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\begin{document}
%%%%% 16.01.1999 %%%%%
\title{A Splitting Theorem\\ for directed minimal Cones}
\author{Oliver~C.~Schn\"urer}
\date{August 7, 1999}
\maketitle

%%%%% 02.08.1999 %%%%%
\begin{abstract}
This paper contains a Splitting Theorem which states
that directed minimal cones in $\r1$ are of the
form $\R^k\times C_0$, where $C_0$ is an undirected
cone or a half-line. The Splitting Theorem can be
applied to prove regularity of minimal cones.
\end{abstract}

%%%%% 02.08.1999 %%%%%
\section{Introduction}
The blow-up cone of a minimal set around a point of the
measure-theoretical boundary is related to the
differentiability of the measure-theoretical boundary
in the following way: The blow-up cone is a half-space
iff the boundary is analytic in a neighborhood of
the respective point.
Therefore the study of minimal cones which arise as
blow-up cones and their structure can be used
to deduce regularity of the measure-theoretical
boundary of minimal sets.

We prove a Splitting Theorem for minimal cones which
describes the structure of minimal cones. In a former
paper \cite{ocs} a regularity theorem for
directed minimal cones is proved; later on it was pointed out to
the author, that this theorem corresponds to a result of
De~Giorgi \cite{dg}.
Using the fact, that the set
of directions of a set together with the origin is a closed
cone which is also closed under addition, the regularity
theorem can be slightly generalized in the following way:

{\it Let $C\subset\r1$ be an open nontrivial minimal cone with
its vertex in the origin. Suppose $C$ has $k$ linearly
independent directions, $k\in\N$. Then $C$ is a half-space
provided $n+1-k\le7$ or
$H^{k}(\partial C\setminus\partial^\star C)=0$.}

The Splitting Theorem extends this result in the
following way: If $C$ as above has $k$ linearly independent
directions, then $C$ is of the form $C_0\times\R^k$, where
$C_0$ is a minimal cone.

The results can be applied to those minimal cones which arise
in \cite{cg} as discussed for the dimension $8$ in \cite{ocs},
so that under the same assumptions as stated in \cite{cg} we
can prove the existence of smooth closed hypersurfaces of
prescribed mean curvature homeomorphic to $S^n$ for $n\le7$,
provided there are barriers.

The author would like to thank Prof.~Dr.~C.~Gerhardt for
many stimulating discussions about minimal cones and
their applications.

This paper is organized as follows:
Section \ref{defkap} contains especially the definition of
a directed set and section \ref{splittingkap} contains
the Main Theorem.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 20.11.98 %%%%%
\section{Preliminaries}\label{defkap}
%%%%% 08.07.1999 %%%%%
\begin{bez}{\rm
The characteristic function of a set $M$ will be
denoted by $\chi_M$. If we are concerned with coordinates in $\R^{n+1}$,
$(\hat x,t)$ stands for $(x^1,\ldots,x^n,t)$, $x=(\hat x,x^{n+1})$
also summarizes the first $n$ coordinates to $\hat x$.
Assume in the whole paper $n\ge2$.}
\end{bez}

%%%%% 17.01.1999 %%%%%
\begin{df}
{\rm The definition of a cone, a subgraph, a minimal set,
a singularity, a singular or regular minimal cone,
the (measure-theoretical) boundary and of the reduced boundary
are as usual, cf.\ e.~g.\ \cite{giusti}.}

A cone $C\subset\r1$ which is different from $\emptyset$ and
$\r1$ is called nontrivial.
\end{df}

%%%%% 29.11.98 %%%%%
\begin{df}[directed set]
Let $E\subset \R^{n+1}$ be a measurable set,
$y\in\R^{n+1}\setminus\{0\}$.
$$D_{\rho}(E, x):=
  \omega_{n+1}^{-1}\rho^{-n-1}\int\limits_{B_{\rho}(x)}\chi_E$$
is called approximative density of $E$ in $x$ with respect
to the radius $\rho$.
\par
A set $E$ is called directed with respect to $y$, if the maps
$$f_{x,\rho}:\R\to [0,1],\quad t\mapsto D_\rho(E,x+ty)$$
are monotone decreasing with respect to $t$ for any
$x\in\R^{n+1}$, $\rho>0$, such $y$ is called direction of the
set.
\end{df}

%%%%% 08.07.1999 %%%%%
\begin{rem}
{\rm For a measurable set $M$ we have the definition
(cf.\ e.~g.\ \cite{giusti})
$$x\in M\Longleftrightarrow \exists\rho>0:\quad
  \int\limits_{B_\rho(x)}\chi_M=\int\limits_{B_\rho(x)}1,$$
so it seems natural to us to introduce the above definition
of a directed set which, however, corresponds to the notation
of a subgraph as stated in the following remark. The proof
of this remark is omitted here, because it is merely technical
and can be found in \cite{ocs}.}
\end{rem}

%%%%% 28.11.98 %%%%%
\begin{bem}\label{subgleichger}{\rm
For a measurable set $M\subset\R^{n+1}$ such that
$H^{n+1}(\partial M)=0$ the following two statements
are equivalent:\\
\quada {\rm (i)} $M$ is the subgraph of a measurable function $u$,\\
\quada {\rm (ii)} $M$ is directed with respect to $e_{n+1}$.
\par
The measurable function $u$ in (i) is given by
$$u(\hat x):=\sup\{t\in\R:(\hat x,t)\in M\}.$$}
\end{bem}

%%%%% 28.11.98 %%%%%
\begin{lem}\label{negkomplementlemma}
Let $M\subset\R^{n+1}$ be a measurable set which is
directed with respect to $v$. Then  $\complement M$ and
$-M\equiv\{x\in\R^{n+1}:-x\in M\}$ are directed
with respect to $-v$. Especially $-\complement M$ is
again directed with respect to $v$.
\end{lem}
\bew
The statement follows at once from the equations
$$D_\rho(M,y)+D_\rho(\complement M,y)=1$$
and
$$D_\rho(-M, -x-\tau v)=D_\rho(M, x+\tau v)$$
which are valid for any $x\in\r1$ and any $\rho>0$.
\qed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 11.01.1999 %%%%%
\section{Splitting Theorem}\label{splittingkap}

%%%%% 17.01.1999 %%%%%
\begin{lem}\label{directionremainslemma}
Let $M\times\R\subset\R^n\times\R$ be a measurable set which
is directed with respect to $(\hat x,t)$.
If $\hat x\not=0$, then $M$ is directed with respect to $\hat x$.
\end{lem}
\bew
Since $M\times\R$ is directed with respect to $(\hat x,t)$, we
deduce in view of \cite[Theorem 1.3.6]{ziemer}
$$\int\limits_{\hat{B}_\rho(\hat z)\times[a,b]}\chi_{M\times\R}\ge
  \int\limits_{\hat{B}_\rho(\hat z+\sigma\hat x)
  \times[a+\sigma t,b+\sigma t]}\chi_{M\times\R}$$
for $\hat z\in\R^n$, $\rho>0$, $-\infty<a<b<+\infty$, $\sigma>0$
and $\hat B_\rho(\hat z)=\{\hat x\in\R^n:|\hat x-\hat z|<\rho\}$.
This inequality implies
$$\int\limits_{\hat{B}_\rho(\hat z)}\chi_{M}\ge
  \int\limits_{\hat{B}_\rho(\hat z+\sigma\hat x)}\chi_{M},$$
because we have
$$(b-a)\int\limits_{\hat{B}_\rho(\hat z)}\chi_{M}=
  \int\limits_{\hat{B}_\rho(\hat z)\times[a,b]}\chi_{M\times\R}$$
and an equality of the same kind is valid
for the other integrals above.
Since $\hat z\in\R^n$, $\rho>0$ and $\sigma>0$ were chosen
arbitrarily, the statement follows.
\qed

%%%%% 11.01.1999 %%%%%
The following Splitting Theorem contains only the case of a
singular cone. For regular cones, however, there is no need
of such a theorem, because regular cones are half-spaces.

%%%%% 02.08.1999 %%%%%
\begin{thm}[Splitting Theorem for directed minimal cones]
\hfill\\
Let $C\subset\r1$ be a singular minimal cone which has
$k$ linearly independent directions. Then there is a singular
minimal cone $C_0\subset\R^{n+1-k}$ such that $C=C_0\times\R^k$
after a suitable rotation and translation of $C$.
\end{thm}
\bew
\OE we may assume that the vertex of the cone and the origin
coincide. In view of Lemma \ref{directionremainslemma}
we have to prove the Theorem only for $k=1$. Then the
statement for $k>1$
follows by induction. Assume \oe that $C$ is directed
with respect to $e_{n+1}$.

Define $u:\R^n\to[-\infty,+\infty]$ as in \cite{mir}
to be a measurable function such that $C=\sub u$.
Then $u$ is positive homogeneous of degree 1, because $C$ is
a cone.
According to \cite[Theorem 15.5]{giusti} the set
$P:=\{\hat x\in\R^n:u(\hat x)=+\infty\}$ is a minimal set.
Since $u$ is positive homogeneous of degree 1 we deduce that
$P$ is a minimal cone with its vertex in the origin. Define
also $N:=\{\hat x\in\R^n:u(\hat x)=-\infty\}$. $N$ is also
a minimal cone with its vertex in the origin.

The case $N=P=\emptyset$ cannot occur:
$N=P=\emptyset$ implies $u\in L^\infty_{loc}(\R^n)$
(\cite[Proposition 16.7]{giusti}). According to
\cite[Theorem~1, p.~317]{gia1} we deduce $u\in BV(\R^n)$.
Finally \cite{cg1} implies $u\in C^{0,1}(\R^n)$ and
$u\in C^\omega(\R^n)$ follows. This would imply that
$C$ is regular, which would be a contradiction.

According to Lemma \ref{negkomplementlemma}
we can assume \oe $P\not=\emptyset$, because in the case
$P\not=\emptyset$ we replace $C$ by $-\complement C$ and
proceed in the same way as we do now.

We also remark that $P$ is different from a half-space;
otherwise we get $P\times\R\subset C$ and in view of
\cite[Theorem 15.5]{giusti} $P\times\R=C$, a contradiction
to the fact that $C$ is a singular cone. In the same
way it can be shown that $N$ is different from a half-space.

$C\not=\r1$ implies
$P\not=\R^n$. According to the definition of
$P$ we deduce $P\times\R\subset C$.
\cite[Example 16.2]{giusti} implies that $P\times\R$ is
minimal and therefore a singular minimal cone.
Now \cite[Theorem 2.4]{omc}, a maximum principle for minimal
cones, states, that two minimal cones are equal if they
have the same vertex and one of them contains the other one.
We apply this theorem and get
$C=P\times\R$. Define $C_0:=P$ and the statement follows.
\qed

%%%%% 17.01.1999 %%%%%
\begin{cor}
Let $C\subset\r1$ be a singular minimal cone with its
vertex in the origin. Then we have, after a suitable rotation,
$C=C_0\times\R^k$, where $k\in\N$ and $C_0$ is a singular minimal
non-directed cone. In the exeptional case, $k=0$, we have of course
$C=C_0$.
\end{cor}

%%%%% 17.01.1999 %%%%%
\begin{rem}
{\rm The Splitting Theorem contains the
Regularity Theorem (\cite{ocs}):}
\end{rem}

%%%%% 17.01.1999 %%%%%
\begin{cor}
Let $C\subset\r1$ be a nontrivial minimal cone with
its vertex in the origin. Suppose $C$ has $k$ linearly
independent directions, $k\in\N$. Then $C$ is a half-space
provided $n+1-k\le7$ or
$H^{k}(\partial C\setminus\partial^\star C)=0$.
\end{cor}
\bew
Assume the statement were false and let $C\subset\r1$ be
a counterexample. Then the Splitting Theorem yields (after
a suitable rotation of $C$) $C=C_0\times\R^k$ where $C_0$
is a singular minimal cone in $\R^{n+1-k}$.

Since $n+1-k\le7$ contradicts the non-existence of singular
minimal cones up to $\R^7$ (\cite{giusti}),
this case does not occur.

In the second case we deduce
$H^0(\partial C_0\setminus\partial^\star C_0)\ge1$, because
$C_0$ is a singular minimal cone. This implies
\begin{eqnarray*}
H^k(\partial(C_0\times\R^k)\setminus\partial^\star(C_0\times\R^k))
&=&H^k((\partial C_0\setminus\partial^\star C_0)\times\R^k)\\
&\ge&H^0(\partial C_0\setminus\partial^\star C_0)\ge1\\
&>&0=H^k(\partial C\setminus\partial^\star C)\\
\end{eqnarray*}
contradicting $C=C_0\times\R^k$.
Thus the statement is proved.
\qed

%%%%% 05.05.1999 %%%%%
\begin{cor}\label{onepointcor}
Let $C\subset\r1$ be an open minimal cone with its vertex in
the origin. If $C$ is nontrivial and directed with respect to
$e_{n+1}$ and there exists a $t>0$ such that $(0,t)\in\complement C$
or $(0,-t)\in C$, then $\partial C$ is a hyperplane.
\end{cor}
\bew
In view of Lemma \ref{negkomplementlemma} we can assume \oe
$(0,-t)\in C$. So there exists $\rho>0$ such that
$B_\rho((0,-t))\subset C$.
Thus $u(\hat x)>-t$ holds for $|\hat x|<\rho$,
where $u$ is defined as in Remark \ref{subgleichger}
such that $\sub u=C$.
In view of the homogeneity of $u$ this implies
$u(\hat x)>-\infty$ for $\hat x\in\R^n$.
Therefore we deduce
$N=\emptyset$, where the notation from the proof of the
Splitting Theorem has been used and will be used in the rest
of this proof. Since $P=\emptyset$ implies the statement as
shown in the proof of the Splitting Theorem, we may assume
$\emptyset\not=P\not=\R^n$. Following again the proof of the
Splitting Theorem we deduce $C=P\times\R$. We choose now
$\hat x\in\complement P$. Since $\complement P$ is a cone,
we have $\tau\hat x\in\complement P$ for $\tau>0$ and
$(\tau\hat x,-t)\in\complement C$, but $(\tau\hat x,-t)$
converges to $(0,-t)\in C$, if $\tau\to0$. This is a
contradiction, because $C$ is an open set.
\qed

%%%%% 05.05.1999 %%%%%
\begin{rem}
{\rm The assumptions of Corollary \ref{onepointcor} imply
also $\langle\nu,e_{n+1}\rangle<0$, where $\nu$ is the inner
normal of $C$.}
\end{rem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 24.11.98 %%%%%
\begin{thebibliography}{9}
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\bibitem{cg} C.~Gerhardt: {\it Closed Hypersurfaces of prescribed
  Mean Curvature in locally conformally flat Riemannian Manifolds,\/}
  J.~Diff.~Geom.~{\bf 48} (1998), 587-613.
\bibitem{cg1} C.~Gerhardt: {\it On the Regularity of Solutions to
  Variational Problems in $BV(\Omega)$,\/} Math.~Z.~{\bf 149} (1976),
  281-286.
\bibitem{gia1} M.~Giaquinta, G.~Modica, J.~Sou\v{c}ek:
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  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3.~Folge,
  Band~37, Springer, Berlin Heidelberg New~York, 1998, 736~pages.
\bibitem{giusti}
  E.~Giusti: {\it Minimal Surfaces and Functions of Bounded Variation,\/}
  Monographs in Math.\ Vol.~80, Birkh\"auser, Boston Basel Stuttgart,
  1984, 240~pages.
\bibitem{omc}
  E.~Gonzalez, U.~Massari, M.~Miranda:
  {\it On minimal Cones,\/} Appl.\ Anal.~{\bf 65} (1997), 135-143.
\bibitem{mir} M.~Miranda:
  {\it Superficie minime illimitate,\/}
  Ann.\ Sc.\ Norm.\ Sup.\ Pisa (4) {\bf 4} (1977), 313-322.
\bibitem{ocs} O.~C.~Schn\"urer:
  {\it Regularity of directed minimal Cones}, 1998, preprint.
\bibitem{ziemer} W.~P.~Ziemer: {\it Weakly differentiable Functions,\/}
  Graduate Texts in Mathe\-ma\-tics Vol. 120, Springer,
  Berlin Heidelberg New~York, 1989, 308~pages.
\end{thebibliography}

%%%%% 08.07.1999 %%%%%
Oliver C.~Schn\"urer\\
Ruprecht-Karls-Universit\"at,
Institut f\"ur Angewandte Mathematik,\\
Im Neuenheimer Feld 294,
D-69120 Heidelberg,
Germany\\
E-mail address: {\tt Oliver.Schnuerer@urz.uni-heidelberg.de}
\end{document}
