeneity) that allows to consider the flows H^{k}, 1 < k ≤ 5, simultaneously. For details, we refer to [54].

7.4. |A|^{2} + βH^{2}-Flow

Theorem 7.10. A smooth compact strictly convex surface in ℝ^{3}, contracting with normal velocity |A|^{2} + βH^{2}, 0 ≤ β ≤ 5, converges to a round point in finite time.

Theorem 7.11. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3}, flowing according to Ẋ = − (|A|^{2} + βH^{2}) ν, 0 ≤ β ≤ 5,

max_{M_{t}} \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{2λ_{1}λ_{2}}

is non-increasing in time.

Proof. Similarly as above, we obtain for F = |A|^{2} + βH^{2}

\frac{d}{dt}H − F^{ij}H_{; ij} = − (|A|^{2})^{2} + 2H tr A^{3} + β|A|^{2}H^{2}
+ 2 ∑ h_{ij; k}^{2} + 2β ∑ h_{ii; k}h_{jj; k},
\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =2|A|^{2} tr A^{3}
+ β (4 (|A|^{2})^{2} H − 2H^{2} tr A^{3})
+ β (−4H ∑ h_{ij; k}^{2} + 4 ∑ h_{ii; k}h_{jj; k}λ_{k}) .

As in the proof of Theorem 4.3, we set

w̃ = log H + log (2|A|^{2} − H^{2}) − log (H^{2} − |A|^{2})

and obtain in a critical point of w̃, where h_{22; 1} = a_{1}h_{11; 1}

−F^{ij}H_{; i}H_{; j} = − 2(λ_{1} + βH)(1 + a_{1})^{2} · h_{11; 1}^{2} + (…) · h_{22; 2}^{2},
−F^{ij} ((|A|^{2})_{; i} H_{; j} + (|A|^{2})_{; j} H_{; i}) = − 8(λ_{1} + βH)(λ_{1} + λ_{2}a_{1})(1 + a_{1}) · h_{11; 1}^{2}
+ (…) · h_{22; 2}^{2},
h_{22; 1} =\frac{λ_{2}}{λ_{1}} \frac{2λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}}{2λ_{2}^{2} + λ_{2}λ_{1} + λ_{1}^{2}} h_{11; 1}.

In a critical point of w̃, we obtain the evolution equation

\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} = (\frac{1}{H} − \frac{2H}{2|A|^{2} − H^{2}} − \frac{2H}{H^{2} − |A|^{2}} ) · (\frac{d}{dt}H − F^{ij}H_{; ij})

+ (\frac{2}{2|A|^{2} − H^{2}} + \frac{1}{H^{2} − |A|^{2}} ) · (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij})

+ (\frac{6}{2|A|^{2} − H^{2}} + \frac{2}{H^{2} − |A|^{2}} ) F^{ij}H_{i}H_{j}

− \frac{2}{H · (2|A|^{2} − H^{2})}F^{ij} ((|A|^{2})_{i} H_{j} + (|A|^{2})_{j} H_{i})

= − 4\frac{K^{2}}{H} − 2βHK

7.4. |A|^{2} + βH^{2}-FLOW 31

− 2\frac{(5λ_{1}^{8} − 4λ_{1}^{7}λ_{2} + 46λ_{1}^{6}λ_{2}^{2} + 48λ_{1}^{5}λ_{2}^{3} + 72λ_{1}^{4}λ_{2}^{4}) λ_{2}}{(λ_{1} + λ_{2}) (λ_{1} − λ_{2})^{2} (λ_{1}^{2} + λ_{1}λ_{2} + 2λ_{2}^{2})^{2} λ_{1}^{3}} h_{11; 1}^{2}

− 2\frac{(44λ_{1}^{3}λ_{2}^{5} + 34λ_{1}^{2}λ_{2}^{6} + 8λ_{1}λ_{2}^{7} + 3λ_{2}^{8}) λ_{2}}{(λ_{1} + λ_{2}) (λ_{1} − λ_{2})^{2} (λ_{1}^{2} + λ_{1}λ_{2} + 2λ_{2}^{2})^{2} λ_{1}^{3}} h_{11; 1}^{2}

+ 2β \frac{(λ_{1}^{4} − 16λ_{1}^{3}λ_{2} − 6λ_{1}^{2}λ_{2}^{2} − 8λ_{1}λ_{2}^{3} − 3λ_{2}^{4}) (λ_{1} + λ_{2})^{3} λ_{2}}{(λ_{1} − λ_{2})^{2} (λ_{1}^{2} + λ_{1}λ_{2} + 2λ_{2}^{2})^{2} λ_{1}^{3}} h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2}.

For β = 5, the factor in front of h_{11; 1}^{2} equals

− \frac{4λ_{2}^{2}}{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2} (λ_{1}^{2} + λ_{1}λ_{2} + 2λ_{2}^{2})^{2} λ_{1}^{3}} ·

· (28λ_{1}^{7} + 183λ_{1}^{6}λ_{2} + 334λ_{1}^{5}λ_{2}^{2} + 371λ_{1}^{4}λ_{2}^{3} + 272λ_{1}^{3}λ_{2}^{4} + 157λ_{1}^{2}λ_{2}^{5} + 54λ_{1}λ_{2}^{6} + 9λ_{2}^{7}) .

Our claim follows.

In order to see that the range for β is sharp for applying the maximum principle, we observe that the terms without derivatives of the second fundamental form require that β ≥ 0. For λ_{2} = 1 and λ_{1} → ∞, the factor in front of h_{11; 1}^{2} behaves like −112λ_{1}^{−3} for β = 5 and like −10λ_{1}^{−2} for β = 0, so we need the upper bound β ≤ 5. □

Lemma 7.12. For a family of compact smooth strictly convex surfaces M_{t} ⊂ ℝ^{3}, 0 ≤ t < T, flowing according to \frac{d}{dt}X = −F ν with F = |A|^{2} +βH^{2}, β ≥ 0, a positive lower bound on the principal curvatures, λ_{1}, λ_{2} ≥ ε > 0, is preserved during the evolution.

Proof. We proceed similarly as in Lemma 7.3. Dropping the term involving second derivatives of F yields

(\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl}) ξ^{i}ξ^{j} ≥2 (tr A^{3} + βH|A|^{2}) ε − (|A|^{2} + βH^{2}) ε^{2}
=ε^{4} + ελ^{3} + ελ^{2}(λ − ε) + β (ε^{4} + ε^{2}λ^{2} + 2ελ^{3}) > 0.
□

Similar calculations as before give an upper bound on the velocity for F = |A|^{2} + βH^{2}, β ≥ 0.

Lemma 7.13. The dual function to F = |A|^{2} + βH^{2}, β ≥ 0, is α-concave for α ≤ −2.

Proof. We have

Φ =\frac{2K − (1 + β)H^{2}}{K^{2}} ,
Φ^{ij} =\frac{1}{K^{2}} ((−2K + 2(1 + β)H^{2}) h̃^{ij} − 2(1 + β)Hg^{ij}) ,
Φ^{ij, kl} =\frac{1}{K^{2}} ((2K − 4(1 + β)H^{2}) h̃^{ij}h̃^{kl} − (−2K + 2(1 + β)H^{2}) h̃^{ik}h̃^{jl}
+4(1 + β)H (g^{ij}h̃^{kl} + g^{kl}h̃^{ij}) − 2(1 + β)g^{ij}g^{kl}) .

We wish to show for α ≤ −2 and for symmetric matrices (η_{ij}), that

Φ^{ij, kl}η_{ij}η_{kl} ≤ \frac{α − 1}{αΦ}Φ^{ij}η_{ij}Φ^{kl}η_{kl}.

Terms involving η_{12}^{2} have the right sign.

Consider α = −2. Then it suffices to prove the inequality

((1 + β)H^{2} − 2K) (\begin{matrix}6\frac{λ_{2}^{2}}{λ_{1}^{2}} + 6β \frac{λ_{2}^{2}}{λ_{1}^{2}} + 4β \frac{λ_{2}}{λ_{1}} & 2β \\ 2β & 6\frac{λ_{1}^{2}}{λ_{2}^{2}} + 6β \frac{λ_{1}^{2}}{λ_{2}^{2}} + 4β \frac{λ_{1}}{λ_{2}} \end{matrix} ) ≥

≥\frac{3}{2} (\begin{matrix}2(1 + β)\frac{λ_{2}^{2}}{λ_{1}} + 2βλ_{2} \\ 2(1 + β)\frac{λ_{1}^{2}}{λ_{2}} + 2βλ_{1} \end{matrix}) ⊗ (\begin{matrix}2(1 + β)\frac{λ_{2}^{2}}{λ_{1}} + 2βλ_{2} \\ 2(1 + β)\frac{λ_{1}^{2}}{λ_{2}} + 2βλ_{1} \end{matrix})

≥\frac{3}{2} (\begin{matrix}(2(1+β) \frac{λ^{2}_{2}}{λ_{1}} +2βλ_{2})^{2} & (2(1+β) \frac{λ^{2}_{1}}{λ_{2}} +2βλ_{1})(2(1+β) \frac{λ^{2}_{2}}{λ_{1}} +2βλ_{2}) \\ (2(1+β) \frac{λ^{2}_{1}}{λ_{2}} +2βλ_{1})(2(1+β) \frac{λ^{2}_{2}}{λ_{1}} +2βλ_{2}) & (2(1+β) \frac{λ^{2}_{1}}{λ_{2}} +2βλ_{1})^{2} \end{matrix})

in order to obtain α-concavity for all α ≤ −2. This inequality is fulfilled, if

\{6λ_{1}λ_{2} + β (4λ_{1}^{2} + 12λ_{1}λ_{2} + 4λ_{2}^{2}) + β^{2} (4λ_{1}^{2} + 8λ_{1}λ_{2} + 4λ_{2}^{2})\} (\begin{matrix}\frac{λ_{2}}{λ_{1}} & −1 \\ −1 & \frac{λ_{1}}{λ_{2}} \end{matrix})

is positive semi-definite. □

We want to derive precise bounds on the principal curvatures. To this end, we use (7.1) and (7.2)

(2 + 4β) ((2 + 4β)3(T − t))^{−2/3} · (1 − c · (T − t)^{1/12})
≤F = |A|^{2} + βH^{2}
≤λ_{1}^{2} + (λ_{1} + |λ_{1} − λ_{2}|)^{2} + β(λ_{1} + (λ_{1} + |λ_{1} − λ_{2}|))^{2}
≤(2 + 4β)λ_{1}^{2} + c · F^{1/2} · F^{1/4} + c · F^{1/2}
≤(2 + 4β)λ_{1}^{2} + c · (T − t)^{−2/3} · ((T − t)^{1/6} + (T − t)^{1/3}) .

We get

λ_{1} ≥ ((2 + 4β)3(T − t))^{−1/3} · (1 + c · (T − t)^{1/12})

and a similar upper bound follows analogously.

We obtain Theorem 7.10.

7.5. tr A^{3}-Flow

Theorem 7.14. A smooth compact strictly convex surface in ℝ^{3}, contracting with normal velocity tr A^{3}, converges to a round point in finite time.

Theorem 7.15. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3}, flowing according to Ẋ = − tr A^{3}ν,

max_{M_{t}} \frac{(3λ_{1}^{2} + 2λ_{1}λ_{2} + 3λ_{2}^{2}) (λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}}

7.5. tr A^{3}-FLOW 33

Proof. Calculations as above yield

w̃ = log (\frac{(3λ_{1}^{2} + 2λ_{1}λ_{2} + 3λ_{2}^{2}) (λ_{1} − λ_{2})^{2}}{2λ_{1}λ_{2}} )

= log (\frac{−H^{4} + 4 (|A|^{2})^{2}}{H^{2} − |A|^{2}} ) ,

\frac{d}{dt}H − F^{ij}H_{; ij} =3 tr A^{4}H − 2 tr A^{3}|A|^{2} + 3 ∑(λ_{i} + λ_{j})h_{ij; k}^{2}
=3 tr A^{4}H − 2 tr A^{3}|A|^{2} + 3 (2λ_{1} + 2λ_{1}a_{1}^{2} + 4λ_{2}a_{1}^{2}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2},

\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =6 tr A^{4}|A|^{2} − 4 (tr A^{3})^{2}

− 6 ∑ λ_{k}^{2}h_{ij; k}^{2} + 6 ∑ λ_{k}(λ_{i} + λ_{j})h_{ij; k}^{2}

=6 tr A^{4}|A|^{2} − 4 (tr A^{3})^{2} − 6 (λ_{1}^{2} + λ_{1}^{2}a_{1}^{2} + 2λ_{2}^{2}a_{1}^{2}) · h_{11; 1}^{2}

+ 6 (2λ_{1}^{2} + 4λ_{1}λ_{2}a_{1}^{2} + 2λ_{2}^{2}a_{1}^{2}) · h_{11; 1}^{2} + (…) · h_{22; 2}^{2},

−F^{ij}H_{; i}H_{; j} = − 3λ_{1}^{2}(1 + a_{1})^{2} · h_{11; 1}^{2} + (…) · h_{22; 2}^{2},
−F^{ij} (|A|^{2})_{;i} (|A|^{2})_{;j} = − 12λ_{1}^{2}(λ_{1} + λ_{2}a_{1})^{2} · h_{11; 1}^{2} + (…) · h_{22; 2}^{2},
h_{22; 1} =\frac{λ_{2}}{λ_{1}} \frac{9λ_{1}^{3} + λ_{1}^{2}λ_{2} + 3λ_{1}λ_{2}^{2} + 3λ_{2}^{3}}{9λ_{2}^{3} + λ_{2}^{2}λ_{1} + 3λ_{2}λ_{1}^{2} + 3λ_{1}^{3}} · h_{11; 1},

\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} = (\frac{−4H^{3}}{−H^{4} + 4 (|A|^{2})^{2}} − \frac{2H}{H^{2} − |A|^{2}} ) · (\frac{d}{dt}H − F^{ij}H_{; ij})

+ (\frac{8|A|^{2}}{−H^{4} + 4 (|A|^{2})^{2}} + \frac{1}{H^{2} − |A|^{2}} ) ·

· (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij})

+ (\frac{−12H^{2}}{−H^{4} + 4 (|A|^{2})^{2}} − \frac{2}{H^{2} − |A|^{2}} ) · (−F^{ij}H_{; i}H_{; j})

+ \frac{8}{−H^{4} + 4 (|A|^{2})^{2}} (−F^{ij} (|A|^{2})_{; i} (|A|^{2})_{; j})

= − \frac{2 (λ_{1}^{4} − 2λ_{1}^{3}λ_{2} + 18λ_{1}^{2}λ_{2}^{2} − 2λ_{1}λ_{2}^{3} + λ_{2}^{4}) λ_{1}λ_{2}}{3λ_{1}^{2} + 2λ_{1}λ_{2} + 3λ_{2}^{2}}

− \frac{6λ_{2}}{(3λ_{1}^{2} + 2λ_{1}λ_{2} + 3λ_{2}^{2}) (λ_{1} − λ_{2})^{2} λ_{1}^{3}} ·

· \frac{1}{(3λ_{1}^{3} + 3λ_{1}^{2}λ_{2} + λ_{1}λ_{2}^{2} + 9λ_{2}^{3})^{2}} ·

· (63λ_{1}^{12} + 381λ_{1}^{11}λ_{2} − 1389λ_{1}^{10}λ_{2}^{2} + 2883λ_{1}^{9}λ_{2}^{3} + 36λ_{1}^{8}λ_{2}^{4}

+1218λ_{1}^{7}λ_{2}^{5} + 2294λ_{1}^{6}λ_{2}^{6} + 582λ_{1}^{5}λ_{2}^{7} + 855λ_{1}^{4}λ_{2}^{8}

+945λ_{1}^{3}λ_{2}^{9} + 135λ_{1}^{2}λ_{2}^{10} + 135λ_{1}λ_{2}^{11} + 54λ_{2}^{12}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2}
≤0.
□

Lemma 7.16. For a family of compact smooth strictly convex surfaces M_{t} ⊂ ℝ^{3}, 0 ≤ t < T, flowing according to \frac{d}{dt}X = −F ν with F = tr A^{α}, α ≥ 2, a positive lower bound on the principal curvatures, λ_{1}, λ_{2} ≥ ε > 0, is preserved during the evolution.

Proof. We proceed similarly as in Lemma 7.3. Once again, the term involving second derivatives of F is non-negative. As before, we obtain

(\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl}) ξ^{i}ξ^{j} ≥α tr A^{α+1}ε − α tr A^{α}ε^{2} − tr A^{α}ε^{2} + 2ε tr A^{α}ε
=ε (α (λ^{α+1} − ελ^{α}) + ε^{α+1} + ελ^{α}) > 0.
□

Similar calculations as in Lemma 7.4, using F + F^{ij}h_{ij} − αF^{ij}h_{i}^{k}h_{kj} ≥ 0 for F = tr A^{β}, β ≥ 2, yield for some c_{β} > 0

\frac{1}{c_{β}} (tr A^{β})^{1/β} ≤ \frac{1}{c_{β}} H ≤ \frac{tr A^{β+1}}{tr A^{β}} ≤ \frac{β + 1}{β} \frac{1}{α}

and an estimate as in Lemma 7.4 follows.

Lemma 7.17. For α > 0, the dual function to F = tr A^{α} = λ_{1}^{α} + λ_{2}^{α} is −αconcave.

Proof. We set Φ = − tr A^{α} · K^{−α} and have to show that

Φ^{ij, kl}η_{ij}η_{kl} ≤ \frac{−α − 1}{−αΦ}Φ^{ij}η_{ij}Φ^{kl}η_{kl}

for symmetric matrices (η_{ij}). Direct computations yield that this inequality is equivalent to

− α(α − 1)K^{−α} (\begin{matrix}η_{11} \\ η_{22} \end{matrix})^{tr} (\begin{matrix}λ_{1}^{α−2} & 0 \\ 0 & λ_{2}^{α−2} \end{matrix}) (\begin{matrix}η_{11} \\ η_{22} \end{matrix}) − 2αK^{−α} ∑_{r=0}^{α−2} λ_{1}^{r}λ_{2}^{α−2−r}η_{12}^{2}

+ α^{2}K^{−α} (\begin{matrix}η_{11} \\ η_{22} \end{matrix})^{tr} (\begin{matrix}2λ_{1}^{α−2} & \frac{1}{λ_{1}λ_{2}} tr A^{α} \\ \frac{1}{λ_{1}λ_{2}} tr A^{α} & 2λ_{2}^{α−2} \end{matrix}) (\begin{matrix}η_{11} \\ η_{22} \end{matrix})

− α^{2} tr A^{α}K^{−α} (\begin{matrix}η_{11} \\ η_{22} \end{matrix})^{tr} (\begin{matrix}\frac{1}{λ_{1}^{2}} & \frac{1}{λ_{1}λ_{2}} \\ \frac{1}{λ_{1}λ_{2}} & \frac{1}{λ_{2}^{2}} \end{matrix} ) (\begin{matrix}η_{11} \\ η_{22} \end{matrix})

− α tr A^{α}K^{−α} (\begin{matrix}η_{11} \\ η_{22} \end{matrix})^{tr} (\begin{matrix}\frac{1}{λ_{1}^{2}} & 0 \\ 0 & \frac{1}{λ_{2}^{2}} \end{matrix} ) (\begin{matrix}η_{11} \\ η_{22} \end{matrix}) − 2α tr A^{α}K^{−α} \frac{1}{λ_{1}λ_{2}} η_{12}^{2}

≤ − \frac{α(α + 1)K^{−α}}{tr A^{α}} (\begin{matrix}η_{11} \\ η_{22} \end{matrix})^{tr} (\begin{matrix}\frac{λ_{2}^{2α}}{λ_{1}^{2}} & λ_{1}^{α−1}λ_{2}^{α−1} \\ λ_{1}^{α−1}λ_{2}^{α−1} & \frac{λ_{1}^{2α}}{λ_{2}^{2}} \end{matrix}) (\begin{matrix}η_{11} \\ η_{22} \end{matrix}) .

Further computations show that this is fulfilled, if

0 ≤ (\begin{matrix}λ_{1}^{α−2}λ_{2}^{α} & −λ_{1}^{α−1}λ_{2}^{α−1} \\ −λ_{1}^{α−1}λ_{2}^{α−1} & λ_{1}^{α}λ_{2}^{α−2} \end{matrix}) .

7.6. tr A^{α}-FLOW 35

□

Then we proceed as before. Similar calculations as for F = |A|^{2} + βH^{2} give for α = 3

λ_{1}, λ_{2} ≤ (2(1 + α) · (T − t))^{− \frac{1}{1+α}} · (1 + c · (T − t)^{\frac{1}{4} \frac{α−1}{α+1}} )

and a corresponding lower bound holds. This estimate holds also for F = tr A^{α}, α = 2, 3, 4, 5, 6.

This finishes the proof of Theorem 7.14.

7.6. tr A^{α}-Flow

Theorem 7.18. A smooth compact strictly convex surface in ℝ^{3}, contracting with normal velocity tr A^{4}, tr A^{5}, or tr A^{6}, converges to a round point in finite time.

Theorem 7.19. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3}, flowing according to Ẋ = − tr A^{α+2}ν, α = 2, 3, 4,

max_{M_{t}} \frac{(λ_{1}^{α} + λ_{2}^{α}) (λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}}

is non-increasing in time.

Proof. One might conjecture, that this quantity is also monotone for other values of α. For α = 0, corresponding to F = |A|^{2}, we have already checked that in Theorem 4.3. Further computations for α = 1, 5, 6, 7 suggest, however, that this quantity is not monotone for these values of α.

We obtain

\frac{d}{dt}H − F^{ij}H_{; ij} =(α + 2) tr A^{α+3}H − (α + 1) tr A^{α+2}|A|^{2}

+ (α + 2) ∑_{r=0}^{α} ∑_{i, j, k=1}^{2} λ_{i}^{r}λ_{j}^{α−r}h_{ij; k}^{2},

\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =2(α + 2) tr A^{α+3}|A|^{2} − 2(α + 1) tr A^{α+2} tr A^{3}

− 2(α + 2) ∑_{i, j, k=1}^{2} λ_{k}^{α+1}h_{ij; k}^{2}

+ 2(α + 2) ∑_{r=0}^{α} ∑_{i, j, k=1}^{2} λ_{i}^{r}λ_{j}^{α−r}h_{ij; k}^{2}λ_{k},

\frac{d}{dt}tr A^{α} − F^{ij}(tr A^{α})_{; ij} =α(α + 2) tr A^{α+3} tr A^{α} − α(α + 1) tr A^{α+2} tr A^{α+1}

− α(α + 2) ∑_{r=0}^{α−2} λ_{k}^{α+1}λ_{i}^{α−2−r}λ_{j}^{r}h_{ij; k}^{2}

+ α(α + 2) ∑_{r=0}^{α} λ_{i}^{r}λ_{j}^{α−r}λ_{k}^{α−1}h_{ij; k}^{2},

w̃ = log (\frac{(λ_{1}^{α} + λ_{2}^{α}) (λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} )
≡ log A + log B + log C − log D,

\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} =\frac{1}{A} (\frac{d}{dt}A − F^{ij}A_{; ij}) + \frac{1}{B} (\frac{d}{dt}B − F^{ij}B_{; ij})

+ \frac{1}{C} (\frac{d}{dt}C − F^{ij}C_{; ij}) − \frac{1}{D} (\frac{d}{dt}D − F^{ij}D_{; ij})

+ \frac{2}{B^{2}} F^{ij}B_{; i}B_{; j} + \frac{2}{C^{2}} F^{ij}C_{; i}C_{; j} + \frac{1}{BC} F^{ij}(B_{; i}C_{; j} + B_{; j}C_{; i})

− \frac{1}{BD} F^{ij}(B_{; i}D_{; j} + B_{; j}D_{; i}) − \frac{1}{CD} F^{ij}(C_{; i}D_{; j} + C_{; j}D_{; i})

=\frac{1}{tr A^{α}} · (\frac{d}{dt}tr A^{α} − F^{ij}(tr A^{α})_{; ij})

+ (\frac{1}{H} − \frac{2H}{2|A|^{2} − H^{2}} − \frac{2H}{H^{2} − |A|^{2}} ) · (\frac{d}{dt}H − F^{ij}H_{; ij})

+ (\frac{2}{2|A|^{2} − H^{2}} + \frac{1}{H^{2} − |A|^{2}} ) · (\frac{d}{dt}|A|^{2} − F^{ij}|A|^{2}_{; ij})

+ (\frac{2}{2|A|^{2} − H^{2}} + \frac{2}{H^{2} − |A|^{2}} + \frac{2}{H^{2}} + \frac{8H^{2}}{(2|A|^{2} − H^{2})^{2}}

−\frac{4H}{H (2|A|^{2} − H^{2})} − \frac{4H}{H (H^{2} − |A|^{2})}

+\frac{8H^{2}}{(2|A|^{2} − H^{2}) (H^{2} − |A|^{2})}) · F^{ij}H_{; i}H_{; j}

+ (\frac{8}{(2|A|^{2} − H^{2})^{2}} + \frac{4}{(2|A|^{2} − H^{2}) (H^{2} − |A|^{2})}) ·

· F^{ij} (|A|^{2})_{; i} (|A|^{2})_{; j}

+ (\frac{1}{H (H^{2} − |A|^{2})} − \frac{8H}{(2|A|^{2} − H^{2})^{2}} + \frac{2}{H (2|A|^{2} − H^{2})}

−\frac{6H}{(2|A|^{2} − H^{2}) (H^{2} − |A|^{2})}) ·

· F^{ij} (H_{; i} (|A|^{2})_{; j} + H_{; j} (|A|^{2})_{; i}) .

Plugging this into a computer algebra program yields for α = 2, corresponding to F = tr A^{4},

h_{22; 1} =\frac{λ_{2}}{λ_{1}} \frac{4λ_{1}^{4} + λ_{1}^{3}λ_{2} + λ_{1}^{2}λ_{2}^{2} + λ_{1}λ_{2}^{3} + λ_{2}^{4}}{4λ_{2}^{4} + λ_{2}^{3}λ_{1} + λ_{2}^{2}λ_{1}^{2} + λ_{2}λ_{1}^{3} + λ_{1}^{4}} h_{11; 1},
\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} =\frac{−24λ_{1}^{4}λ_{2}^{4}}{(λ_{1} + λ_{2}) (λ_{1}^{2} + λ_{2}^{2})}

+ \frac{−4λ_{2}}{(λ_{1} − λ_{2})^{2} (λ_{1} + λ_{2}) λ_{1}^{3} (λ_{1}^{2} + λ_{2}^{2})} ·

· \frac{1}{(λ_{1}^{4} + λ_{1}^{3}λ_{2} + λ_{1}^{2}λ_{2}^{2} + λ_{1}λ_{2}^{3} + 4λ_{2}^{4})^{2}} ·

7.6. tr A^{α}-FLOW 37

· (11λ_{1}^{16} + 24λ_{1}^{15}λ_{2} + 39λ_{1}^{14}λ_{2}^{2} − 328λ_{1}^{13}λ_{2}^{3} + 482λ_{1}^{12}λ_{2}^{4} +192λ_{1}^{11}λ_{2}^{5} + 215λ_{1}^{10}λ_{2}^{6} + 236λ_{1}^{9}λ_{2}^{7} + 432λ_{1}^{8}λ_{2}^{8} + 200λ_{1}^{7}λ_{2}^{9} +173λ_{1}^{6}λ_{2}^{10} + 144λ_{1}^{5}λ_{2}^{11} + 158λ_{1}^{4}λ_{2}^{12} + 32λ_{1}^{3}λ_{2}^{13} + 21λ_{1}^{2}λ_{2}^{14} +12λ_{1}λ_{2}^{15} + 5λ_{2}^{16}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2},

for α = 3 (F = tr A^{5})

h_{22; 1} =\frac{λ_{2}}{λ_{1}} \frac{5λ_{1}^{4} − 4λ_{1}^{3}λ_{2} + 2λ_{1}^{2}λ_{2}^{2} + λ_{2}^{4}}{5λ_{2}^{4} − 4λ_{2}^{3}λ_{1} + 2λ_{2}^{2}λ_{1}^{2} + λ_{1}^{4}} h_{11; 1},
\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} =\frac{−4 (2λ_{1}^{4} − 7λ_{1}^{3}λ_{2} + 12λ_{1}^{2}λ_{2}^{2} − 7λ_{1}λ_{2}^{3} + 2λ_{2}^{4}) λ_{1}^{2}λ_{2}^{2}}{(λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2})}

+ \frac{−10λ_{2}}{λ_{1}^{3} (λ_{1} − λ_{2})^{2} (λ_{1}^{4} + 2λ_{1}^{2}λ_{2}^{2} − 4λ_{1}λ_{2}^{3} + 5λ_{2}^{4})^{2} (λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2})}·

· (7λ_{1}^{16} − 65λ_{1}^{15}λ_{2} + 397λ_{1}^{14}λ_{2}^{2} − 1295λ_{1}^{13}λ_{2}^{3} + 2464λ_{1}^{12}λ_{2}^{4} −2981λ_{1}^{11}λ_{2}^{5} + 2645λ_{1}^{10}λ_{2}^{6} − 2007λ_{1}^{9}λ_{2}^{7} + 1510λ_{1}^{8}λ_{2}^{8} −1011λ_{1}^{7}λ_{2}^{9} + 583λ_{1}^{6}λ_{2}^{10} − 309λ_{1}^{5}λ_{2}^{11} + 176λ_{1}^{4}λ_{2}^{12} − 71λ_{1}^{3}λ_{2}^{13} +23λ_{1}^{2}λ_{2}^{14} − 5λ_{1}λ_{2}^{15} + 3λ_{2}^{16}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2},

and for α = 4 (F = tr A^{6})

h_{22; 1} =\frac{λ_{2}}{λ_{1}} \frac{6λ_{1}^{6} + λ_{1}^{5}λ_{2} − 3λ_{1}^{4}λ_{2}^{2} + 2λ_{1}^{2}λ_{2}^{4} + λ_{1}λ_{2}^{5} + λ_{2}^{6}}{6λ_{2}^{6} + λ_{2}^{5}λ_{1} − 3λ_{2}^{4}λ_{1}^{2} + 2λ_{2}^{2}λ_{1}^{4} + λ_{2}λ_{1}^{5} + λ_{1}^{6}} h_{11; 1},
\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} =\frac{−10 (λ_{1}^{8} − 2λ_{1}^{6}λ_{2}^{2} + 6λ_{1}^{4}λ_{2}^{4} − 2λ_{1}^{2}λ_{2}^{6} + λ_{2}^{8}) λ_{1}^{2}λ_{2}^{2}}{(λ_{1} + λ_{2}) (λ_{1}^{4} + λ_{2}^{4})}

+ \frac{−6λ_{2}}{λ_{1}^{3} (λ_{1} − λ_{2})^{2} (λ_{1} + λ_{2}) (λ_{1}^{4} + λ_{2}^{4})} ·

· \frac{1}{(λ_{1}^{6} + λ_{1}^{5}λ_{2} + 2λ_{1}^{4}λ_{2}^{2} − 3λ_{1}^{2}λ_{2}^{4} + λ_{1}λ_{2}^{5} + 6λ_{2}^{6})^{2}} ·

· (17λ_{1}^{24} − 124λ_{1}^{23}λ_{2} + 218λ_{1}^{22}λ_{2}^{2} + 646λ_{1}^{21}λ_{2}^{3} − 642λ_{1}^{20}λ_{2}^{4} −2586λ_{1}^{19}λ_{2}^{5} + 2536λ_{1}^{18}λ_{2}^{6} + 3576λ_{1}^{17}λ_{2}^{7} − 2411λ_{1}^{16}λ_{2}^{8} −2928λ_{1}^{15}λ_{2}^{9} + 1524λ_{1}^{14}λ_{2}^{10} + 1724λ_{1}^{13}λ_{2}^{11} + 548λ_{1}^{12}λ_{2}^{12} −276λ_{1}^{11}λ_{2}^{13} − 696λ_{1}^{10}λ_{2}^{14} − 8λ_{1}^{9}λ_{2}^{15} + 499λ_{1}^{8}λ_{2}^{16} + 236λ_{1}^{7}λ_{2}^{17} +146λ_{1}^{6}λ_{2}^{18} − 66λ_{1}^{5}λ_{2}^{19} − 2λ_{1}^{4}λ_{2}^{20} + 46λ_{1}^{3}λ_{2}^{21} + 48λ_{1}^{2}λ_{2}^{22} +16λ_{1}λ_{2}^{23} + 7λ_{2}^{24}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2}.

In each case, Sturm’s algorithm yields, that the right-hand side is non-positive. □ Theorem 7.18 follows.

7.7. H|A|^{2}-Flow

Theorem 7.20. A smooth compact strictly convex surface in ℝ^{3}, contracting with normal velocity H|A|^{2}, converges to a round point in finite time.

Theorem 7.21. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3}, flowing according to \frac{d}{dt}X = −H|A|^{2}ν,

max_{M_{t}} \frac{(λ_{1} + λ_{2})^{2}(λ_{1} − λ_{2})^{2}}{2λ_{1}λ_{2}}

is non-increasing in time.

Proof. We proceed as above.

w = log (\frac{(λ_{1} + λ_{2})^{2}(λ_{1} − λ_{2})^{2}}{2λ_{1}λ_{2}} )
= log (\frac{−H^{4} + 2|A|^{2}H^{2}}{H^{2} − |A|^{2}} ) ,
h_{22; 1} =\frac{3λ_{1}^{2} + λ_{2}^{2}}{3λ_{2}^{2} + λ_{1}^{2}} \frac{λ_{2}}{λ_{1}} h_{11; 1},
\frac{d}{dt}H − F^{ij}H_{; ij} =2H^{2} tr A^{3} − H (|A|^{2})^{2}

+ 2 ∑(λ_{i} + λ_{j})h_{ii; k}h_{jj; k} + 2H ∑ h_{ij; k}^{2}

=2H^{2} tr A^{3} − H (|A|^{2})^{2}

+ 4(λ_{1} + λ_{2}a_{1})(1 + a_{1}) · h_{11; 1}^{2} + 2H(1 + 3a_{1}^{2}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2},

\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =2 (|A|^{2})^{3} − 2|A|^{2} ∑ h_{ij; k}^{2} + 4 ∑(λ_{i} + λ_{j})λ_{k}h_{ii; k}h_{jj; k}
=2 (|A|^{2})^{3} − 2|A|^{2}(1 + 3a_{1}^{2}) · h_{11; 1}^{2}

+ 8λ_{1}(λ_{1} + λ_{2}a_{1})(1 + a_{1}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2},

−F^{ij}H_{; i}H_{; j} = − ∑ (|A|^{2} + 2Hλ_{k}) h_{ii; k}h_{jj; k}
= − |A|^{2}(1 + a_{1})^{2} · h_{11; 1}^{2} − 2Hλ_{1}(1 + a_{1})^{2} · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2},

−F^{ij} ((|A|^{2})_{; i} H_{; j} + (|A|^{2})_{; j} H_{; i}) = − 2 ∑ (|A|^{2} + 2Hλ_{k}) (λ_{i} + λ_{j})h_{ii; k}h_{jj; k}
= − 4|A|^{2}(λ_{1} + λ_{2}a_{1})(1 + a_{1}) · h_{11; 1}^{2}

− 8H(λ_{1} + λ_{2}a_{1})(1 + a_{1})λ_{1}h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2},

7.7. H|A|^{2}-FLOW 39

\frac{d}{dt}w − F^{ij}w_{; ij} = (\frac{−4H^{3} + 4|A|^{2}H}{−H^{4} + 2|A|^{2}H^{2}} − \frac{2H}{H^{2} − |A|^{2}} ) (\frac{d}{dt}H − F^{ij}H_{; ij})

+ (\frac{2H^{2}}{−H^{4} + 2|A|^{2}H^{2}} + \frac{1}{H^{2} − |A|^{2}} ) (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij})

+ (\frac{−12H^{2} + 4|A|^{2}}{−H^{4} + 2|A|^{2}H^{2}} − \frac{2}{H^{2} − |A|^{2}} ) (−F^{ij}H_{; i}H_{; j})

+ \frac{4H}{−H^{4} + 2|A|^{2}H^{2}} (−F^{ij} ((|A|^{2})_{; i} H_{; j} + (|A|^{2})_{; j} H_{; i}))

= − 8λ_{1}^{2}λ_{2}^{2}

− \frac{4λ_{2}}{(λ_{1}^{2} + 3λ_{2}^{2})^{2} (λ_{1} − λ_{2})^{2}λ_{1}^{3}} ·

· (6λ_{1}^{8} − 39λ_{1}^{7}λ_{2} + 91λ_{1}^{6}λ_{2}^{2} + λ_{1}^{5}λ_{2}^{3} + 91λ_{1}^{4}λ_{2}^{4} + 3λ_{1}^{3}λ_{2}^{5}
+33λ_{1}^{2}λ_{2}^{6} + 3λ_{1}λ_{2}^{7} + 3λ_{2}^{8}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2}

≤0.
□

Lemma 7.22. For a family of compact smooth strictly convex surfaces M_{t} ⊂ ℝ^{3}, 0 ≤ t < T, flowing according to \frac{d}{dt}X = −F ν with F = H|A|^{2}, a positive lower bound on the principal curvatures, λ_{1}, λ_{2} ≥ ε > 0, is preserved during the evolution.

Proof. We proceed similarly as in Lemma 7.3 and compute

F^{kl, rs}h_{kl; 1}h_{rs; 1} =(6λ_{1} + 2λ_{2}) · h_{11; 1}^{2} + 4(λ_{1} + λ_{2}) · h_{11; 1}h_{22; 1}
+ 4(λ_{1} + λ_{2}) · h_{11; 2}^{2} + (2λ_{1} + 6λ_{2}) · h_{22; 1}^{2} ≥ 0,
(\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl}) ξ^{i}ξ^{j} ≥ ((|A|^{2})^{2} + 2H tr A^{3}) · h_{ij}ξ^{i}ξ^{j}
− 4H|A|^{2}h_{i}^{k}h_{kj}ξ^{i}ξ^{j} + 2εH|A|^{2}h_{ij}ξ^{i}ξ^{j}
=ε^{5} + 3ελ^{4} > 0.
□

Similar calculations as in Lemma 7.4 using F + F^{ij}h_{ij} − αF^{ij}h_{i}^{k}h_{kj} ≥ 0 for F = H|A|^{2} give

α ((|A|^{2})^{2} + 2H tr A^{3}) ≤4H|A|^{2},
\frac{1}{c} (H|A|^{2})^{1/3} ≤ \frac{1}{2}H ≤ \frac{|A|^{2}}{H} ≤\frac{4}{α}

and an estimate as in Lemma 7.4 follows.

Lemma 7.23. The dual function to F = H|A|^{2} is −3-concave.

Proof. We set Φ = −H|A|^{2}K^{−3} and want to prove that

Φ^{ij, kl}η_{ij}η_{kl} ≤ \frac{4}{3Φ}Φ^{ij}η_{ij}Φ^{kl}η_{kl}.

We compute

Φ^{ij} = − |A|^{2}K^{−3}g^{ij} − 2HK^{−3}h^{ij} + 3H|A|^{2}K^{−3}h̃^{ij},

40 7. OTHER NORMAL VELOCITIES

Φ^{ij, kl} = − 2K^{−3} (g^{ij}h^{kl} + h^{ij}g^{kl}) + 3|A|^{2}K^{−3} (g^{ij}h̃^{kl} + h̃^{ij}g^{kl})
+ 6HK^{−3} (h^{ij}h̃^{kl} + h̃^{ij}h^{kl})
− 2HK^{−3}g^{ik}g^{jl} − 9H|A|^{2}K^{−3}h̃^{ij}h̃^{kl} − 3H|A|^{2}K^{−3}h̃^{ik}h̃^{jl}.

We have to check that

− 2H|A|^{2} (\begin{matrix}2λ_{1} & λ_{1} + λ_{2} \\ λ_{1} + λ_{2} & 2λ_{2} \end{matrix}) + 3H (|A|^{2})^{2} (\begin{matrix}\frac{2}{λ_{1}} & \frac{1}{λ_{1}} + \frac{1}{λ_{2}} \\ \frac{1}{λ_{1}} + \frac{1}{λ_{2}} & \frac{2}{λ_{2}} \end{matrix} )

+ 6H^{2}|A|^{2} (\begin{matrix}2 & \frac{λ_{1}}{λ_{2}} + \frac{λ_{2}}{λ_{1}} \\ \frac{λ_{1}}{λ_{2}} + \frac{λ_{2}}{λ_{1}} & 2 \end{matrix} ) − 2H^{2}|A|^{2} (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix})

− 9H^{2} (|A|^{2})^{2} (\begin{matrix}\frac{1}{λ_{1}^{2}} & \frac{1}{λ_{1}λ_{2}} \\ \frac{1}{λ_{1}λ_{2}} & \frac{1}{λ_{2}^{2}} \end{matrix} ) − 3H^{2} (|A|^{2})^{2} (\begin{matrix}\frac{1}{λ_{1}^{2}} & 0 \\ 0 & \frac{1}{λ_{2}^{2}} \end{matrix} )

≤ − \frac{4}{3} (\begin{matrix}(2λ_{2}^{2}+λ_{1}λ_{2}+ \frac{3λ^{3}_{2}}{λ_{1}} )^{2} & (2λ_{2}^{2}+λ_{1}λ_{2}+ \frac{3λ^{3}_{2}}{λ_{1}} )·(2λ_{1}^{2}+λ_{1}λ_{2}+ \frac{3λ^{3}_{1}}{λ_{2}} ) \\ (2λ_{2}^{2}+λ_{1}λ_{2}+ \frac{3λ^{3}_{2}}{λ_{1}} )·(2λ_{1}^{2}+λ_{1}λ_{2}+ \frac{3λ^{3}_{1}}{λ_{2}} ) & (2λ_{1}^{2}+λ_{1}λ_{2}+ \frac{3λ^{3}_{1}}{λ_{2}} )^{2} \end{matrix}) .

This is equivalent to

0 ≤ (2λ_{1}^{4} + \frac{20}{3}λ_{1}^{3}λ_{2} + \frac{44}{3}λ_{1}^{2}λ_{2}^{2} + \frac{20}{3}λ_{1}λ_{2}^{3} + 2λ_{2}^{4}) (\begin{matrix}\frac{λ_{2}}{λ_{1}} & −1 \\ −1 & \frac{λ_{1}}{λ_{2}} \end{matrix}) .

□

Calculations as before show that (16(T −t))^{−1/4}·(1 − c · (T − t)^{1/8}) ≤ λ_{1}, λ_{2} ≤ (16(T −t))^{−1/4}·(1 + c · (T − t)^{1/8}) . This finishes the proof of Theorem 7.20.

7.8. |A|^{4}-Flow Theorem 7.24. A smooth compact strictly convex surface in ℝ^{3}, contracting

with normal velocity |A|^{4}, converges to a round point in finite time.

Theorem 7.25. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3}, flowing according to \frac{d}{dt}X = − |A|^{4} ν = − (|A|^{2})^{2} ν,

max_{M_{t}} \frac{(λ_{1}^{4} + 2λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} + 2λ_{1}λ_{2}^{3} + λ_{2}^{4}) (λ_{1} − λ_{2})^{2}}{(λ_{1} + λ_{2})λ_{1}λ_{2}}

is non-increasing in time.

Proof. We calculate

w̃ = log (\frac{(λ_{1}^{4} + 2λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} + 2λ_{1}λ_{2}^{3} + λ_{2}^{4}) (λ_{1} − λ_{2})^{2}}{(λ_{1} + λ_{2})λ_{1}λ_{2}} )
= log (\frac{−H^{6} + 2|A|^{2}H^{4} − (|A|^{2})^{2} H^{2} + 2 (|A|^{2})^{3}}{H^{3} − |A|^{2}H} ) ,
\frac{d}{dt}H − F^{ij}H_{; ij} =4H|A|^{2} tr A^{3} − 3 (|A|^{2})^{3}
+ 8 ∑ λ_{i}λ_{j}h_{ii; k}h_{jj; k} + 4|A|^{2} ∑ h_{ij; k}^{2},

7.8. |A|^{4}-FLOW 41

\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} =2 (|A|^{2})^{2} tr A^{3} + 16 ∑ λ_{i}λ_{j}λ_{k}h_{ii; k}h_{jj; k},

\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} = (\frac{−6H^{5} + 8|A|^{2}H^{3} − 2 (|A|^{2})^{2} H}{−H^{6} + 2|A|^{2}H^{4} − (|A|^{2})^{2} H^{2} + 2 (|A|^{2})^{3}} − \frac{3H^{2} − |A|^{2}}{H^{3} − |A|^{2}H} ) ·

· (\frac{d}{dt}H − F^{ij}H_{; ij})

+ (\frac{2H^{4} − 2|A|^{2}H^{2} + 6 (|A|^{2})^{2}}{−H^{6} + 2|A|^{2}H^{4} − (|A|^{2})^{2} H^{2} + 2 (|A|^{2})^{3}} − \frac{−H}{H^{3} − |A|^{2}H} ) ·

· (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij})

+ (\frac{−30H^{4} + 24|A|^{2}H^{2} − 2 (|A|^{2})^{2}}{−H^{6} + 2|A|^{2}H^{4} − (|A|^{2})^{2} H^{2} + 2 (|A|^{2})^{3}} − \frac{6H}{H^{3} − |A|^{2}H} ) ·

(−F^{ij}H_{; i}H_{; j})

+ \frac{−2H^{2} + 12|A|^{2}}{−H^{6} + 2|A|^{2}H^{4} − (|A|^{2})^{2} H^{2} + 2 (|A|^{2})^{3}} ·

· (−F^{ij} (|A|^{2})_{; i} (|A|^{2})_{; j})

+ (\frac{8H^{3} − 4|A|^{2}H}{−H^{6} + 2|A|^{2}H^{4} − (|A|^{2})^{2} H^{2} + 2 (|A|^{2})^{3}} − \frac{−1}{H^{3} − |A|^{2}H} ) ·

· (−F^{ij} (H_{; i} (|A|^{2})_{; j} + H_{; j} (|A|^{2})_{; i})) .

We use a computer algebra program and obtain

h_{22; 1} =\frac{λ_{2}}{λ_{1}} \frac{4λ_{1}^{6} + 9λ_{1}^{5}λ_{2} + 11λ_{1}^{4}λ_{2}^{2} + 10λ_{1}^{3}λ_{2}^{3} + 2λ_{1}^{2}λ_{2}^{4} + 3λ_{1}λ_{2}^{5} + λ_{2}^{6}}{4λ_{2}^{6} + 9λ_{1}λ_{2}^{5} + 11λ_{1}^{2}λ_{2}^{4} + 10λ_{1}^{3}λ_{2}^{3} + 2λ_{1}^{4}λ_{2}^{2} + 3λ_{1}^{5}λ_{2} + λ_{1}^{6}} · h_{11; 1},
\frac{d}{dt}w̃ − F^{ij}w̃_{; ij} =\frac{−12 (3λ_{1}^{2} + 4λ_{1}λ_{2} + 3λ_{2}^{2}) (λ_{1}^{2} + λ_{2}^{2}) λ_{1}^{3}λ_{2}^{3}}{(λ_{1} + λ_{2}) (λ_{1}^{4} + 2λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} + 2λ_{1}λ_{2}^{3} + λ_{2}^{4})}

+ \frac{−4λ_{2}}{(λ_{1}^{4} + 2λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} + 2λ_{1}λ_{2}^{3} + λ_{2}^{4}) (λ_{1} − λ_{2})^{2} (λ_{1} + λ_{2})λ_{1}^{3}} ·

· \frac{1}{(λ_{1}^{6} + 3λ_{1}^{5}λ_{2} + 2λ_{1}^{4}λ_{2}^{2} + 10λ_{1}^{3}λ_{2}^{3} + 11λ_{1}^{2}λ_{2}^{4} + 9λ_{1}λ_{2}^{5} + 4λ_{2}^{6})^{2}} ·

· (11λ_{1}^{22} + 90λ_{1}^{21}λ_{2} − 113λ_{1}^{20}λ_{2}^{2} − 840λ_{1}^{19}λ_{2}^{3} − 1507λ_{1}^{18}λ_{2}^{4}

+66λ_{1}^{17}λ_{2}^{5} + 7465λ_{1}^{16}λ_{2}^{6} + 23136λ_{1}^{15}λ_{2}^{7} + 45494λ_{1}^{14}λ_{2}^{8}

+70100λ_{1}^{13}λ_{2}^{9} + 84982λ_{1}^{12}λ_{2}^{10} + 85120λ_{1}^{11}λ_{2}^{11} + 70882λ_{1}^{10}λ_{2}^{12}

+52148λ_{1}^{9}λ_{2}^{13} + 33938λ_{1}^{8}λ_{2}^{14} + 20928λ_{1}^{7}λ_{2}^{15} + 11263λ_{1}^{6}λ_{2}^{16}

+5490λ_{1}^{5}λ_{2}^{17} + 2363λ_{1}^{4}λ_{2}^{18} + 744λ_{1}^{3}λ_{2}^{19} + 193λ_{1}^{2}λ_{2}^{20}

+42λ_{1}λ_{2}^{21} + 5λ_{2}^{22}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}.

We apply Sturm’s algorithm and obtain that the right-hand side is non-positive. □

Lemma 7.26. For a family of compact smooth strictly convex surfaces M_{t} ⊂ ℝ^{3}, 0 ≤ t < T, flowing according to \frac{d}{dt}X = −F ν with F = |A|^{4}, a positive lower bound on the principal curvatures, λ_{1}, λ_{2} ≥ ε > 0, is preserved during the evolution.

Proof. The term involving second derivatives of F is non-negative, so we have

(\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl}) ξ^{i}ξ^{j} ≥|A|^{2}ε (4 tr A^{3} − 3ε|A|^{2})
=|A|^{2}ε (λ^{3} + 3λ^{2}(λ − ε) + ε^{3}) > 0.
□

Similar calculations as in Lemma 7.4 using F + F^{ij}h_{ij} − αF^{ij}h_{i}^{k}h_{kj} ≥ 0 for F = |A|^{4} give

5|A|^{2} − 4α tr A^{3} ≥0,

\frac{1}{c} (|A|^{4})^{1/4} ≤ \frac{tr A^{3}}{|A|^{2}} ≤ \frac{5}{4α}

and an estimate as in Lemma 7.4 follows.

The following lemma implies that the dual function to F = |A|^{4} is −4-concave.

Lemma 7.27. If the dual function to F is α-concave for some α < 0, then the dual function to F^{β} is α · β-concave for β > 0.

Proof. We use similar notation as before. Assume that

−G^{ij, kl}η_{ij}η_{kl} ≤ \frac{α − 1}{α(−G)}G^{ij}η_{ij}G^{kl}η_{kl}.

We have to show for Φ = −G^{β}

Φ^{ij, kl}η_{ij}η_{kl} ≤ \frac{αβ − 1}{αβΦ}Φ^{ij}η_{ij}Φ^{kl}η_{kl}.

Direct calculations yield

Φ^{ij} = − βG^{β−1}G^{ij},
Φ^{ij, kl} = − βG^{β−1}G^{ij, kl} − β(β − 1)G^{β−2}G^{ij}G^{kl},
Φ^{ij, kl}η_{ij}η_{kl} = − βG^{β−1}G^{ij, kl}η_{ij}η_{kl} − β(β − 1)G^{β−2} (G^{ij}η_{ij})^{2}
≤ − β \frac{α − 1}{α}G^{β−2} (G^{ij}η_{ij})^{2} − β(β − 1)G^{β−2} (G^{ij}η_{ij})^{2}
= − \frac{αβ − 1}{αβG^{β}}β^{2}G^{2β−2} (G^{ij}η_{ij})^{2}
=\frac{αβ − 1}{αβΦ} (Φ^{ij}η_{ij})^{2} .
□

Calculations as before show that

λ_{1}, λ_{2} ≤ (20(T − t))^{−4/5} · (1 + c · (T − t)^{3/20}) .

A corresponding lower estimate is also true. Theorem 7.24 follows.

Part 3

Surfaces Expanding by the Inverse Gauß Curvature Flow

CHAPTER 8 Introduction

We consider a family of compact strictly convex surfaces M_{t} in ℝ^{3} that expand by the inverse Gauß curvature flow

(8.1)  \frac{d}{dt}X = \frac{1}{K} ν.

This is a parabolic flow equation. We obtain a solution on a maximal time interval [0, T), 0 < T < ∞. For t ↑ T, the surfaces converge to infinity. After appropriate rescaling, they converge to a round sphere. We say that the surfaces M_{t} converge to “round spheres at infinity”. The key step in the proof, Theorem 9.1, is to show that

(8.2)  max_{M_{t}} (\frac{(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}} )

is non-increasing in time.

Our main theorem in this part is

Theorem 8.1. For any smooth strictly convex closed surface M in ℝ^{3}, there exists a smooth family of surfaces M_{t}, t ∈ [0, T), solving (8.1) with M_{0} = M. For t ↑ T, M_{t} converges to infinity. The rescaled surfaces M_{t} ·(T −t) converge smoothly to the unit sphere 𝕊^{2}.

We will also consider other normal velocities for which similar results hold. Therefore, we have to find quantities like (8.2) that are monotone during the flow. In the following Table, we have collected some normal velocities F (1^{st} column) and quantities w (2^{nd} column) such that max_{M_{t}} w is non-increasing in time for surfaces contracting with normal velocity F. It is common to use positive functions F for contracting flows. Thus the negative sign corresponds to the fact that these surfaces expand. In each case, we obtain convergence to round spheres at infinity for smooth compact strictly convex initial surfaces M_{0}.

This part concerns mainly the expansion of surfaces by the inverse Gauß curvature flow. This flow is homogeneous of degree minus two in the principal curvatures. We want to stress that we don’t have to assume any pinching condition on the initial surface. To our knowledge, the expansion of hypersurfaces by normal velocities homogeneous in the principal curvatures of degree smaller than minus one has not been studied before.

\begin{matrix}−\frac{1}{K} & \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}} \\ −\frac{H^{2}}{K^{2}} & \frac{(λ_{1} − λ_{2})^{2}}{(λ_{1} + λ_{2})λ_{1}λ_{2}} \\ −\frac{|A|^{2}}{K^{2}} & \frac{(λ_{1} − λ_{2})^{2}}{(λ_{1} + λ_{2})λ_{1}λ_{2}} \\ −\frac{H^{3}}{K^{3}} & \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{(λ_{1}^{2} + λ_{2}^{2}) (λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) λ_{1}^{3}λ_{2}^{3}} \end{matrix}

CHAPTER 9 Inverse Gauß Curvature Flow

9.1. A Monotone Quantity

Theorem 9.1. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3} flowing according to Ẋ = \frac{1}{K} ν,

(9.1)  max_{M(t)} \frac{(λ_{1} − λ_{2})^{2}}{4λ_{1}^{2}λ_{2}^{2}} = max_{M(t)} \frac{2|A|^{2} − H^{2}}{(H^{2} − |A|^{2})^{2}} ≡ max_{M(t)} w

is non-increasing in time.

An immediate consequence of this theorem is

Corollary 9.2. The only homothetically expanding smooth compact strictly convex surfaces M_{t}, solving the flow equation Ẋ = \frac{1}{K} ν in ℝ^{3}, are spheres.

Proof. The quantity \frac{(λ_{1}−λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}} is negative homogeneous of degree two in the principal curvatures and non-negative. If M is homothetically expanding, Theorem 9.1 implies that (λ_{1} − λ_{2})^{2} = 0 everywhere. Thus M(t) is umbilic and [57, Lemma 7.1] implies that M(t) is a sphere. □

Proof of Theorem 9.1. First and second derivatives of F = −1/K with respect to the second fundamental form are given by

\begin{matrix}F^{ij} = & \frac{1}{K} h̃^{ij}, \\ F^{ij, kl} = & − \frac{1}{K} (h̃^{ij}h̃^{kl} + h̃^{ik}h̃^{jl}) . \end{matrix} (9.2)

Lemma 2.1 yields

(9.3)  \frac{d}{dt}H − F^{ij}H_{; ij} =\frac{1}{K} (H^{2} − 3|A|^{2}) − \frac{1}{K} ∑ \frac{1}{λ_{i}λ_{j}} (h_{ii; k}h_{jj; k} + h_{ij; k}^{2})

and

\begin{aligned}\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} = & \frac{1}{K} (2H|A|^{2} − 6 tr A^{3}) − \frac{1}{K} ∑ \frac{2}{λ_{k}} h_{ij; k}^{2} \\ & − \frac{1}{K} ∑ \frac{2λ_{k}}{λ_{i}λ_{j}} (h_{ii; k}h_{jj; k} + h_{ij; k}^{2}) . \end{aligned} (9.4)

For the rest of the proof, we consider a critical point of w|_{M_{t}} for some t > 0, where w > 0 and λ_{1} ≠ λ_{2}. It suffices to show that w is non-increasing in such a point. Then our theorem follows.

We differentiate w

w =\frac{−H^{2} + 2|A|^{2}}{(H^{2} − |A|^{2})^{2}} ,

w_{; i} =\frac{2H (H^{2} − 3|A|^{2}) H_{; i} + 2|A|^{2} (|A|^{2})_{; i}}{(H^{2} − |A|^{2})^{3}} ,
w_{; ij} =\frac{1}{(H^{2} − |A|^{2})^{4}} \{2H (H^{2} − |A|^{2}) (H^{2} − 3|A|^{2}) H_{; ij}

+2|A|^{2} (H^{2} − |A|^{2}) (|A|^{2})_{; ij}

(6 (H^{2} − |A|^{2})^{2} − 12H^{2} (H^{2} − 3|A|^{2})) H_{; i}H_{; j}

+2 (H^{2} + 2|A|^{2}) (|A|^{2})_{; i} (|A|^{2})_{; j}

−12H|A|^{2} (H_{; i} (|A|^{2})_{; j} + H_{; j} (|A|^{2})_{; i})} .

In a critical point of w with λ_{1} ≠ λ_{2}, we get

0 =4λ_{2}^{2}(λ_{1} − λ_{2})h_{11; k} + 4λ_{1}^{2}(λ_{2} − λ_{1})h_{22; k} for k = 1, 2,
(9.5)  h_{22; 1} =\frac{λ_{2}^{2}}{λ_{1}^{2}} h_{11; 1} ≡ a_{1}h_{11; 1}^{2} and h_{11; 2} = \frac{λ_{1}^{2}}{λ_{2}^{2}} h_{22; 2} ≡ a_{2}h_{22; 2}^{2}.

The evolution equation of w is

(H^{2} − |A|^{2})^{4} · (\frac{d}{dt}w − F^{ij}w_{; ij}) =
=2H (H^{2} − |A|^{2}) (H^{2} − 3|A|^{2}) (\frac{d}{dt}H − F^{ij}H_{; ij})

+ 2|A|^{2} (H^{2} − |A|^{2}) (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij})

+ (6 (H^{2} − |A|^{2})^{2} − 12H^{2} (H^{2} − 3|A|^{2})) (−F^{ij}H_{; i}H_{; j})

+ 2 (H^{2} + 2|A|^{2}) (−F^{ij} (|A|^{2})_{; i} (|A|^{2})_{; j})

− 12H|A|^{2} (−F^{ij} (H_{; i} (|A|^{2})_{; j} + H_{; j} (|A|^{2})_{; i})) .

We consider the terms on the right-hand side individually and use (9.3), (9.4), and (9.5)

\frac{d}{dt}H − F^{ij}H_{; ij} = − \frac{2 (λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2})}{λ_{1}λ_{2}}

− \frac{2 (λ_{1}^{2}a_{1}^{2} + λ_{1}λ_{2}a_{1}^{2} + λ_{1}λ_{2}a_{1} + λ_{2}^{2})}{λ_{1}^{3}λ_{2}^{3}}h_{11; 1}^{2}

− \frac{2 (λ_{2}^{2}a_{2}^{2} + λ_{2}λ_{1}a_{2}^{2} + λ_{2}λ_{1}a_{2} + λ_{1}^{2})}{λ_{2}^{3}λ_{1}^{3}}h_{22; 2}^{2}

= − \frac{2 (λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2})}{λ_{1}λ_{2}}

− \frac{2(λ_{1} + λ_{2}) (λ_{1}^{2} + λ_{2}^{2})}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) ,

2H (H^{2} − |A|^{2}) (H^{2} − 3|A|^{2}) · (\frac{d}{dt}H − F^{ij}H_{; ij}) =

9.1. A MONOTONE QUANTITY 49

= − 8(λ_{1} + λ_{2}) (λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2}) λ_{1}λ_{2} (\frac{d}{dt}H − F^{ij}H_{; ij})
=16(λ_{1} + λ_{2}) (λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2})^{2}

+ 16(λ_{1} + λ_{2})^{2} (λ_{1}^{2} − λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1}^{2} + λ_{2}^{2}) ·

· (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2})

=8(λ_{1} + λ_{2}) (2λ_{1}^{4} − 4λ_{1}^{3}λ_{2} + 6λ_{1}^{2}λ_{2}^{2} − 4λ_{1}λ_{2}^{3} + 2λ_{2}^{4})

+ (4λ_{1}^{7}λ_{2} + 4λ_{1}^{6}λ_{2}^{2} + 4λ_{1}^{5}λ_{2}^{3} + 8λ_{1}^{4}λ_{2}^{4}

+4λ_{1}^{3}λ_{2}^{5} + 4λ_{1}^{2}λ_{2}^{6} + 4λ_{1}λ_{2}^{7}) ·

· \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) ,

\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij} = − \frac{2(λ_{1} + λ_{2}) (2λ_{1}^{2} − 3λ_{1}λ_{2} + 2λ_{2}^{2})}{λ_{1}λ_{2}}

− \frac{2 (2λ_{1}^{2}a_{1}^{2} + 2λ_{1}λ_{2}a_{1}^{2} + 3λ_{2}^{2}a_{1}^{2} + 2λ_{1}λ_{2}a_{1} + 3λ_{2}^{2})}{λ_{1}^{2}λ_{2}^{3}}h_{11; 1}^{2}

− \frac{2 (2λ_{2}^{2}a_{2}^{2} + 2λ_{2}λ_{1}a_{2}^{2} + 3λ_{1}^{2}a_{2}^{2} + 2λ_{2}λ_{1}a_{2} + 3λ_{1}^{2})}{λ_{2}^{2}λ_{1}^{3}}h_{22; 2}^{2}

= − \frac{2(λ_{1} + λ_{2}) (2λ_{1}^{2} − 3λ_{1}λ_{2} + 2λ_{2}^{2})}{λ_{1}λ_{2}}

− \frac{2 (3λ_{1}^{4} + 2λ_{1}^{3}λ_{2} + 2λ_{1}^{2}λ_{2}^{2} + 2λ_{1}λ_{2}^{3} + 3λ_{2}^{4})}{λ_{1}λ_{2}}·

· (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) ,

2|A|^{2} (H^{2} − |A|^{2}) · (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij}) =
=4 (λ_{1}^{2} + λ_{2}^{2}) λ_{1}λ_{2} (\frac{d}{dt}|A|^{2} − F^{ij}(|A|^{2})_{; ij})
= − 8(λ_{1} + λ_{2}) (λ_{1}^{2} + λ_{2}^{2}) (2λ_{1}^{2} − 3λ_{1}λ_{2} + 2λ_{2}^{2})

− 8 (λ_{1}^{2} + λ_{2}^{2}) (3λ_{1}^{4} + 2λ_{1}^{3}λ_{2} + 2λ_{1}^{2}λ_{2}^{2} + 2λ_{1}λ_{2}^{3} + 3λ_{2}^{4}) ·

· (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2})

= − 8(λ_{1} + λ_{2}) (2λ_{1}^{4} − 3λ_{1}^{3}λ_{2} + 4λ_{1}^{2}λ_{2}^{2} − 3λ_{1}λ_{2}^{3} + 2λ_{2}^{4})

− (6λ_{1}^{7}λ_{2} + 4λ_{1}^{6}λ_{2}^{2} + 10λ_{1}^{5}λ_{2}^{3} + 8λ_{1}^{4}λ_{2}^{4} + 10λ_{1}^{3}λ_{2}^{5}

+4λ_{1}^{2}λ_{2}^{6} + 6λ_{1}λ_{2}^{7}) \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) .

For the remaining terms, we employ (9.2)

−F^{ij}H_{; i}H_{; j} = − \frac{(a_{1} + 1)^{2}}{λ_{1}^{2}λ_{2}} h_{11; 1}^{2} − \frac{(a_{2} + 1)^{2}}{λ_{2}^{2}λ_{1}} h_{22; 2}^{2}

50 9. INVERSE GAUSS CURVATURE FLOW

= − \frac{(λ_{1}^{2} + λ_{2}^{2})^{2}}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) ,

(6 (H^{2} − |A|^{2})^{2} − 12H^{2} (H^{2} − 3|A|^{2})) (−F^{ij}H_{; i}H_{; j}) =

= − (λ_{1}^{4} + λ_{1}^{3}λ_{2} + λ_{1}^{2}λ_{2}^{2} + λ_{1}λ_{2}^{3} + λ_{2}^{4}) ·

· 6 (λ_{1}^{2} + λ_{2}^{2})^{2} \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2})

= − (6λ_{1}^{8} + 6λ_{1}^{7}λ_{2} + 18λ_{1}^{6}λ_{2}^{2} + 18λ_{1}^{5}λ_{2}^{3} + 24λ_{1}^{4}λ_{2}^{4} + 18λ_{1}^{3}λ_{2}^{5}

+18λ_{1}^{2}λ_{2}^{6} + 6λ_{1}λ_{2}^{7} + 6λ_{2}^{8}) ·

· \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) ,

−F^{ij} (|A|^{2})_{; i} (|A|^{2})_{; j} = − \frac{4(λ_{2}a_{1} + λ_{1})^{2}}{λ_{1}^{2}λ_{2}} h_{11; 1}^{2} − \frac{4(λ_{1}a_{2} + λ_{2})^{2}}{λ_{2}^{2}λ_{1}} h_{22; 2}^{2}
= − (λ_{1}^{3} + λ_{2}^{3})^{2} · \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) ,
2 (H^{2} + 2|A|^{2}) · (−F^{ij} (|A|^{2})_{; i} (|A|^{2})_{; j}) =
= − 2 (3λ_{1}^{2} + 2λ_{1}λ_{2} + 3λ_{2}^{2}) (λ_{1}^{3} + λ_{2}^{3})^{2} ·

· \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2})

= − (6λ_{1}^{8} + 4λ_{1}^{7}λ_{2} + 6λ_{1}^{6}λ_{2}^{2} + 12λ_{1}^{5}λ_{2}^{3} + 8λ_{1}^{4}λ_{2}^{4} + 12λ_{1}^{3}λ_{2}^{5}

+6λ_{1}^{2}λ_{2}^{6} + 4λ_{1}λ_{2}^{7} + 6λ_{2}^{8}) \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) ,

− F^{ij} (H_{;i} (|A|^{2})_{; j} + H_{;j} (|A|^{2})_{; i}) =

= − \frac{4(λ_{2}a_{1} + λ_{1})(a_{1} + 1)}{λ_{1}^{2}λ_{2}}h_{11; 1}^{2} − \frac{4(λ_{1}a_{2} + λ_{2})(a_{2} + 1)}{λ_{2}^{2}λ_{1}}h_{22; 2}^{2}
= − (λ_{1}^{3} + λ_{2}^{3}) (λ_{1}^{2} + λ_{2}^{2}) · \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) ,

− 12H|A|^{2} (−F^{ij} (H_{; i} (|A|^{2})_{; j} + H_{; j} (|A|^{2})_{; i})) =

=12(λ_{1} + λ_{2}) (λ_{1}^{2} + λ_{2}^{2})^{2} (λ_{1}^{3} + λ_{2}^{3}) ·

· \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2})

= (12λ_{1}^{8} + 12λ_{1}^{7}λ_{2} + 24λ_{1}^{6}λ_{2}^{2} + 36λ_{1}^{5}λ_{2}^{3} + 24λ_{1}^{4}λ_{2}^{4} + 36λ_{1}^{3}λ_{2}^{5}

+24λ_{1}^{2}λ_{2}^{6} + 12λ_{1}λ_{2}^{7} + 12λ_{2}^{8}) ·

· \frac{4}{λ_{1}λ_{2}} (\frac{1}{λ_{1}^{5}} h_{11; 1}^{2} + \frac{1}{λ_{2}^{5}} h_{22; 2}^{2}) .

Combining these expressions, we see that many terms cancel. We obtain

(H^{2} − |A|^{2})^{4} (\frac{d}{dt}w − F^{ij}w_{; ij}) =

= − 8(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}λ_{1}λ_{2} − \frac{32λ_{2}^{3}}{λ_{1}^{2}} h_{11; 1}^{2} − \frac{32λ_{1}^{3}}{λ_{2}^{2}} h_{22; 2}^{2},
\frac{d}{dt}w − F^{ij}w_{; ij} = − \frac{(λ_{1} + λ_{2})(λ_{1} − λ_{2})^{2}}{2λ_{1}^{3}λ_{2}^{3}} − \frac{2}{λ_{1}^{6}λ_{2}} h_{11; 1}^{2} − \frac{2}{λ_{1}λ_{2}^{6}} h_{22; 2}^{2}
≤0.

We finally apply the maximum principle and our theorem follows. □

9.2. Convergence to Infinity

It is known, that (8.1) is a parabolic evolution equation for strictly convex initial data and that it has a solution on a maximal time interval [0, T). Here, we want to show that some points on M_{t} converge to infinity for t ↑ T, i. e.

lim_{t↑T} sup_{M_{t}} |X| = ∞.

We show that the principal curvatures of M_{t} stay uniformly bounded above.

Lemma 9.3. For a smooth compact strictly convex surface M in ℝ^{3}, flowing according to Ẋ = \frac{1}{K} ν, the maximum of the principal curvatures is non-increasing.

Proof. Consider M_{ij} = h_{ij} − μg_{ij} with μ > 0 so large that M_{ij} is negative semi-definite for some time t_{0}. We wish to show that M_{ij} is negative semi-definite for t > t_{0}. Lemma 2.1 gives

\frac{d}{dt}h_{ij} − F^{kl}h_{ij; kl} = \frac{1}{K} Hh_{ij} − \frac{1}{K} h_{i}^{k}h_{kj} − \frac{1}{K} (h̃^{kl}h̃^{rs} + h̃^{kr}h̃^{ls}) h_{kl; i}h_{rs; j}.

In the evolution equation for M_{ij}, we drop the negative definite terms involving derivatives of the second fundamental form

\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl} ≤ \frac{1}{K} (Hh_{ij} − h_{i}^{k}h_{kj} − 2μh_{ij}) .

Let ξ be a zero eigenvalue of M_{ij} with |ξ| = 1, M_{ij}ξ^{j} = h_{ij}ξ^{j} − μg_{ij}ξ^{j} = 0. So we obtain in a point with M_{ij} ≤ 0

(Hh_{ij} − h_{i}^{k}h_{kj} − 2μh_{ij}) ξ^{i}ξ^{j} = Hμ − 3μ^{2} ≤ 2μμ − 3μ^{2} ≤ 0

and the maximum principle for tensors [18, 29] implies the claimed result. □

We obtain a pinching estimate

Lemma 9.4. For a smooth compact strictly convex surface M_{t} in ℝ^{3}, flowing according to Ẋ = \frac{1}{K} ν, there exists c = c(M_{0}) such that 0 < \frac{1}{c} ≤ \frac{λ_{1}}{λ_{2}} ≤ c.

Proof. Choose μ > 0 such that λ_{1}, λ_{2} ≤ μ at t = 0. Theorem 9.1 and Lemma 9.3 imply that

\frac{1}{μ^{2}} \frac{(\frac{λ_{1}}{λ_{2}} − 1)^{2}}{\frac{λ_{1}}{λ_{2}}} = \frac{1}{μ^{2}} \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} ≤ \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}} ≤ c.

We obtain the bound on \frac{λ_{1}}{λ_{2}} claimed above. □

It is only here that we use the monotone quantity of Theorem 9.1. For our purposes, this quantity is better than scaling invariant. As it becomes apparent in the proof of Lemma 9.10, however, the Aleksandrov reflection principle can be used instead for the rest of the proof. This simplifies the proof compared to [12] or Part 2, where similar monotone quantities are used. Later on, see Theorem 9.12 and Theorem 9.17, we will use monotone quantities to improve the convergence rate that follows from Aleksandrov reflection.

The next result shows that K stays uniformly bounded below by a positive constant as long as M_{t} is enclosed by a ball of fixed positive radius. A similar estimate is used in [59].

Lemma 9.5. For a strictly convex solution of (8.1), K is uniformly bounded below by a positive constant in terms of the radius R of an enclosing sphere B_{R}(x_{0}) and max_{M_{0}} \frac{K^{−1}}{2R−⟨X−x_{0}, ν⟩}. More precisely, we have everywhere

(9.6)  K ≥ \frac{1}{3R} (max_{M_{0}} \frac{K^{−1}}{2R − ⟨X − x_{0}, ν⟩})^{−1} or H ≥ \frac{1}{2R}.

In both cases, we obtain a positive lower bound on the principal curvatures.

Proof. We may assume that x_{0} = 0. Observe that R is a positive lower bound for 2R − ⟨X, ν⟩. Thus \frac{K^{−1}}{2R−⟨X, ν⟩} is finite for strictly convex surfaces. If \frac{K^{−1}}{2R−⟨X, ν⟩} is bounded above by its maximum over M_{0}, we get

inf_{t} inf_{M_{t}} K ≥ \frac{1}{3R} (max_{M_{0}} \frac{K^{−1}}{2R − ⟨X, ν⟩})^{−1}

and (9.6) follows. According to Lemma 2.1, we get

\frac{d}{dt}X^{β} − F^{ij}X_{; ij}^{β} =\frac{3}{K} ν^{β},
\frac{d}{dt}ν^{β} − F^{ij}ν_{; ij}^{β} =\frac{H}{K} ν^{β},
\frac{d}{dt}⟨X, ν⟩ − F^{ij}⟨X, ν⟩_{; ij} = − \frac{1}{K} + \frac{H}{K} ⟨X, ν⟩,
\frac{d}{dt} \frac{1}{K} − F^{ij}(\frac{1}{K} )_{; ij} =\frac{H}{K^{2}} .

In a critical point of \frac{K^{−1}}{2R−|X, ν|}, we obtain

\frac{d}{dt} log \frac{K^{−1}}{2R − ⟨X, ν⟩} − F^{ij} (log \frac{K^{−1}}{2R − ⟨X, ν⟩})_{; ij} = \frac{1}{2R − ⟨X, ν⟩} \frac{1}{K} (2RH − 1).

We only have to prove that we preserve the bound in Equation (9.6) for t > 0, when

max_{M_{t}} \frac{K^{−1}}{2R − ⟨X, ν⟩}

increases. Then we have 0 ≤ 2RH−1 at a point, where max_{M_{t}} \frac{K^{−1}}{2R−⟨X, ν⟩} is attained. Finally, the positive lower bound on K or H and our pinching estimate, Lemma 9.4, imply a positive lower bound on the principal curvatures. □

Let us recall a form of the maximum principle for evolving hypersurfaces.

Lemma 9.6. Let M_{t} and M̃_{t} be two smooth compact strictly convex solutions to (8.1) on some time interval [0, T^{∗}). If M_{0} encloses M̃_{0}, then M_{t} encloses M̃_{t} at any time t ∈ [0, T^{∗}), for which both solutions exist.

Proof. This is a standard consequence of the maximum principle. □

The next result describes the evolution of spheres.

Lemma 9.7. Spheres ∂B_{r(t)}(x_{0}) solve (8.1) for t ∈ [0, T) with r(t) = (T − t)^{−1} and T = r^{−1}(0).

Proof. The evolution equation for the radius of a sphere is

ṙ(t) = r^{2}(t).
□

Lemma 9.8. Let M_{t} be a family of smooth compact strictly convex solutions to (8.1) on a maximal time interval [0, T). Then T < ∞.

Proof. Spheres solving (8.1) tend to infinity in finite time, see Lemma 9.7. So Lemma 9.6 implies that T < ∞. □

Lemma 9.9. Let M_{t} be a family of smooth compact strictly convex surfaces solving (8.1) on a maximal time interval [0, T). Then

lim_{t↑T} sup_{M_{t}} |X| = ∞.

Proof. Assume that M_{t} ⊂ B_{R}(0), 0 ≤ t < T, for some R > 0. Then Lemmata 9.3 and 9.5 imply that the principal curvatures of M_{t} stay uniformly bounded above and below by positive constants. Thus (8.1) can be rewritten as a uniformly parabolic equation and the estimates of Krylov, Safonov, Evans (see also [4]), and Schauder imply uniform a priori estimates up to t = T. This allows to continue the solution M_{t} smoothly past t = T, a contradiction. Note finally, that (8.1) is an expanding flow, so sup_{M_{t}} |X| is monotone in t and the limit exists. □

9.3. Convergence to a Sphere

Lemma 9.10. Under the assumptions of Theorem 8.1, we get

lim_{t↑T} inf_{M_{t}} |X| = ∞,

more precisely, there exists c = c(M_{0}) such that

sup_{M_{t}} |X| − c ≤ (T − t)^{−1} ≤ inf_{M_{t}} |X| + c

and

(9.7)  M_{t} · (T − t) ⊂ B_{1+c(T −t)}(0) \ B_{1−c(T −t)}(0),

so the rescaled surfaces M_{t} · (T − t) converge to the unit sphere 𝕊^{2} in Hausdorff distance.

Proof. We may shift the origin such that 0 lies inside M_{0}. This does not affect the convergence rate claimed above.

Define the support function u : 𝕊^{2} × [0, T) → ℝ_{+} for a convex surface by

u(z, t) = ⟨X (ν^{−1}(z), t) , z⟩.

It fulfills the evolution equation, see e. g. [10],

u̇ = \frac{det(u_{; ij} + uσ_{ij})}{det(σ_{ij})},

where u_{; ij} denotes covariant derivatives on 𝕊^{2} and σ_{ij} is the standard metric on 𝕊^{2}.

We apply the Aleksandrov reflection principle of Bennett Chow, Robert Gulliver [17], and James McCoy [44, Theorem 3.1] and obtain a uniform bound on the oscillation (and the gradient) of u(·, t) for all t ∈ [0, T), that depends only on the initial data.

As sup_{M_{t}} |X| → ∞ for t ↑ T, we obtain that inf_{M_{t}} |X| → ∞ for t ↑ T, more precisely, we have

sup_{M_{t}} |X| ≤ inf_{M_{t}} |X| + c.

It remains to show that

inf_{M_{t}} |X| ≤ (T − t)^{−1} ≤ sup_{M_{t}} |X|.

Consider the surfaces ∂M_{(T −t)^{−1}}(0), solving (8.1). For t = T, M_{t} and ∂B_{(T −t)^{−1}}(0) converge to infinity. We claim that for all t ∈ [0, T), ∂B_{(T −t)^{−1}}(0) ∩ M_{t} ≠ ∅. Otherwise, for some t_{0}, M_{t_{0}} encloses ∂B_{(T −t)^{−1}}(0) or is contained in B_{(T −t)^{−1}}(0). Both cases are similar. We only consider the first case. Choose ε > 0 such that M_{t_{0}} encloses also ∂B_{(T −t_{0}−ε)^{−1}}(0), a slightly larger sphere. For t ∈ [t_{0}, T − ε), ∂B_{(T −t−ε)^{−1}}(0) solves (8.1). As ∂B_{(T −t−ε)^{−1}}(0) converges to infinity for t ↑ T − ε, Lemma 9.6 implies that M_{t} has to converge to infinity for t ↑ T −ε, a contradiction.

□

In terms of the support function u, this Lemma implies that

(9.8)  u(x, t) − c ≤ (T − t)^{−1} ≤ u(x, t) + c.

Note that this estimate is sharp for spheres ∂B_{(T −t)^{−1}}(Q) solving (8.1), if Q is different from the origin. The method of [12], see also Part 2, where the origin is replaced by some q(t), can be adapted to the present situation. In order to improve the estimate (9.8), however, we need a monotone quantity similar to (8.2) with a better scaling behavior. We address this issue in Section 9.5.

9.4. Smooth Convergence to a Sphere

Lemma 9.11. Under the assumptions of Theorem 8.1, there exists c = c(M_{0}), such that

K ≤ c · (T − t)^{2}.

Proof. For μ ≫ 1 to be fixed below, we consider

w := |X|^{2} − μ\frac{1}{K} .

We may assume that μ is so large that w < 0 on M_{0}. Our aim is to show that w stays negative during the flow. We use the evolution equations of the proof of Lemma 9.5. The evolution equation of w is given by

\frac{d}{dt}w − F^{ij}w_{; ij} = 6\frac{1}{K} ⟨X, ν⟩ − 2\frac{H}{K^{2}} − μ\frac{H}{K^{2}} .

Let t_{0} ∈ [0, T) be minimal such that max_{M_{t}} w = 0. Choose x_{0} ∈ M_{t} such that w(x_{0}, t_{0}) = 0. At this point, we apply the parabolic maximum principle and obtain

0 ≤6K⟨X, ν⟩ − (2 + μ)H
≤6K|X| − (2 + μ)H
=6\sqrt{K}\sqrt{μ} − (2 + μ)H  as w(x_{0}, t_{0}) = 0,
≤(6\sqrt{μ} − 2 − μ)H  as K ≤ H^{2}.

We fix μ sufficiently large and obtain that w ≤ 0 during the flow. In view of Lemma 9.10, this implies the upper bound on the Gauß curvature claimed above. □

Combining this result with the Lemmata 9.4, 9.5, and 9.10, we obtain

(9.9)  \frac{1}{c} · (T − t)^{2} ≤ K ≤ c · (T − t)^{2}.

We rescale our surfaces similarly as in [7]. Consider the embeddings X̃(·, t),

X̃(z, t) := (T − t) · X(z, t).

Define a new time function

τ(t) := − log (\frac{T − t}{T}) .

We use a tilde to denote geometric quantities of the rescaled surfaces. For X̃, we obtain the evolution equation

\frac{d}{dτ} X̃ = \frac{1}{K̃} ν̃ − X̃.

Our a priori estimates and the estimates of Krylov, Safonov, Evans, and Schauder imply uniform bounds on all derivatives of the support function ũ of X̃. Applying interpolation inequalities as in [52, Lemma C.2] to

|ũ − 1| ≤c · (T − t),
|D^{k}ũ| ≤c_{k},

we get

|D^{k}ũ| ≤c(k, ε) · (T − t)^{1−ε}

for any ε > 0.

This finishes the proof of Theorem 8.1.

9.5. Improved Convergence Rate

Theorem 9.12. For a family of smooth compact strictly convex surfaces M_{t} in ℝ^{3} flowing according to Ẋ = \frac{1}{K} ν,

(9.10)  max_{M(t)} \frac{(λ_{1}^{2} + λ_{2}^{2}) (λ_{1} − λ_{2})^{2}}{8(λ_{1} + λ_{2})λ_{1}^{3}λ_{2}^{3}} ≡ max_{M(t)} w ≡ max_{M_{t}} \frac{|A|^{2} (2|A|^{2} − H^{2})}{H (H^{2} − |A|^{2})^{3}}

Proof. We use Section 6 and obtain in a critical point of w

\frac{d}{dt}w − F^{ij}w_{; ij} =\frac{−3 (λ_{1}^{4} + 2λ_{1}^{3}λ_{2} − 2λ_{1}^{2}λ_{2}^{2} + 2λ_{1}λ_{2}^{3} + λ_{2}^{4}) (λ_{1} − λ_{2})^{2}}{8(λ_{1} + λ_{2})^{2}λ_{1}^{4}λ_{2}^{4}}

− \frac{1}{(3λ_{1}^{3} + 3λ_{1}^{2}λ_{2} − λ_{1}λ_{2}^{2} + 3λ_{2}^{3})^{2} (λ_{1} + λ_{2})^{2}λ_{1}^{7}λ_{2}^{2}} ·

· (18λ_{1}^{9} − 9λ_{1}^{8}λ_{2} + 12λ_{1}^{7}λ_{2}^{2} + 72λ_{1}^{6}λ_{2}^{3} − 12λ_{1}^{5}λ_{2}^{4} + 70λ_{1}^{4}λ_{2}^{5}

+60λ_{1}^{3}λ_{2}^{6} + 18λ_{1}λ_{2}^{8} + 27λ_{2}^{9}) h_{11; 1}^{2}

− \frac{1}{(3λ_{2}^{3} + 3λ_{2}^{2}λ_{1} − λ_{2}λ_{1}^{2} + 3λ_{1}^{3})^{2} (λ_{2} + λ_{1})^{2}λ_{2}^{7}λ_{1}^{2}} ·

· (18λ_{2}^{9} − 9λ_{2}^{8}λ_{1} + 12λ_{2}^{7}λ_{1}^{2} + 72λ_{2}^{6}λ_{1}^{3} − 12λ_{2}^{5}λ_{1}^{4} + 70λ_{2}^{4}λ_{1}^{5}

+60λ_{2}^{3}λ_{1}^{6} + 18λ_{2}λ_{1}^{8} + 27λ_{1}^{9}) h_{22; 2}^{2}
≤0.

We finally apply the maximum principle. □

This allows to improve our bound on |λ_{1} − λ_{2}|.

Lemma 9.13. For a smooth compact strictly convex surface M_{t} in ℝ^{3}, flowing according to Ẋ = \frac{1}{K} ν, there exists a constant c = c(M_{0}) such that |λ_{1} − λ_{2}| ≤ c · K^{5/4} ≤ c · (T − t)^{5/2}.

Proof. This is a direct consequence of Theorem 9.12, Lemma 9.4 and Lemma 9.11. □

We now closely follow the corresponding sections of [12] and Part 2.

Proposition 9.14. Define q(t) := \frac{1}{4π} ∫_{M_{t}} KX. Then

|⟨X − q, ν⟩ − \frac{1}{8π} ∫_{M_{t}} H| ≤ \frac{1}{4π} · sup_{M_{t}} |λ_{1} − λ_{2}| · H^{2}(M_{t}),

where H^{2}(M_{t}) denotes the area of M_{t}.

Proof. This is [12, Proposition 4]. □

We will call q(t) the pseudocenter of M_{t}. We define r_{+}(t) to be the minimal radius of a sphere, centered at q(t), that

encloses M_{t}. Similarly, we define r_{−}(t) to be the maximal radius of a sphere, centered at q(t), that is enclosed by M_{t}. Let ρ_{−}(t) be the maximal radius of a sphere (with arbitrary center) enclosed by M_{t} and ρ_{+}(t) be the minimal radius of spheres enclosing M_{t}.

Lemma 9.15. Under the assumptions of Theorem 8.1, for T − t sufficiently small, r_{+} and r_{−} are estimated as follows

(T − t)^{−1} · (1 − c · (T − t)^{3/2}) ≤ r_{−}(t) ≤ r_{+}(t) ≤ (T − t)^{−1} · (1 + c · (T − t)^{3/2}) .

9.5. IMPROVED CONVERGENCE RATE 57

Proof. Denote the bounded component of ℝ^{3} \M_{t} by E_{t}. The transformation formula for integrals implies that

\frac{1}{4π} ∫_{M_{t}} KX = \frac{1}{4π} ∫_{𝕊^{2}} X (ν^{−1}(·)) .

So we see that q(t) ∈ E_{t}. We have

r_{+} = max_{M_{t}} ⟨X − q(t), ν⟩,  r_{−} = min_{M_{t}} ⟨X − q(t), ν⟩,
ρ_{+} = min_{p∈ℝ^{3}} max_{M_{t}} ⟨X − p, ν⟩, and ρ_{−} = max_{p∈E_{t}} min_{M_{t}} ⟨X − p, ν⟩.

Recall the first variation formula for a vector field Y along M_{t}

∫_{M_{t}} H⟨Y, ν⟩ = ∫_{M_{t}} div_{M_{t}} Y

and get for p ∈ E_{t} such that ρ_{+} = max_{M_{t}}⟨X − p, ν⟩

∫_{M_{t}} H ≥ \frac{1}{ρ_{+}} ∫_{M_{t}} H · ⟨X − p, ν⟩ = \frac{1}{ρ_{+}} ∫_{M_{t}} div_{M_{t}} X = \frac{1}{ρ_{+}} ∫_{M_{t}} 2 = \frac{2}{ρ_{+}} H^{2}(M_{t}).

We employ Proposition 9.14 and deduce that

r_{−} ≥\frac{1}{8π} ∫_{M_{t}} H · \{1 − 2 (∫_{M_{t}} H)^{−1} · sup_{M_{t}} |λ_{1} − λ_{2}| · H^{2}(M_{t})\}
≥\frac{1}{8π} ∫_{M_{t}} H · \{1 − ρ_{+} · sup_{M_{t}} |λ_{1} − λ_{2}|\}
≥\frac{1}{8π} ∫_{M_{t}} H · \{1 − c · (T − t)^{3/2}\} ,

where we have used the Lemmata 9.13 and 9.10. So we obtain

(9.11)  r_{−}(t) ≥ \frac{1}{8π} ∫_{M_{t}} H · (1 − c · (T − t)^{3/2})

and similarly

(9.12)  r_{+}(t) ≤ \frac{1}{8π} ∫_{M_{t}} H · (1 + c · (T − t)^{3/2}) .

As in the proof of Lemma 9.10, we get

r_{−} ≤ ρ_{−} ≤ (T − t)^{−1} ≤ ρ_{+} ≤ r_{+}

and

(T − t)^{−1} · (1 − c · (T − t)^{3/2}) ≤ \frac{1}{8π} ∫_{M_{t}} H ≤ (T − t)^{−1} · (1 + c · (T − t)^{3/2}) .

Using (9.11) and (9.12) gives the claimed estimates on r_{−} and r_{+}. □

Lemma 9.16. Under the assumptions of Theorem 8.1, q(t) as defined in Proposition 9.14 is a smooth function of t in [0, T) and converges to some point Q ∈ ℝ^{3} for t ↑ T,

|q(t) − Q| ≤ c · (T − t)^{1/2}.

Proof. The definition of q(t) involves only quantities that depend smoothly on t, so it remains to prove convergence for t ↑ T.

For 0 < t_{1} < t_{2} < T, we want to estimate |q(t_{1}) − q(t_{2})| from above. We may assume that q(t_{1}) ≠ q(t_{2}). Consider the line passing through q(t_{1}) and q(t_{2}). It intersects the surface M_{t_{2}} in two points, denoted by p_{l}(t_{2}) and p_{r}(t_{2}).

p^{l}(t^{2})  q(t^{1})q(t^{2})  p^{r}(t^{2})

Figure: Convergence of pseudocenters

We may assume that

⟨p_{r}(t_{2}) − q(t_{2}), q(t_{2}) − q(t_{1})⟩ > 0.

This corresponds to p_{r}(t_{2}) and q(t_{2}) lying on the same side of q(t_{1}) as shown in the figure. We estimate

2c(T − t_{1})^{1/2} ≥r_{+}(t_{1}) − r_{−}(t_{1})  by Lemma 9.15
= osc u_{q(t_{1})}(·, t_{1})
≥ osc u_{q(t_{1})}(·, t_{2})  by [44, Theorem 3.1]
≥ |p_{r}(t_{2}) − q(t_{1})| − |p_{l}(t_{2}) − q(t_{1})|
= |p_{r}(t_{2}) − q(t_{2})| + |q(t_{2}) − q(t_{1})|
− (|p_{l}(t_{2}) − q(t_{2})| − |q(t_{2}) − q(t_{1})|)
≥2 |q(t_{2}) − q(t_{1})| + r_{−}(t_{2}) − r_{+}(t_{2})
≥2 |q(t_{2}) − q(t_{1})| − 2c(T − t_{2})^{1/2},

where u_{q(t_{1})}(·, t_{2}) is the support function of M_{t_{2}} − q(t_{1}). Thus we can apply Cauchy’s convergence criterion. Finally, we let t_{2} ↑ T and the claimed bound follows. □

This allows to improve our convergence result.

Theorem 9.17. Under the assumptions of Theorem 8.1, there exists Q ∈ ℝ^{3} such that the Hausdorff distance of M_{t} to a family of expanding spheres around Q is bounded as follows

d_{H}(M_{t}, ∂B_{(T −t)^{−1}}(Q)) ≤ c · (T − t)^{1/2}.

Proof. Combine the Lemmata 9.16 and 9.15. □

Remark 9.18. Theorem 9.17 implies also better estimates for rescaled surfaces,

d_{H} ((M_{t} − Q) · (T − t), 𝕊^{2}) ≤ c · (T − t)^{3/2}.

As above, this implies that all derivatives of the support function decay,

∥u_{Q}(·, t) − 1∥_{C^{k}} ≤ c(k, ε) · (T − t)^{3/2−ε}

for any ε > 0, where u_{Q}(·, t) is the support function of M_{t} − Q. This implies for the principal curvatures λ̃_{i}, i = 1, 2, of (M_{t} − Q) · (T − t)

1 − c(ε) · (T − t)^{3/2−ε} ≤ λ̃_{i} ≤ 1 + c(ε) · (T − t)^{3/2−ε}.

In the proof of Theorem 8.1, we have used (8.2) only to prove that surfaces stay uniformly pinched, i. e. that λ_{1}/λ_{2} is uniformly bounded. If we use it to bound |λ_{1} − λ_{2}|, and then r_{+} and r_{−} as above, we don’t get more than in (9.7), where we used the oscillation estimates of [44]. Our computer program, however, did not yield a scaling invariant quantity that implies uniform pinching. This is similar to [5]. It might be possible to find a monotone quantity that allows to further improve this convergence rate.

9.6. Convergence Rate

In order to find out what the optimal convergence rate might be, we proceed as in Part 2, use the same notation, and compute the linearized equation corresponding to

\frac{d}{dt}X = \frac{1}{K} ν − X or \frac{∂u}{∂t} = \frac{1}{K} w − u

as

\frac{∂v}{∂t} = Δv + v.

As in the contracting case, we only need to consider eigenvalues −l(l + 1) of the laplacian on the sphere for l ∈ ℕ_{+}. For l = 1, we cannot expect convergence rates better than

r_{+} ≤ (T − t)^{−1} · (1 + c · (T − t)),

if we fix q(t) arbitrarily. This estimate is sharp, if we do not adjust q(t). The corresponding eigenfunctions induce translations of the surface. Considering l = 2, we expect that we cannot obtain convergence rates better than

r_{+}(t) ≤ (T − t)^{−1} · (1 + c · (T − t)^{5}) .

CHAPTER 10 Finding Monotone Quantities

10.1. The Algorithm

We use a sieve algorithm and start with symmetric rational functions of the principal curvatures as candidates for test functions, e. g.

w = \frac{p_{1}(λ_{1}, λ_{2})}{p_{2}(λ_{1}, λ_{2})} = \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}} .

Here, p_{1} ≠ 0 and p_{2} ≠ 0 are homogeneous polynomials.

In the end, we want to find functions w such that W := sup_{M_{t}} w is monotone and ensures convergence to round spheres.

We check, whether these test functions w fulfill the following conditions.

(1) (a) p_{1}(λ_{1}, λ_{2}), p_{2}(λ_{1}, λ_{2}) ≥ 0 for 0 < λ_{1}, λ_{2}, (b) p_{1}(λ_{1}, λ_{2}) = 0 for λ_{1} = λ_{2} > 0.

(2) deg p_{1} < deg p_{2}.

(3) \frac{∂w(1,λ_{2})}{∂λ_{2}} < 0 for 0 < λ_{2} < 1 and \frac{∂w(1,λ_{2})}{∂λ_{2}} > 0 for λ_{2} > 1.

(4) \frac{d}{dt}w − F^{ij}w_{; ij} ≤ 0 (a) for terms without derivatives of (h_{ij}),

(b) for terms involving derivatives of (h_{ij}), if w_{; i} = 0 for i = 1, 2.

10.2. Motivation and Randomized Tests

For all flow equations considered, spheres contract to points and stay spherical. So we can only find monotone quantities, if deg p_{1} ≥ deg p_{2} or p_{1}(λ, λ) = 0.

If deg p_{1} ≥ deg p_{2}, we obtain that W is non-increasing on any self-similarly expanding surface. So this does not imply convergence to a sphere.

Condition (3) ensures that the quantity decreases, if the eigenvalues approach each other.

In step (4a) and (4b), we check that we can apply the maximum principle. Here, we have to use various differentiation rules.

In steps (1), (2), and (4), inequalities are tested by evaluating both sides at random numbers.

CHAPTER 11 Other Normal Velocities

11.1. Homogeneity less than minus one

In this section, we prove

Theorem 11.1. For any smooth compact strictly convex surface M in ℝ^{3}, there exists a smooth family of surfaces M_{t}, t ∈ [0, T), solving one of the following flow equations

• \frac{d}{dt}X = \frac{H^{2}}{K^{2}} ν,

• \frac{d}{dt}X = \frac{|A|^{2}}{K^{2}} ν,

• \frac{d}{dt}X = \frac{H^{3}}{K^{3}} ν,

with M_{0} = M. For t ↑ T, M_{t} converges to infinity. The rescaled surfaces M_{t}·r^{−1}(t) converge smoothly to the unit sphere 𝕊^{2}, where r(t) is the radius of an expanding sphere that converges to infinity for t ↑ T, more precisely, r(t) is as follows

• \frac{d}{dt}X = \frac{H^{2}}{K^{2}} ν : r(t) = (4(T − t))^{−1},

• \frac{d}{dt}X = \frac{|A|^{2}}{K^{2}} ν : r(t) = (2(T − t))^{−1},

• \frac{d}{dt}X = \frac{H^{3}}{K^{3}} ν : r(t) = (16(T − t))^{−1/2}.

In Part 2, we have studied convex surfaces contracting according to

\frac{d}{dt}X = −F ν,

where F is positive homogeneous of some degree larger than or equal to two. There, we got the impression, that appropriate monotone quantities that assure convergence to a sphere after rescaling are available for almost every normal velocity considered. In contrast to this, such monotone quantities seem to be rare objects for expanding surfaces with normal velocity of homogeneity less than or equal to minus two.

In both cases, we restricted our attention to normal velocities and possible candidates for monotone quantities in ℤ(λ_{1}, λ_{2}) with small coefficients which are symmetric in λ_{1} and λ_{2}.

In the case of expanding surfaces, however, we also find monotone quantities for surfaces expanding with a normal velocity that is positive homogeneous of degree minus one in the principal curvatures.

The proof of Theorem 11.1 is similar to the proof of Theorem 8.1. Therefore, we will present in the following only those results that are not almost identical to their counterparts in the proof of Theorem 8.1.

Theorem 11.2. For a family M_{t} of smooth compact strictly convex surfaces in ℝ^{3}, flowing according to \frac{d}{dt}X = \frac{H^{2}}{K^{2}} ν,

max_{M_{t}} \frac{(λ_{1} − λ_{2})^{2}}{2(λ_{1} + λ_{2})λ_{1}λ_{2}} ≡ max_{M_{t}} w

is non-increasing in time.

Proof. According to Section 6, we obtain in a critical point of w

\frac{d}{dt}w − F^{ij}w_{; ij} = − \frac{5(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}}

− \frac{128}{(λ_{1} + 3λ_{2})^{2}λ_{1}^{4}} · h_{11; 1}^{2} + (…) · h_{22; 2}^{2}.

Here and in the following, we write (…) to denote a term that equals the factor in front of h_{11; 1}^{2} with λ_{1} and λ_{2} interchanged.

We finally apply the maximum principle. □

The factor 2 in the denominator is useful to rewrite w in terms of the algebraic basis consisting of H and |A|^{2}

\frac{(λ_{1} − λ_{2})^{2}}{2(λ_{1} + λ_{2})λ_{1}λ_{2}} = \frac{2|A|^{2} − H^{2}}{H · (H^{2} − |A|^{2})}.

Theorem 11.3. For a family M_{t} of smooth compact strictly convex surfaces in ℝ^{3}, flowing according to \frac{d}{dt}X = \frac{|A|^{2}}{K^{2}} ν,

max_{M_{t}} \frac{(λ_{1} − λ_{2})^{2}}{2(λ_{1} + λ_{2})λ_{1}λ_{2}} ≡ max_{M_{t}} w

is non-increasing in time.

Proof. According to Section 6, we obtain in a critical point of w

\frac{d}{dt}w − F^{ij}w_{; ij} = − \frac{2 (2λ_{1}^{2} + λ_{1}λ_{2} + 2λ_{2}^{2}) (λ_{1} − λ_{2})^{2}}{(λ_{1} + λ_{2})^{2}λ_{1}^{2}λ_{2}^{2}}

− \frac{4 (21λ_{1}^{4} + 24λ_{1}^{3}λ_{2} + 18λ_{1}^{2}λ_{2}^{2} + λ_{2}^{4})}{(λ_{1} + 3λ_{2})^{2}(λ_{1} + λ_{2})^{2}λ_{1}^{6}} · h_{11; 1}^{2} + (…) · h_{22; 2}^{2}

and apply the maximum principle. □

Theorem 11.4. For a family M_{t} of smooth compact strictly convex surfaces in ℝ^{3}, flowing according to \frac{d}{dt}X = \frac{H^{3}}{K^{3}} ν,

max_{M_{t}} \frac{(λ_{1} + λ_{2})^{6}(λ_{1} − λ_{2})^{2}}{16 (λ_{1}^{2} + λ_{2}^{2}) (λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2}) λ_{1}^{3}λ_{2}^{3}} ≡ max_{M_{t}} w

Proof. According to Section 6, we obtain in a critical point of w

\frac{d}{dt}w − F^{ij}w_{; ij} = − \frac{(λ_{1} + λ_{2})^{8}(λ_{1} − λ_{2})^{2}}{4 (λ_{1}^{2} + λ_{2}^{2})^{2} (λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2})^{2} λ_{1}^{6}λ_{2}^{6}} ·

· (λ_{1}^{6} − λ_{1}^{5}λ_{2} + 8λ_{1}^{4}λ_{2}^{2} + 2λ_{1}^{3}λ_{2}^{3} + 8λ_{1}^{2}λ_{2}^{4} − λ_{1}λ_{2}^{5} + λ_{2}^{6})

− \frac{3(λ_{1} + λ_{2})^{8}}{8 (3λ_{1}^{6} + 11λ_{1}^{4}λ_{2}^{2} + 2λ_{1}^{3}λ_{2}^{3} + 9λ_{1}^{2}λ_{2}^{4} − 2λ_{1}λ_{2}^{5} + λ_{2}^{6})^{2}} ·

· \frac{1}{(λ_{1}^{2} + λ_{1}λ_{2} + λ_{2}^{2})^{2} (λ_{1}^{2} + λ_{2}^{2})^{2} λ_{1}^{8}λ_{2}^{6}} ·

· (2λ_{1}^{18} − 4λ_{1}^{17}λ_{2} + 57λ_{1}^{16}λ_{2}^{2} − 108λ_{1}^{15}λ_{2}^{3} + 508λ_{1}^{14}λ_{2}^{4}

−428λ_{1}^{13}λ_{2}^{5} + 2152λ_{1}^{12}λ_{2}^{6} − 156λ_{1}^{11}λ_{2}^{7} + 4784λ_{1}^{10}λ_{2}^{8}

+172λ_{1}^{9}λ_{2}^{9} + 4942λ_{1}^{8}λ_{2}^{10} − 612λ_{1}^{7}λ_{2}^{11} + 2676λ_{1}^{6}λ_{2}^{12}

−772λ_{1}^{5}λ_{2}^{13} + 872λ_{1}^{4}λ_{2}^{14} − 340λ_{1}^{3}λ_{2}^{15} + 126λ_{1}^{2}λ_{2}^{16}

−56λ_{1}λ_{2}^{17} + 9λ_{2}^{18}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2}

and apply the maximum principle. □

Lemma 11.5. Under the assumptions of Theorem 11.1, \frac{d}{dt}X = −F ν, the maximum principal curvature of M_{t} is non-increasing in time.

Proof. Consider M_{ij} := h_{ij} − μg_{ij}, where μ is chosen such that M_{ij} ≤ 0 initially. We compute according to (2.5) and Lemma 2.1

\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl} =F^{kl}h_{k}^{a}h_{al} · h_{ij} − F^{kl}h_{kl} · h_{i}^{a}h_{aj} − F h_{i}^{k}h_{kj} + 2μF h_{ij}
+ F^{kl, rs}h_{kl; i}ξ^{i}h_{rs; j}ξ^{j}.

Assume that ξ is a zero eigenvalue of (M_{ij}), h_{ij}ξ^{j} = μg_{ij}ξ^{j}, with g_{ij}ξ^{i}ξ^{j} = 1. We may assume, that in our coordinate system, we have ξ = (1, 0) and (h_{ij}) = ( \begin{matrix}μ & 0 \\ 0 & λ \end{matrix} ) with 0 < λ ≤ μ. Normal velocities F as in Theorem 11.1 are concave, so

F^{kl, rs}h_{kl; i}ξ^{i}h_{rs; j}ξ^{j} ≤ 0.

We estimate

• (\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl}) ξ^{i}ξ^{j} ≤ −\frac{(λ+μ)(3μ−λ)}{λ^{2}} ≤ 0 for F = −\frac{H^{2}}{K^{2}} ,
• (\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl}) ξ^{i}ξ^{j} ≤ −\frac{−2λμ+λ^{2}+3μ^{2}}{λ^{2}} ≤ 0 for F = −\frac{|A|^{2}}{K^{2}} ,
• (\frac{d}{dt}M_{ij} − F^{kl}M_{ij; kl}) ξ^{i}ξ^{j} ≤ −\frac{2(λ+μ)^{2}(2μ−λ)}{μλ^{3}} ≤ 0 for F = −\frac{H^{3}}{K^{3}} .

Now the claim follows directly from the maximum principle for tensors. □

Lemma 11.6. Under the assumptions of Theorem 11.1, we obtain a lower bound on the principal curvatures similarly to Lemma 9.5.

Proof. We proceed as in the proof of Lemma 9.5. In a critical point of \frac{−F}{2R−⟨X, ν⟩}, we compute the following evolution equation for a family of surfaces

\begin{matrix}−\frac{1}{H} & \frac{(λ_{1} − λ_{2})^{2}}{(λ_{1} + λ_{2})λ_{1}λ_{2}} \\ −\frac{1}{H} & \frac{(λ_{1}^{2} + λ_{2}^{2}) (λ_{1} − λ_{2})^{2}}{(λ_{1} + λ_{2})λ_{1}^{3}λ_{2}^{3}} \\ −\frac{H}{K} & \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}} \end{matrix}

Table 1. More monotone quantities

flowing according to \frac{d}{dt}X = −F ν

\frac{d}{dt} log \frac{−F}{2R − ⟨X, ν⟩} − F^{ij} (log \frac{−F}{2R − ⟨X, ν⟩})_{; ij} =\frac{1}{2R − ⟨X, ν⟩}·
· (2R · F^{ij}h_{i}^{k}h_{kj} − F^{ij}h_{ij} − F ) .

We compute this explicitly for F as in Theorem 11.1 and obtain in an increasing maximum of \frac{−F}{2R−⟨X, ν⟩}

• 8R − \frac{H}{K} ≥ 0 for F = −\frac{H^{2}}{K^{2}} ,
• 4HR − \frac{|A|^{2}}{K} ≥ 0 for F = −\frac{|A|^{2}}{K},
• 6R − \frac{H}{K} ≥ 0 for F = −\frac{H^{3}}{K^{3}} .

Our monotone quantities and Lemma 11.5 imply that our surfaces are pinched, i. e. that \frac{λ_{1}}{λ_{2}} + \frac{λ_{2}}{λ_{1}} is uniformly bounded above. So we obtain there that λ_{i} ≥ \frac{1}{cR}. □

Lemma 11.7. Consider one of the flow equations of Theorem 11.1, \frac{d}{dt}X = −F ν, and a solution as in this theorem. Let γ be such that F is positive homogeneous of degree −γ. Then there exists a constant α ≫ 1 such that |F | ≥ \frac{1}{α}|X|^{γ} during the flow. More precisely, there exists α ≫ 1 such that |X|^{γ} + αF remains non-positive during the flow, if this quantity is negative initially.

Proof. For F positive homogeneous of degree minus two, we get F^{ij}h_{ij} = −2F and obtain at a point, where |X|^{2} + αF = 0,

\frac{d}{dt} (|X|^{2} + αF ) − F^{ij} (|X|^{2} + αF )_{; ij} = − 6F ⟨X, ν⟩ − 2F^{ij}g_{ij} + αF F^{ij}h_{i}^{k}h_{kj}
≤ − 6F \sqrt{α}\sqrt{−F} + αF F^{ij}h_{i}^{k}h_{kj}.

It is straightforward to check that for α ≫ 1 sufficiently large and F as assumed above, the right-hand side is non-positive.

If F = −\frac{H^{3}}{K^{3}} , we obtain similarly as above

\frac{d}{dt} (|X|^{3} + αF ) − F^{ij} (|X|^{3} + αF )_{; ij} ≤12|X|^{2} \frac{H^{3}}{K^{3}} − 6α\frac{H^{5}}{K^{5}} ≤ 0.

In both cases, the lemma follows from the maximum principle. □

11.2. Homogeneity minus one

In Table 1, we have collected some normal velocities F, positive homogeneous of degree minus one, and test functions w, such that max_{M_{t}} w is non-increasing during the flow of a closed strictly convex surface flowing according to

\frac{d}{dt}X = −F ν.

Theorem 11.8. For a family M_{t} of smooth compact strictly convex surfaces in ℝ^{3}, flowing according to \frac{d}{dt}X = \frac{1}{H} ν,

max_{M_{t}} \frac{(λ_{1} − λ_{2})^{2}}{2(λ_{1} + λ_{2})λ_{1}λ_{2}} ≡ max_{M_{t}} w

and

max_{M_{t}} \frac{(λ_{1}^{2} + λ_{2}^{2}) (λ_{1} − λ_{2})^{2}}{8(λ_{1} + λ_{2})λ_{1}^{3}λ_{2}^{3}} ≡ max_{M_{t}} ŵ

are non-increasing in time.

Proof. According to Section 6, we obtain in a critical point of w

\frac{d}{dt}w − F^{ij}w_{; ij} = − \frac{(λ_{1}^{2} + 4λ_{1}λ_{2} + λ_{2}^{2}) (λ_{1} − λ_{2})^{2}}{2(λ_{1} + λ_{2})^{3}λ_{1}λ_{2}}

− \frac{2 (5λ_{1}^{2} + 2λ_{1}λ_{2} + λ_{2}^{2}) λ_{2}}{(λ_{1} + 3λ_{2})^{2}(λ_{1} + λ_{2})λ_{1}^{5}} · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2}

and apply the maximum principle.

Similarly, we obtain in a critical point of ŵ

\frac{d}{dt}ŵ − F^{ij}ŵ_{; ij} = − \frac{(3λ_{1}^{4} − 2λ_{1}^{2}λ_{2}^{2} + 3λ_{2}^{4}) (λ_{1} − λ_{2})^{2}}{8(λ_{1} + λ_{2})^{3}λ_{1}^{3}λ_{2}^{3}}

− \frac{1}{2 (3λ_{1}^{3} + 3λ_{1}^{2}λ_{2} − λ_{1}λ_{2}^{2} + 3λ_{2}^{3})^{2} (λ_{1} + λ_{2})^{3}λ_{1}^{7}λ_{2}} ·

· (9λ_{1}^{10} − 9λ_{1}^{8}λ_{2}^{2} + 96λ_{1}^{7}λ_{2}^{3} − 38λ_{1}^{6}λ_{2}^{4} + 96λ_{1}^{5}λ_{2}^{5} + 30λ_{1}^{4}λ_{2}^{6}

+45λ_{1}^{2}λ_{2}^{8} + 27λ_{2}^{10}) · h_{11; 1}^{2}

+ (…) · h_{22; 2}^{2}

and apply the maximum principle once again. □

Theorem 11.9. For a family M_{t} of smooth compact strictly convex surfaces in ℝ^{3}, flowing according to \frac{d}{dt}X = \frac{H}{K} ν,

max_{M_{t}} \frac{(λ_{1} − λ_{2})^{2}}{4λ_{1}^{2}λ_{2}^{2}} ≡ max_{M_{t}} w

is non-increasing in time.

Proof. According to Section 6, we obtain in a critical point of w

\frac{d}{dt}w − F^{ij}w_{; ij} = − \frac{(λ_{1} − λ_{2})^{2}}{λ_{1}^{2}λ_{2}^{2}} − \frac{2}{λ_{1}^{6}} · h_{11; 1}^{2} + (…) · h_{22; 2}.

and apply the maximum principle. □

For these flow equations, convergence to infinity and convergence to a sphere after rescaling have been proved before for hypersurfaces [26, 60, 61]. Monotone quantities as mentioned above might at most be useful to improve the convergence rate. Inverse mean curvature was used to prove the Penrose inequality in general relativity [34]. Our techniques might also apply to surfaces expanding in the asymptotically flat manifolds considered there.

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