STABILITY OF EUCLIDEAN SPACE UNDER RICCI FLOW

OLIVER C. SCHNÜRER, FELIX SCHULZE, AND MILES SIMON

Abstract. We study the Ricci flow for initial metrics which are C^{0} small perturbations of the Euclidean metric on ℝ^{n}. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow.

1. Introduction

In this paper we investigate the evolution of a family of complete non-compact Riemannian manifolds (ℝ^{n}, g(t)) under Ricci flow

(RF)  \{ \begin{matrix}\frac{∂}{∂t}g(t) = −2 Ric(t) & in ℝ^{n} × (0, ∞), \\ g(t) → g_{0} & in C_{loc}^{0} as t ↘ 0, \end{matrix}

where g_{0} is a given initial metric on ℝ^{n}. We study the long-term behavior as t → ∞ of solutions to (RF) for initial metrics g_{0} which are C^{0}-close to the standard Euclidean metric h, in standard coordinates h_{ij} = δ_{ij}. For analytic reasons, it is convenient to study the Ricci harmonic map heat flow which is a variant of the Ricci-DeTurck flow

(1.1)  \{ \begin{matrix}\frac{∂}{∂t}g_{ij}(x, t) = −2R_{ij}(x, t) + ∇_{i}V_{j} + ∇_{j}V_{i} & in ℝ^{n} × (0, ∞), \\ g_{ij}(t) → (g_{0})_{ij} & in C_{loc}^{0} (ℝ^{n}) as t ↘ 0, \end{matrix}

where V_{i} = g_{ik} (^{g}Γ_{rs}^{k} −^{h}Γ_{rs}^{k}) g^{rs}, for a flat metric h, V_{i} = g_{ik}^{g}Γ_{rs}^{k}g^{rs}. If a family of metrics (g(t))_{t∈(0, ∞)} solves (1.1), we call it a solution to the h-flow with initial metric g_{0}. Precise definitions are to be found in Section 2. Note that the h-flow (1.1) and the Ricci flow (RF) are equivalent up to diffeomorphisms [12] in the case that the initial metric is smooth, and M is compact.

Definition 1.1. Let g_{1} and g_{2} denote two Riemannian metrics on a given manifold. We say that g_{1} is δ-fair to g_{2}, if

δ^{−1}g_{2} ≤ g_{1} ≤ δ g_{2} .

We call g_{1} ε-close to g_{2}, if g_{1} is (1 + ε)-fair to g_{2}.

Date: June 2007, revised January 2008.

2000 Mathematics Subject Classification. 53C44, 35B35.

Key words and phrases. Stability, Ricci flow.

We denote with M^{k}(ℝ^{n}, I) the space of families (g(t))_{t∈I} of sections in the space of Riemannian metrics on ℝ^{n} which are C^{k} on ℝ^{n} × I. Similarly, we define M^{∞}, M_{loc}^{k} and use M^{k}(ℝ^{n}) if the metric does not depend on t. We wish to point out that we use C^{k} on non-compact sets to denote the space, where derivatives of order up to k are in L^{∞}. We also use C_{loc}^{k}.

Our existence result reads as follows

Theorem 1.2. For every ε > 0 there exists 0 < ε_{0}(ε, n) ≤ ε, such that the following is true: If g_{0} ∈ M^{0}(ℝ^{n}) is a Riemannian metric on ℝ^{n} which is ε_{0}-close to the standard metric h = δ, then there exists a solution g ∈ M^{∞}(ℝ^{n}, (0, ∞)) ∩ M_{loc}^{0}(ℝ^{n}, [0, ∞)) to (1.1) such that g(t) is ε-close to h for all t. As t → ∞, the metrics g(t) converge subsequentially in M_{loc}^{∞}(ℝ^{n}) to a complete flat metric.

In this situation, our main theorem addresses convergence to the Euclidean background metric.

Theorem 1.3. Let g ∈ M^{∞}(ℝ^{n}, (0, ∞)) ∩ M_{loc}^{0}(ℝ^{n}, [0, ∞)) be a solution to (1.1). Assume that g(t) is ε̃(n)-close to the standard metric h for all t ≥ 0, for some ε̃(n) chosen sufficiently small, and that g_{0} is ε(r)-close to h on ℝ^{n} \ B_{r}(0) with ε(r) → 0 as r → ∞. Then

g(t) → h in C^{∞} as t → ∞.

That is:

sup_{ℝ^{n}} |g(t) − h| → 0

as t → ∞, and

sup_{ℝ^{n}} |^{h}∇^{k}g(t)| → 0

as t → ∞ for all k ∈ {1, 2 …}.

In this paper we will be concerned with the geometric quantities

(1.2)  φ_{m} := g^{a_{1}b_{1}}h_{b_{1}a_{2}}g^{a_{2}b_{2}}h_{b_{2}a_{3}} ⋯ g^{a_{m}b_{m}}h_{b_{m}a_{1}}

and

(1.3)  ψ_{m} := g_{a_{1}b_{1}}h^{b_{1}a_{2}}g_{a_{2}b_{2}}h^{b_{2}a_{3}} ⋯ g_{a_{m}b_{m}}h^{b_{m}a_{1}}.

We use the Einstein summation convention, i. e. we sum over repeated upper and lower indices like a_{j}, b_{j}, 1 ≤ j ≤ m, from 1 to n. If (λ_{i}) are the eigenvalues of (g_{ij}) with respect to (h_{ij}), we get

φ_{m} = ∑_{i=1}^{n} \frac{1}{λ_{i}^{m}} and ψ_{m} = ∑_{i=1}^{n} λ_{i}^{m}.

In particular

(1.4)  φ_{m} + ψ_{m} − 2n = ∑_{i=1}^{n} (\frac{1}{λ_{i}^{m}} + λ_{i}^{m} − 2) = ∑_{i=1}^{n} \frac{1}{λ_{i}^{m}} (λ_{i}^{m} − 1)^{2} ≥ 0

is non-negative and vanishes precisely when λ_{i} = 1 for all i ∈ {1, …, n}.

Theorem 1.4. Let n ≥ 3 and g ∈ M_{loc}^{∞}(ℝ^{n}, [0, ∞)) be a solution to (1.1), satisfying all the conditions of Theorem 1.3. Then there exist m = m(n) ∈ ℕ and ε̃(n) > 0, such that the following holds. If 1 ≤ p < \frac{n}{2} can be chosen such that initially

(1.5)  ∫_{ℝ^{n}} (φ_{m} + ψ_{m} − 2n)^{p} < ∞,

or, equivalently,

∥g(0) − h∥_{L^{2p}} < ∞,

then there exists a smooth family (φ_{t})_{t≥0} of diffeomorphisms of ℝ^{n}, φ_{0} = id_{ℝ^{n}}, such that for g̃(t) := φ_{t}^{∗}g(t) the family (g̃(t))_{t≥0} is a smooth solution to (RF) satisfying

g̃(t) → (φ_{∞})^{∗}h in M^{∞}(ℝ^{n}) as t → ∞

for some smooth diffeomorphism φ_{∞} of ℝ^{n} which satisfies φ_{t} → φ_{∞} in C^{∞} (ℝ^{n}, ℝ^{n}) as t → ∞ and

|φ_{∞}(x) − x| → 0 as |x| → ∞.

Rewriting (1.5) in terms of a decay condition on ε(r) in the closeness condition for g_{0}, see Theorem 1.3, yields

Corollary 1.5. Let n ≥ 3 and g ∈ M_{loc}^{∞}(ℝ^{n}, [0, ∞)) be a solution to (1.1), satisfying all the conditions of Theorem 1.3. If g_{0} is ε_{0}-close to h for some ε_{0} = ε_{0}(n) > 0 sufficiently small, and in addition there exist constants C > 0 and ζ > 0 such that g_{0} is Cr^{−1−ζ}-close to h on ℝ^{n} \ B_{r}(0), then the same conclusions as in Theorem 1.4 hold.

Note that the solutions constructed in Theorem 1.2 satisfy the conditions of Theorem 1.3, but are not necessarily ε(r)-close to h at infinity. We point out that these solutions may depend on the subsequence chosen in the construction in Section 4. Note, however, that it is not clear, whether solutions to the initial value problem (1.1) are unique, as in general g_{0} does not have bounded curvature or a well-defined Riemannian curvature tensor. Even if g(t) is ε(n)-close to h on ℝ^{n} for all t > 0, we do not know, whether such solutions are unique. If we study only solutions as constructed in Section 4, we can replace the ε(r)-closeness condition imposed on g_{0} by a considerably weaker integrability condition.

Theorem 1.6. Fix p ≥ 1, m = m(n) ∈ ℕ sufficiently large, and ε_{0} = ε_{0}(n) > 0 sufficiently small. Let g_{0} ∈ M^{0}(ℝ^{n}) be a Riemannian metric which is ε_{0}-close to the standard metric h. Let g ∈ M_{loc}^{∞} (ℝ^{n}, (0, ∞)) be a solution to (1.1) as constructed in Section 4. If for every δ > 0 the integral I_{δ}^{m,p}(0), as defined in Theorem 6.1, is finite, then

g(t) → h in M^{∞} (ℝ^{n}) as t → ∞.

In the situation of Theorem 1.6, if I_{0}^{m,p}(0) is finite, we can show that the diffeomorphisms φ_{t} stay bounded and converge in C_{loc}^{∞} to a limiting diffeomorphism φ_{∞}.

Outline. In order to prove stability of Euclidean ℝ^{n} under the h-flow, we proceed as follows.

If we perturb the metric on ℝ^{n}, such that the perturbed metric is C^{0}-close to our original metric and the perturbation is small enough, this is preserved for all times. The perturbation is measured in terms of the deviation of the eigenvalues (λ_{i}) of the perturbed metric with respect to the Euclidean metric from one. We have

(1 + ε)^{−1} ≤ λ_{i} ≤ 1 + ε for i = 1, …, n

everywhere for all times and 0 < ε = ε(n) provided that initially (1 + ε_{0})^{−1} ≤ λ_{i} ≤ 1 + ε_{0} for some 0 < ε_{0} ≤ ε_{0}(n).

We have interior estimates for the gradient of the metric evolving under DeTurck flow. If we assume that λ_{1} ≤ … ≤ λ_{n}, we get the estimate

|λ_{i}(x, t) − λ_{i}(y, t)| ≤ \frac{c}{\sqrt{t}} · d_{Eucl.}(x, y).

Thus, for large times, the eigenvalues λ_{i} are almost spatially constant in any Euclidean ball. It is, however, possible that these spatial constants vary in time. In particular, it is not clear up until now whether these constants converge as time tends to infinity.

Our perturbation of the Euclidean metric is small near infinity, i. e. all eigenvalues λ_{i} approach one near infinity. Note, however, that we do not impose a decay rate on our initial metric, at which the eigenvalues converge to one as we approach spatial infinity. Still, we obtain for all positive times that the eigenvalues converge to one at spatial infinity.

In order to get uniform control on the eigenvalues, we use the quantity considered in (1.4) for some m ∈ ℕ. It vanishes precisely when λ_{i} = 1 for all i and measures the deviation from λ_{i} = 1. The functions φ_{m} and ψ_{m} were initially considered by W.-X. Shi in [12]. The key estimate is to show essentially that ∫ (φ_{m} + ψ_{m} − 2n) is non-increasing in time. Here we integrate over the manifold at a fixed time. For technical reasons we have to consider a more complicated quantity. For details we refer to the respective proofs in Section 6.

This implies convergence of the eigenvalues λ_{i}(x, t) to 1 as t → ∞, uniformly in x. For if this were not the case, we could pick points (x_{k}, t_{k}) with t_{k} → ∞ such that at least one λ_{i}(x_{k}, t_{k}) differs significantly from 1. If t_{k} is large enough, we find a big ball, where at least one eigenvalue differs considerably from 1. This yields an arbitrarily large contribution to ∫ (φ_{m} + ψ_{m} − 2n) if t_{k} is big compared to max_{i} |λ_{i}(x_{k}, t_{k}) − 1|, contradicting the monotonicity of ∫ (φ_{m} + ψ_{m} − 2n). Thus λ_{i}(x, t) → 1, uniformly in x as t → ∞, follows.

Remark 1.7. Note that all the above results also hold if we replace ℝ^{n} by 𝕋^{k}×ℝ^{n−k} for n − k ≥ 1, where 𝕋^{k} is a flat k-dimensional torus.

In two space dimensions, Ricci flow is a conformal flow given by the evolution equation

\frac{∂}{∂t}g = −Rg,

where R is the scalar curvature, see [6]. In this situation, we obtain (for details see Appendix A)

Theorem 1.8. Let g_{0} = e^{−u_{0}}h, where h = δ is the standard Euclidean metric, and u_{0} ∈ C^{0} (ℝ^{2}) such that

sup_{ℝ^{2}\B_{r}(0)} |u| → 0 as r → ∞.

Then there exists a smooth solution (g(t))_{t∈(0,∞)}, g(t) = e^{−u(·,t)}h, to Ricci flow such that u(·, t) → u_{0} in C_{loc}^{0} as t ↘ 0. Furthermore, as t → ∞, we obtain that u(·, t) → 0 in C^{∞} (ℝ^{2}).

For two dimensional Ricci flow, L.-F. Wu studied long time behavior and convergence of solutions [16]. She studied initial complete metrics g_{0} = e^{u_{0}}h which have bounded curvature and satisfy e^{−u_{0}}|Du_{0}|^{2} < ∞. In this case, she showed that a long time solution to Ricci flow exists and that it “converges smoothly in the sense of modified subsequences” to a smooth limiting metric as t → ∞. A smooth family of metrics (g(t))_{t∈[0,∞)} on ℝ^{n} “converges smoothly in the sense of modified subsequences” to a metric l on ℝ^{n} as t → ∞ if there exist diffeomorphisms φ_{i} : ℝ^{n} → ℝ^{n} and a sequence t_{1} < t_{2} < … with t_{i} → ∞ as i → ∞, such that (φ_{i})^{∗}(g(t_{i})) → l smoothly on any fixed compact subset of ℝ^{n}. In particular, this does not imply uniform convergence on all of ℝ^{n}.

By imposing strong decay conditions on the curvature tensor at infinity, stability of flat space under Ricci flow was proved by W.-X. Shi [13]. Stability of compact flat manifolds under Ricci flow was studied by C. Guenther, J. Isenberg, D. Knopf and by N. Šešum [4, 15]. In the rotationally symmetric situation stability of flat Euclidean space was investigated by T. Oliynyk and E. Woolgar [10]. A. Chau and the first author obtained a stability result for the Kähler potential of stationary rotationally symmetric solitons of positive holomorphic bisectional curvature under Kähler-Ricci flow [1]. Uniqueness of solutions to Ricci flow with bounded curvature on non-compact manifolds is discussed in [2, 8]. J. Clutterbuck and the first two authors proved stability of convex rotationally symmetric translating solutions to mean curvature flow in [3]. Short time existence results for C^{0}-metrics were shown in [14] using similar techniques to this paper.

The rest of the paper is organized as follows. We introduce our notation in Section 2 and recall some evolution equations in Section 3. Is Section 4 we show the existence of solutions for all times. We prove interior closeness and a priori estimates in Section 5. In Section 6 we show that an integral quantity based on the expression in (1.4) is monotonically decreasing along the flow. Combining this with the interior gradient estimates, we obtain in Section 7 that the eigenvalues of g_{ij}(x, t) with respect to the background metric converge uniformly to one as t → ∞. An iteration scheme allows us to improve our gradient estimates for large times, see Section 8. In Section 9, we study the diffeomorphisms relating (RF) and (1.1), and show their convergence for large times. We address the proof of Theorem 1.6 in Section 10. Stability results for Ricci flow in two dimensions are addressed in Appendix A.

We want to thank Gerhard Huisken for fruitful discussions. The third author acknowledges support from SFB 647/B3 during his visits in Berlin. This paper is part of the research project “Stability of non-compact manifolds under curvature flows” of the first two authors within the priority program “Global Differential Geometry” SPP 1154 of the German research foundation DFG.

2. Notation

Let g = (g_{ij})_{1≤i, j≤n} be a Riemannian metric. By (g(t))_{t∈(0, ∞)}, we denote a family of metrics. We denote the inverse of (g_{ij}) by (g^{ij}) and use the Einstein summation convention for repeated upper and lower indices. If g_{1} and g_{2} are two metrics such that (g_{1})_{ij}ξ^{i}ξ^{j} ≤ (g_{2})_{ij}ξ^{i}ξ^{j} for all (ξ^{i}) ∈ ℝ^{n}, we denote this by g_{1} ≤ g_{2}. Unless otherwise stated, geometric quantities like covariant derivatives ∇_{i} and Christoffel symbols Γ_{ij}^{k} are computed with respect to the evolving metrics g(t). We use indices h to denote quantities depending on the metric h, e. g. ^{h}∇g denotes the covariant derivative of g with respect to h, i. e. a partial derivative. In short formulae, we also use ∇g instead. With the exception of the beginning of Section 4, h will always denote the standard metric on ℝ^{n}. On ℝ^{n} we will always use coordinates such that h_{ij} = δ_{ij}. A ball of radius r, centered at x, is denoted by B_{r}(x). We will only use Euclidean balls. The norm ^{h}| · | is computed with respect to the flat background metric h. We also denote it by | · |. The letter c denotes generic constants.

In formulae, where we use λ_{i} to denote the eigenvalues of g_{ij} with respect to h_{ij}, we will always assume that h_{ij} = δ_{ij} in the coordinate system chosen. Several times, we will use that ^{h}|g_{ij}(x, t) − g_{ij}(y, t)| ≤ A implies that |λ_{i}(x, t) − λ_{i}(y, t)| ≤ A if we assume that λ_{1}(·, t) ≤ … ≤ λ_{n}(·, t). Similarly, ^{h}|g_{ij}(x, t) − h_{ij}| ≤ A implies |λ_{i}(x, t) − 1| ≤ A.

3. Evolution Equations

In this section, we collect some evolution equations from [12] and state some direct consequences.

Assume that in appropriate coordinates, we have at a fixed point and at a fixed time h_{ij} = δ_{ij}, g_{ij} = diag(λ_{1}, λ_{2}, …, λ_{n}), λ_{i} > 0.

The evolution equation for the metric, computed with respect to a flat background metric, is, see [12, Lemma 2.1],

\frac{∂}{∂t}g_{ij} = g^{ab}^{h}∇_{a}^{h}∇_{b}g_{ij}
+ \frac{1}{2}g^{ab}g^{pq} (^{h}∇_{i}g_{pa}^{h}∇_{j}g_{qb} + 2^{h}∇_{a}g_{jp}^{h}∇_{q}g_{ib} − 2^{h}∇_{a}g_{jp}^{h}∇_{b}g_{iq}
−2^{h}∇_{j}g_{pa}^{h}∇_{b}g_{iq} − 2^{h}∇_{i}g_{pa}^{h}∇_{b}g_{jq}) .

Consider φ_{m} as defined in (1.2). The evolution equation of φ_{m} for a flat background metric h_{ij} is, see [12, Lemma 2.2, Equation (70)],

\frac{∂}{∂t}φ_{m} = g^{ab}^{h}∇_{a}^{h}∇_{b}φ_{m}

− m ∑_{i,j,a} \frac{1}{λ_{a}} [∑_{k=2}^{m} (\frac{1}{λ_{i}} )^{k} (\frac{1}{λ_{j}} )^{m+2−k}] (^{h}∇_{a}g_{ij})^{2}

− ∑_{i,q,k} \frac{m}{2λ_{i}^{m+1}λ_{q}λ_{k}} (^{h}∇_{k}g_{iq} +^{h}∇_{q}g_{ik} −^{h}∇_{i}g_{qk})^{2} .

We deduce for g(t) that is ε-close to h that

\frac{∂}{∂t}φ_{m} ≤ g^{ij}^{h}∇_{i}^{h}∇_{j}φ_{m}

− m(m − 1)(1 + ε)^{−(m+3)} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} .

From the evolution equation of the metric, we get for ψ_{m}, as introduced in (1.3), the evolution equation

\frac{∂}{∂t}ψ_{m} = mh^{a_{1}b_{1}}g_{b_{1}a_{2}}h^{a_{2}b_{2}}g_{b_{2}a_{3}} ⋯ h^{a_{m}b_{m}} \frac{∂}{∂t}g_{b_{m}a_{1}}
= mh^{a_{1}b_{1}}g_{b_{1}a_{2}}h^{a_{2}b_{2}}g_{b_{2}a_{3}} ⋯ h^{a_{m}b_{m}} (g^{ij}^{h}∇_{i}^{h}∇_{j}g_{b_{m}a_{1}})

+ c(n)m \underbrace{h^{−1} ∗ ⋯ ∗ h^{−1}}_{m} ∗ \underbrace{g ∗ ⋯ ∗ g}_{m−1} ∗g^{−1} ∗ g^{−1} ∗^{h}∇g ∗^{h}∇g

= g^{ij}^{h}∇_{i}^{h}∇_{j}ψ_{m}

− m ∑_{i, j, a} [ ∑_{k=0}^{m−2} λ_{i}^{k}λ_{j}^{m−2−k}] \frac{1}{λ_{a}} (^{h}∇_{a}g_{ij})^{2}

+ c(n)m \underbrace{h^{−1} ∗ ⋯ ∗ h^{−1}}_{m} ∗ \underbrace{g ∗ ⋯ ∗ g}_{m−1} ∗g^{−1} ∗ g^{−1} ∗^{h}∇g ∗^{h}∇g,

where ∗ indicates contractions and linear combinations of contractions. The factor c(n) indicates that the number of the respective terms depends only on n. Therefore, we deduce that

\frac{∂}{∂t}ψ_{m} ≤ g^{ij}^{h}∇_{i}^{h}∇_{j}ψ_{m}

− m(m − 1)(1 + ε)^{−(m−1)} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2}

+ c(n)m(1 + ε)^{m+1} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} .

4. Existence

In this section we prove long time existence of the h-flow for a smooth initial metric g_{0} which is ε-close to h = δ. Short time existence for Ricci-DeTurck flow on noncompact manifolds was first proved in [12]. Short time existence for an arbitrary smooth background metric h with bounded curvature, when g_{0} is ε-close to h, was proved in [14], using similar techniques to those of W.-X. Shi. We also reprove the short time existence here for completeness. (note: some of the citations in [14] in the proof of the short time existence were incorrect.)

Lemma 4.1. Let (g(t))_{t∈[0,T)} be a smooth ε-close solution to the h-flow with flat background metric h = δ on a closed ball D ⊂ ℝ^{n} with 0 < ε ≤ ε(n) and g(·, t)|_{∂D} = h(·)|_{∂D}. Then

sup_{D×[0,T)} |^{h}∇^{m}g|^{2} ≤ c(n, m, D, T, g(0)).

Proof. We proceed exactly as in the proof of Shi [12, Lemma 3.1]: where he uses g̃, we use h. The only other minor difference is that the dependence on g(0) does not explicitly appear in Shi’s paper [12, Lemma 3.1]. This is because he has g̃ = g(0). □

Lemma 4.2. Let (g(t))_{t∈[0,T)} be a smooth solution to the h-flow on B_{i}, i ∈ ℕ, where h is the standard metric on ℝ^{n}. Then for all ε > 0 there exists 0 < ε_{0}(n, ε) < ε, such that the following holds. If g(0) is ε_{0}-close to h and g(t) = h on ∂B_{i} × [0, T), then g(t) is ε-close to h for all t ∈ [0, T).

Proof. Consider Φ := φ_{m} + ψ_{m} − 2n as in (1.4). It vanishes on ∂D × [0, T). The evolution equations in Section 3 imply that

(4.1)  \frac{∂}{∂t}Φ ≤ g^{ij}^{h}∇_{i}^{h}∇_{j}Φ

as long as g(t) is ε̃-close to h for ε̃ > 0 satisfying

c(n)(1 + ε̃)^{2m} ≤ (m − 1).

Choose ε̃ = ε̃(n) and m = m(n) accordingly.

Fix δ > 0 so that Φ ≤ 2δ implies, see (1.4), that the metric is ε-close to h, in particular ε̃-close to h. Now fix ε_{0} = ε_{0}(n) > 0 such that ε_{0} < ε and so that Φ ≤ δ for every metric which is ε_{0}-close to h.

Consider the maximal time interval I ⊂ [0, T) on which Φ ≤ 2δ. We may assume that I = [0, τ] with τ < T. This implies that (g(t))_{t∈[0,τ]} is ε̃-close to h (even ε-close to h). According to (4.1) and the maximum principle, max_{x∈D} Φ(x, t) is non-increasing

in t for all t ∈ [0, τ]. Thus Φ(τ) ≤ Φ(0) ≤ δ. This contradicts the choice of τ. Therefore I = [0, T) and (g(t))_{t∈[0,T)} is ε-close to h. □

In the remainder of this section, we will denote with ε_{0}(n, ε) > 0 the constant in Lemma 4.2. Note that although we allow for arbitrary ε > 0, it follows from the proof that ε_{0}(n, ε) ≤ \overline{ε} for some small \overline{ε} = \overline{ε}(n) > 0.

Theorem 4.3. Let ε > 0 be given, and g_{0} be smooth and ε_{0}(ε, n)-close to h = δ on a closed ball D ⊂ ℝ^{n}. Assume that g_{0} = h near ∂D. Then there exists a unique smooth solution (g(t))_{t∈[0,∞)} to the h-flow with g(0) = g_{0} and g(t)|_{∂D} = h|_{∂D}. Furthermore, (g(t))_{t∈[0,∞)} is ε-close to h.

Proof. Lemma 4.2 implies that a prospective solution g(t) is ε-close to h as long as it exists. Therefore, we can apply the a priori bounds of Lemma 4.1 on a bounded time interval. We may then use the same arguments as in [9, Theorem 7.1, Chapter VII] to show that a smooth solution exists (the argument used there is based on the Leray-Schauder fixed point argument of [9, Theorem 6.1 of Chapter V]). Note that this argument works for every finite time interval. □

Theorem 4.4. Let g ∈ M^{∞} (B_{R}, [0, T]) be ε̃-close to h, solving (1.1), where ε̃ = ε̃(n) > 0 is sufficiently small and h = δ. Then

sup_{B_{\frac{R}{2}} ×[0,T]} |^{h}∇^{m}g|^{2} ≤ c(m, n, g(0)|_{B_{R}}, T).

Proof. This follows using the arguments of Shi [12, Lemma 4.1 and Lemma 4.2]. □

Theorem 4.5. Let ε > 0 be given, and g_{0} be smooth and ε_{0}(n, ε)-close to the standard Riemannian metric h = δ on ℝ^{n}. Then there exists a ε-close solution g ∈ M_{loc}^{∞} (ℝ^{n}, [0, ∞)) to the h-flow (1.1).

Proof. Let B_{i} be the balls of radius i and center 0 (with respect to the Euclidean metric). Set ^{i}g_{0} = η_{i}g_{0} + (1 − η_{i})h, where η_{i} : ℝ^{n} → ℝ is smooth, and satisfies η_{i} = 0 on B_{i} − B_{i−1}, η_{i} = 1 on B_{i−2}, 0 ≤ η ≤ 1 and |∇^{m}η_{i}|^{2} ≤ c(m, n). Note that ^{i}g_{0} is ε_{0}-close to h. Let ^{i}g(t) ∈ M^{∞} (B_{i}, [0, ∞)) be the solution coming from the local existence theorem (Theorem 4.3) above. Using the interior estimates of Theorem 4.4 above, we see that the solutions all satisfy

sup_{B_{j}×[0,T]} |∇^{m} (\frac{d}{dt})^{k} (^{i}g)|^{2} ≤ c(j, g_{0}|_{B_{2j}}, m, k, n, T),

for all i big enough, and so, using a diagonal subsequence argument, we can find a subsequence which converges to a solution g ∈ M_{loc}^{∞} (ℝ^{n}, [0, ∞)) where the convergence ^{i}g → g is uniform on B_{j} × [0, T] for every fixed j ∈ ℕ and T ∈ (0, ∞) and g(0) = g_{0}. □

We do not know, whether the solution obtained in Theorem 4.5 is unique or depends on the subsequence chosen.

5. Interior Estimates

Note that the following lemma does not imply λ_{i}(x, t)−1 → 0 for |x| → ∞ uniformly in t, i. e.

lim_{R→∞} sup_{x∈ℝ^{n}\B_{R}} sup_{i∈{1, ..., n}} sup_{t∈[0,∞)} |λ_{i}(x, t) − 1|

may be nonzero.

Lemma 5.1. Let g ∈ M_{loc}^{∞} (ℝ^{n}, (0, ∞)) be a solution to (1.1) with g_{0} as in Theorem 1.3. Then the eigenvalues (λ_{i}(x, t)) of (g_{ij}(x, t)) with respect to (h_{ij}(x)) uniformly tend to 1 in bounded time intervals,

lim_{|x|→∞} λ_{i}(x, t) = 1.

This Lemma is a direct consequence of the following interior closeness estimate.

Lemma 5.2. Let g ∈ M_{loc}^{∞} (ℝ^{n}, (0, ∞)) be a solution to (1.1) which is ε̃(n)-close to the Euclidean background metric h. Assume that g_{0} is ε-close to h on some ball B_{R}(x_{0}) for some ε ≤ ε̃(n)/2. Then there is a constant γ(n) > 0 such that g(t) is 2ε-close to h on B_{R/2}(x_{0}) for t ∈ [0, γR^{2}].

Proof. According to the estimates in Section 3, there exists ε(n) > 0 and m(n) ∈ ℕ such that

\frac{∂}{∂t}(φ_{m} + ψ_{m} − 2n) ≤ g^{ab}^{h}∇_{a}^{h}∇_{b}(φ_{m} + ψ_{m} − 2n) − ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} .

If the background metric h is the standard flat metric on ℝ^{n}, we define for λ > 0

g_{ij}^{λ}(x, t) = g_{ij} (\frac{x}{λ}, \frac{t}{λ^{2}} ) .

If g is a solution to the h-flow, g^{λ} is also a solution to the h-flow as h is the standard metric on ℝ^{n}. Moreover, g^{λ} is ε-close to h, if g is ε-close to h.

By scaling with the factor λ = 1/R and translation we can assume that R = 1 and x_{0} = 0. Let η ∈ C_{c}^{∞}(B_{1}) be such that 0 ≤ η ≤ 1 and η ≡ 1 on B_{1/2} and let

ζ := η · (φ_{m} + ψ_{m} − 2n) .

We compute

\frac{∂}{∂t}ζ ≤ g^{ab}^{h}∇_{a}^{h}∇_{b}ζ − 2g^{ab}^{h}∇_{a}η^{h}∇_{b}(φ_{m} + ψ_{m} − 2n)

− (φ_{m} + ψ_{m} − 2n)g^{ab}^{h}∇_{a}^{h}∇_{b}η − η ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2}

≤ g^{ab}^{h}∇_{a}^{h}∇_{b}ζ + c ,

since |^{h}∇(φ_{m} + ψ_{m} − 2n)|^{2} ≤ c ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} and |∇η|^{2}/η ≤ C(n, ∥η∥_{C^{2}}). So by

the maximum principle the maximum of ζ grows at most linearly, which implies the stated estimate. □

By scaling we want to extend a priori derivative estimates for the metric to balls of any radius. If the background metric h is not necessarily flat, the third author [14] obtained the following

Theorem 5.3. Let R > 0. Let h be a complete background metric of bounded curvature. Fix T = T (n, R, h) > 0 sufficiently small. Fix a point x_{0}. Let g be a solution to the h-flow on ^{h}B_{R}(x_{0}) × (0, T) which is ε̃-close to the background metric h for ε̃ = ε̃(n) > 0 fixed sufficiently small, where ^{h}B_{R}(x_{0}) denotes a geodesic ball of radius R with respect to the metric h. Then

|^{h}∇^{i}g(x, t)| ≤ \frac{c(n, i, R)}{t^{i/2}}

for all (x, t) ∈ ^{h}B_{R/2}(x_{0}) × (0, T) and all i ∈ ℕ.

Applying Theorem 5.3 to g^{λ} as defined in the proof of Lemma 5.2, we get directly

Corollary 5.4. Let h denote the standard flat metric on ℝ^{n} and assume that ε = ε(n) > 0 and γ = γ(n) > 0 are chosen sufficiently small. If g solves the h-flow in B_{R}(x_{0}) × (0, γR^{2}) for some x_{0} ∈ ℝ^{n}, R > 0, and if g is ε-close to h on B_{R}(x_{0}) × (0, γR^{2}), then we get the a priori estimates

|^{h}∇^{i}g(x, t)| ≤ \frac{c(n, i)}{t^{i/2}}

for all (x, t) ∈ B_{R/2}(x_{0}) × (0, γR^{2}) and all i ∈ ℕ. We set c_{1} := c(n, 1).

Proof of Theorem 1.2. Let us first assume that g_{0} ∈ M_{loc}^{∞} (ℝ^{n}). Theorem 4.5 implies that an ε-close solution g ∈ M_{loc}^{∞} (ℝ^{n}, [0, ∞)) to (1.1) exists. The interior decay estimates of Corollary 5.4 and Arzelà-Ascoli imply that the metrics g(t) converge subsequentially in M_{loc}^{∞} to a complete flat metric as t → ∞.

We approximate g_{0} ∈ M_{0} (ℝ^{n}) by smooth metrics ^{i}g_{0} preserving the ε_{0}-closeness. Thus we obtain solutions ^{i}g ∈ M_{loc}^{∞} which are ε-close to h. In view of the interior a priori estimates we obtain a limiting solution g ∈ M_{loc}^{∞} (ℝ^{n}, (0, ∞)).

Note that g(t) → g_{0} in M_{loc}^{0} as t ↘ 0: Fix a point x_{0} ∈ ℝ^{n} and use g_{0}(x_{0}) as a flat background metric on ℝ^{n}. Thus the metrics ^{i}g are again solutions to the h-flow with h = g_{0}(x_{0}). Hence we can apply the interior closeness estimates of Lemma 5.2 in order to see that all metrics ^{i}g attain their initial values uniformly in i. That implies that g(t) → g_{0} as t ↘ 0 locally uniformly. □

6. Integral Estimates

In this section, we are once again concerned with the quantity φ_{m} + ψ_{m} − 2n as introduced in (1.4).

Theorem 6.1. Fix δ > 0, m = m(n) ∈ ℕ, and p ≥ 1. Let g ∈ M_{loc}^{∞} (ℝ^{n}, (0, ∞)) be a solution to (1.1) which is ε̃(n, m)-close to the standard Euclidean metric h for some 1 ≥ ε̃ = ε̃(n, m) > 0 sufficiently small. Let g_{0} be as in Theorem 1.3. Then the integral

I(t) ≡ I_{δ}^{m,p}(t) := \frac{1}{p} ∫_{ℝ^{n}} (φ_{m} + ψ_{m} − 2n − δ)_{+}^{p} dx ≡ \frac{1}{p} ∫_{ℝ^{n}} Φ_{m,δ}^{p}

is non-increasing in time.

Proof. As g is ε̃-close to h, φ_{m} + ψ_{m} − 2n is uniformly bounded above. According to Lemma 5.1, we see that φ_{m} + ψ_{m} − 2n → 0 for |x| → ∞ and t bounded above. Thus the positive part of φ_{m} + ψ_{m} − 2n − δ is contained in a compact set for every bounded time interval and I(t) is finite there.

Assume that in appropriate coordinates, we have at a fixed point h_{ij} = δ_{ij}, g_{ij} = diag(λ_{1}, λ_{2}, …, λ_{n}), λ_{i} > 0.

Recall from Section 3 that we can estimate in the evolution equation of φ_{m} + ψ_{m} for metrics ε-close to h as follows

\frac{∂}{∂t}(φ_{m} + ψ_{m}) ≤ g^{ij}^{h}∇_{i}^{h}∇_{j}(φ_{m} + ψ_{m})

− m(m − 1)(1 + ε)^{−(m−1)} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2}

+ c(n)m(1 + ε)^{m+1} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} .

We now want to show that I(t) is decreasing in time for m and ε chosen properly. Note that Φ_{m,δ}^{p} is Lipschitz continuous in space and time, and the support of Φ_{m,δ}^{p} is contained in B_{R}(0) for some R > 0 on bounded time intervals. This yields that I(t) is Lipschitz continuous in time as well and thus also absolutely continuous. By Sard’s theorem we know that for almost every δ > 0 the sets

U_{δ} ≡ U_{δ}(t) := {x ∈ ℝ^{n} | φ_{m}(x, t) + ψ_{m}(x, t) − 2n > δ}

have a smooth boundary for almost every t. For such a δ and 0 ≤ t_{1} < t_{2}, we can compute

I(t_{2}) − I(t_{1}) = ∫_{t_{1}}^{t_{2}} \frac{1}{p} \frac{d}{dτ} ∫_{ℝ^{n}} Φ_{m,δ}^{p} dx dτ

= ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} Φ_{m,δ}^{p−1} \frac{∂}{∂τ} (φ_{m} + ψ_{m}) dx dτ ,

as Φ_{m,δ} is compactly supported as long as t is finite. Since the boundary of U_{δ} is smooth we may estimate and integrate by parts

I(t_{2}) − I(t_{1}) ≤ ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} Φ_{m,δ}^{p−1} (g^{ij}^{h}∇_{i}^{h}∇_{j}(φ_{m} + ψ_{m})) dx dτ

− ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} m(m − 1)(1 + ε)^{−(m−1)}Φ_{m,δ}^{p−1} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} dx dτ

+ ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} c(n)m(1 + ε)^{m+1}Φ_{m,δ}^{p−1} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} dx dτ

= ∫_{t_{1}}^{t_{2}} ∫_{∂U_{δ}} Φ_{m,δ}^{p−1}ν_{i}g^{ij}^{h}∇_{j}Φ_{m,δ} dσ dt

− ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} (p − 1)Φ_{m,δ}^{p−2}g^{ij}^{h}∇_{i}Φ_{m,δ}^{h}∇_{j}Φ_{m,δ} dx dτ

− ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} Φ_{m,δ}^{p−1}(^{h}∇_{i}g^{ij})^{h}∇_{j}Φ_{m,δ} dx dτ

+ ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} (−m(m − 1))(1 + ε)^{−m+1}Φ_{m,δ}^{p−1} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} dx dτ

+ ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} c(n)m(1 + ε)^{m+1}Φ_{m,δ}^{p−1} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} dx dτ

≤ m ∫_{t_{1}}^{t_{2}} ∫_{U_{δ}} Φ_{m,δ}^{p−1} ∑_{i, j, k} (^{h}∇_{k}g_{ij})^{2} ·

· (−(m − 1)(1 + ε)^{−m+1} + c(n)(1 + ε)^{m+3}) dx dτ , where we used generic constants c(n), the estimates

h |^{h}∇g^{ij}| ≤ c(1 + ε)^{2} · \sqrt{∑_{i,j,k} (^{h}∇_{k}g_{ij})^{2}},

h |^{h}∇Φ_{m,δ}| ≤ c m (1 + ε)^{m+1} · \sqrt{∑_{i,j,k} (^{h}∇_{k}g_{ij})^{2}}

and the fact that ^{h}∇Φ_{m,δ} is antiparallel to the outer unit normal ν of U_{δ} along ∂U_{δ}. Note that the integration by parts involving the exponent p − 2 above is justified by applying the divergence theorem on sets U_{δ_{i}} for a sequence δ_{i} ↘ δ, where we can assume by Sard’s theorem that ∂U_{δ_{i}} is smooth for all i and that ∂U_{δ_{i}} → ∂U_{δ}. If m and ε are such that

1 + c(n)(1 + ε)^{2m+2} ≤ m,

we obtain that the right hand side is nonpositive and our theorem follows for such a δ. Fix ε̃ > 0 accordingly. Since

I_{m,p}^{δ_{i}}(t) → I_{m,p}^{δ}(t)

for δ_{i} → δ we obtain the above monotonicity for all δ. □

Corollary 6.2. Let g ∈ M_{loc}^{∞} (ℝ^{n}, (0, ∞)) be a solution to (1.1) with g_{0} as in Theorem 1.3. Let m, p, and ε̃ be as in Theorem 6.1. If I_{0}^{m,p}(0) is finite, then I_{0}^{m,p}(t) is non-increasing and thus

I^{m,p}(t) := \frac{1}{p} ∫_{ℝ^{n}} (φ_{m} + ψ_{m} − 2n)^{p} dx = I_{0}^{m,p}(t) ≤ I_{0}^{m,p}(0) < ∞.

Proof. Theorem 6.1 implies that

I_{δ}^{m,p}(t) ≤ I_{δ}^{m,p}(0) ≤ I_{0}^{m,p}(0)

for every δ > 0. For δ ↘ 0, we get

I_{δ}^{m,p}(t) → I_{0}^{m,p}(t)

and our claim follows. □

7. Convergence of Eigenvalues

Lemma 7.1. Let g ∈ M_{loc}^{∞} (ℝ^{n}, (0, ∞)) be a solution to (1.1) with g_{0} as in Theorem 1.3, such that g(t) is ε̃(n)-close to h for all t. Then the eigenvalues (λ_{i})_{1≤i≤n} of g(t) with respect to the background metric h converge uniformly to one as t → ∞,

sup_{x∈ℝ^{n}} |λ_{i}(x, t) − 1| → 0 as t → ∞.

Proof. Denote the eigenvalues of g_{ij} with respect to h_{ij} by (λ_{i}). It is convenient to assume that λ_{1} ≤ … ≤ λ_{n}.

Assume that the lemma were false. Then we could find ζ > 0 and a sequence (x_{k}, t_{k})_{k∈ℕ} in ℝ^{n} × [0, ∞) with t_{k} → ∞ so that

(7.1)  max_{i∈{1, ..., n}} |λ_{i}(x_{k}, t_{k}) − 1| ≥ 2ζ.

The strategy of the proof is to use Corollary 5.4 with i = 1 to show that we find a controlled spatial neighborhood of (x_{k}, t_{k}) such that (7.1) is fulfilled with 2ζ replaced by ζ, i. e. the eigenvalues are not all close to one there. Therefore, φ_{m} +ψ_{m} −2n is estimated from below by a positive constant in that neighborhood. For large values of t, these neighborhoods can be chosen arbitrarily large. Thus I_{δ}^{m,p} becomes large, if δ is chosen small enough. This contradicts Theorem 6.1.

Here we present the details: let B_{R}(x_{k}) denote a Euclidean ball around x_{k}. According to Corollary 5.4, applied with i = 1, we obtain for x ∈ B̃_{R}(x_{k}) that the eigenvalues (λ_{i}(x, t_{k})) differ at most by c·R·t_{k}^{−1/2} from the eigenvalues (λ_{i}(x_{k}, t_{k})),

sup_{i∈\{1, ..., n\}} |λ_{i}(x_{k}, t_{k}) − λ_{i}(x, t_{k})| ≤ \frac{c_{1}R}{\sqrt{t_{k}}} .

We deduce that in a ball of radius at least R around x_{k} with R := \frac{1}{c_{1}} ζ\sqrt{t_{k}}, we have

max_{i∈{1, ..., n}} |λ_{i}(x, t_{k}) − 1| ≥ ζ.

Note that this implies

max_{i∈{1, ..., n}} |λ_{i}^{m}(x, t) − 1| ≥ ζ.

So at least one eigenvalue differs significantly from one in that ball. As our solution is ε̃(n)-close to h with 0 < ε̃(n) ≤ 1, we still have

\frac{1}{2} ≤ (1 + ε̃)^{−1} ≤ λ_{i}, λ_{i}^{−1} ≤ 1 + ε̃ ≤ 2.

We obtain in B_{R}(x_{k}) × {t_{k}}

φ_{m} + ψ_{m} − 2n = ∑_{i=1}^{m} \frac{1}{λ_{i}^{m}} (λ_{i}^{m} − 1)^{2} ≥ \frac{1}{2^{m}} ζ^{2}.

We define δ := \frac{1}{2} \frac{1}{2^{m}} ζ^{2} and get in B_{R}(x_{k}) × \{t_{k}\}

φ_{m} + ψ_{m} − 2n ≥ 2δ,
(φ_{m} + ψ_{m} − 2n − δ)_{+} ≥ δ.

This allows to estimate I_{δ}^{m,p}(t_{k}) from below as follows

I_{δ}^{m,p}(t_{k}) ≥ \frac{1}{p} ∫_{B_{\frac{1}{c_{1}} ζ\sqrt{t_{k}}} (x_{k})} (φ_{m} + ψ_{m} − 2n − δ)_{+}^{p}

≥ \frac{1}{p} · δ^{p} · c(n) · (\frac{1}{c_{1}} ζ\sqrt{t_{k}})^{n} .

We deduce that I_{δ}^{m,p}(t_{k}) → ∞ for t_{k} → ∞. This contradicts Theorem 6.1 which implies that

I_{δ}^{m,p}(t_{k}) ≤ I_{δ}^{m,p}(0)

as our initial decay assumption guarantees that for fixed m ∈ ℕ_{+} as in Theorem 6.1, p ≥ 1, and δ > 0, the integral I_{δ}^{m,p}(0) is finite. □

Proof of Theorem 1.3. Lemma 7.1 implies that g(t) → h in C^{0} as t → ∞. The convergence of the derivatives ^{h}∇^{m}g(t) for all m ∈ ℕ is a consequence of Corollary 5.4. □

If I_{0}^{m,p}(0) is finite for some m, p as in Section 6, such a calculation implies even a decay rate in time of

sup_{x∈ℝ^{n}} |λ_{i}(x, t) − 1|.

Lemma 7.2. Let g ∈ M_{loc}^{∞} (ℝ^{n}, (0, ∞)) be a solution to (1.1) with g_{0} as in Theorem 1.3, such that g(t) is ε̃(n)-close to h for all t. Then the eigenvalues (λ_{i})_{1≤i≤n} of g(t) with respect to the flat background metric h converge to 1 as t → ∞:

sup_{x∈ℝ^{n}} sup_{i∈\{1, ..., n\}} |λ_{i}(x, t) − 1| ≤ c(n, m, p, c_{1}) · (I_{0}^{m,p}(0))^{\frac{1}{2p+n}} · (\frac{1}{t})^{\frac{n}{2(2p+n)}}

if I_{0}^{m,p}(0) < ∞.

Proof. Fix x_{0} ∈ ℝ^{n}, t_{0} > 0, and i_{0} ∈ {1, …, n}. Define

ζ := \frac{1}{2}|λ_{i_{0}}(x_{0}, t_{0}) − 1|.

We want to estimate ζ from above. As in the proof of Lemma 7.1, we obtain that

ζ ≤ max_{i∈\{1, ..., n\}} max_{x∈B_{\frac{1}{c_{1}} ζ\sqrt{t_{0}}} (x_{0})} |λ_{i}(x, t_{0}) − 1|,

φ_{m} + ψ_{m} − 2n ≥ \frac{1}{2^{m}} ζ^{2},
I_{0}^{m,p}(0) ≥ I_{0}^{m,p}(t_{0})
≥ \frac{1}{p}  ∫_{B_{\frac{1}{c_{1}} ζ\sqrt{t_{0}}} (x_{0})} (φ_{m} + ψ_{m} − 2n)^{p}

≥ \frac{1}{p} (\frac{1}{2^{m}} ζ^{2})^{p} · c(n) · (\frac{1}{c_{1}} ζ\sqrt{t_{0}})^{n} ,

and thus

(2ζ)^{2p+n} ≤ c(n, m, p, c_{1}) · t_{0}^{−n/2} · I_{0}^{m,p}(0).

Our claim follows. □

8. Improved C^{1}-Estimates

Based on an iteration of two steps, we can improve our a priori estimates. So far, we have obtained a priori estimates of the form

|g − h|_{C^{0}} ≤ \frac{c (n, m, p, c_{1}(n), I_{0}^{m,p})}{t^{\frac{n}{2(2p+n)}}} ≡ \frac{c_{0}}{t^{\frac{n}{2(2p+n)}}} ,

see Lemma 7.2, and

(8.1)  |^{h}∇^{i}g| ≤ \frac{c(i, n)}{t^{i/2}} ≡ \frac{c_{i}}{t^{i/2}}

for every i ∈ ℕ, see Lemma 5.4. The two steps to improve these are:

Step 1: Interpolation inequalities of the form ∥Du∥_{L^{∞}}^{2} ≤ c · ∥u∥_{L^{∞}} · ∥D^{2}u∥_{L^{∞}} and an iteration argument as in [11, Lemma C.2] can be applied to a metric g satisfying (8.1) and

|g − h|_{C^{0}} ≤ \frac{a}{t^{α}}

for some constant a. We obtain for any 0 < β < α the estimate

|^{h}∇^{i}g| ≤ \frac{c((c_{k})_{1≤k≤k_{0}(i,α,β)}, a, α, β)}{t^{\frac{i}{2} +β}},

c_{k} as in (8.1). Note that if β gets closer to α, the upper bound depends on more and more constants c_{k} as in (8.1).

Step 2: Recall that for solutions of the h-flow satisfying

|^{h}∇g| ≤ \frac{c}{t^{γ}} ,

arguing as in Lemma 7.2 yields

|g − h|_{C^{0}} ≤ \frac{c}{t^{γ \frac{n}{2p+n}}} .

Iteration: We start with

|^{h}∇g| ≤ \frac{c}{t^{γ_{0}}} .

According to Corollary 5.4 (with i = 1), we may take γ_{0} = \frac{1}{2}. So step 2 yields

|g − h|_{C^{0}} ≤ \frac{C_{1}}{t^{α_{1}}}

with α_{1} = \frac{n}{2(2p+n)}. By combining steps 1 and 2, we get

|g − h|_{C^{0}} ≤ \frac{C_{2}(δ)}{t^{α_{2}}}

with α_{2} = (\frac{1}{2} + (1 − δ)α_{1}) \frac{n}{2p+n} for any fixed 0 < δ < 1. We also get

|g − h|_{C^{0}} ≤ \frac{C_{k}(δ)}{t^{α_{k}}}

with α_{k+1} = (\frac{1}{2} + (1 − δ)α_{k}) \frac{n}{2p+n} for all k ∈ ℕ.

It is easy to check that

\frac{n}{2(2p + n)} = α_{1} < α_{2} < … < \frac{n}{2(2p + δn)}.

As α_{k} is increasing in k, we see that lim α_{k} exists. Passing to the limit in the defining equation for α_{k} yields

lim_{k→∞} α_{k} = \frac{n}{2(2p + δn)}.

Now δ > 0 can be chosen arbitrarily small. So we obtain for any 0 < ζ ≪ 1

|g − h|_{C^{0}} ≤ \frac{c(ζ)}{t^{\frac{n}{4p} −ζ}} .

We can finally apply step 1 once again in order to obtain a derivative bound.

Thus we have proved the following Lemma 8.1. Let g be as in Lemma 7.2. Then we obtain for every 0 < ζ ≪ 1 the estimates

|g − h|_{C^{0}} ≤ \frac{c(ζ)}{t^{\frac{n}{4p} −ζ}}

and for every i ∈ ℕ

(8.2)  |^{h}∇^{i}g| ≤ \frac{c(ζ)}{t^{\frac{i}{2} + \frac{n}{4p} −ζ}} .

9. Construction of a Solution to Ricci Flow

In this section we construct a solution to Ricci flow for smooth initial metrics g_{0} on ℝ^{n} which are ε-close to the standard metric δ. It is well known that on a compact manifold, we can recover the Ricci flow from a solution to the h-flow using time dependent diffeomorphisms [7, 14]. We use the a priori estimates of the previous section and construct diffeomorphisms in our set up which allow us to construct a solution to the Ricci flow (similar to the compact case).

Lemma 9.1. Let g ∈ M_{loc}^{∞} (ℝ^{n}, [0, ∞)) be a solution to (1.1) which is ε̃(n)-close to h (for all t with ε̃ as in Theorem 1.2). Then there exists a smooth family of diffeomorphisms (φ(·, t))_{t∈[0,∞)}, φ_{t} ≡ φ(·, t) : ℝ^{n} → ℝ^{n}, with φ(x, 0) = x, such that (φ_{t})^{∗}(g(t)) solves (RF).

Proof. Let ^{i}φ : B_{i} × [0, ∞) → B_{i} ⊂ ℝ^{n} be the solutions to the ordinary differential equations

\begin{cases}\frac{∂}{∂t}(^{i}φ)^{α}(x, t) = η_{i}(φ(x, t))V^{α}(φ(x, t), t), & (x, t) ∈ B_{i} × [0, ∞), \\ ^{i}φ(x, 0) = x, & x ∈ B_{i}, \end{cases}

where V^{α}(y, t) := g^{βγ} (^{g}Γ_{βγ}^{α} −^{h}Γ_{βγ}^{α}) (y, t), and η_{i} : ℝ^{n} → ℝ is a smooth cut-off function 0 ≤ η ≤ 1 with η|_{B_{\frac{i}{2}}} ≡ 1 and η|_{ℝ^{n}−B_{i}} ≡ 0. Notice that^{i}V := η_{i}V is smooth and compactly supported in B_{i}. Hence the φ_{i}(·, t) : B_{i} → B_{i} are diffeomorphisms.

Using again Corollary 5.4 we see that for x ∈ B_{i/2} and small times

|^{h}∇g(x, t)| ≤ \frac{c(n)}{\sqrt{t}.}

This implies that

|\frac{∂}{∂t}(^{i}φ)^{α}(x, t)| = |^{i}V^{α}(^{i}φ(x, t), t)| ≤ \frac{c(n)}{\sqrt{t}}

(here we have used that g(t) is ε-close to h). In particular, this gives us

|(^{i}φ)^{α}(x, t) − x| ≤ ∫_{0}^{t} |\frac{∂}{∂τ} (^{i}φ)^{α}(x, τ)| dτ ≤ c(n)\sqrt{t}.

This implies that ^{i}φ(·, t)|_{B_{j}} is independent of i for i big enough, j fixed and t bounded above. In particular we obtain, taking a (diagonal sub-sequence) limit, a well defined solution φ : ℝ^{n} × [0, ∞) → ℝ^{n} to the equation

(9.1)  \begin{cases}\frac{∂}{∂t}(φ)^{α}(x, t) = V^{α}(φ(x, t), t), & (x, t) ∈ ℝ^{n} × [0, ∞), \\ φ(x, 0) = x, & x ∈ ℝ^{n}, \end{cases}

and thus φ_{t} = φ(t, ·) : ℝ^{n} → ℝ^{n} is a diffeomorphism for all t. According to [12] (φ_{t})^{∗}(g(t)) solves (RF). □

In the situation that we have the improved decay estimates of Lemma 8.1 we show that the diffeomorphisms φ_{t} converge to a diffeomorphism φ_{∞} : ℝ^{n} → ℝ^{n} as t → ∞ for n ≥ 3 as long as p ≥ 1 is small enough.

Theorem 9.2. Let g ∈ M_{loc}^{∞} (ℝ^{n}, [0, ∞)) be a solution to (1.1) and m, p, and ε̃(m, n) all be as in Theorem 6.1. Assume that (1.5) holds. For n ≥ 3, φ_{t} converges in C_{loc}^{∞} to a diffeomorphism φ_{∞} : ℝ^{n} → ℝ^{n} as t → ∞.

Proof. Step 1: Fix p with \frac{n}{2} > p ≥ 1 and ζ > 0 such that α := \frac{1}{2} + \frac{n}{4p} − ζ > 1. In the following, our constants c may depend on n, p, ζ, m, and I_{0}^{m,p} as defined in Theorem 6.1.

We wish to show that

|φ_{t}(x) − x| ≤ c ∀ t ∈ [0, ∞).

Corollary 5.4 implies that

|\frac{d}{dt}φ_{t}(x)| ≤ \frac{c}{t^{1/2}} .

Hence |φ_{t}(x)−x| ≤ c for all 0 ≤ t ≤ 1. So we may assume that t > 1. The evolution equation (9.1) for φ_{t} and the fact that |∇g| ≤ \frac{c}{t^{α}} imply that

|\frac{d}{dt}φ_{t}(x)| ≤ \frac{c}{t^{α}} .

As α > 1, we see that (upon integrating from 1 to t > 1) |φ_{1}(x) − φ_{t}(x)| ≤ c. By the triangle inequality we get

|φ_{t}(x) − x| ≤ |φ_{t}(x) − φ_{1}(x)| + |φ_{1}(x) − x| ≤ c.

For later use, note that this implies for F_{t} : ℝ^{n} → ℝ^{n} defined by F_{t} := φ_{t}^{−1}

|F_{t}(x) − x| ≤ c.

Step 2: Using the estimate |\frac{d}{dt}φ_{t}(x)| ≤ \frac{c}{t^{α}} from Step 1, we see that for fixed x ∈ ℝ^{n}, φ_{t}(x) → y as t → ∞ for some y ∈ ℝ^{n}. We define φ_{∞}(x) = lim_{t→∞} φ_{t}(x). All the derivatives of φ_{t} (for t > 1) satisfy

\frac{∂}{∂t} log (1 + |^{h}∇^{m}φ_{t}(x)|^{2}) ≤ \frac{c (φ_{1}|_{B_{r}(y)}, n, m, p, ζ, (c_{k}), I_{0}^{m,p})}{t^{α}} ∀ x ∈ B_{r}(y)

for some α > 1 and

sup_{x∈B_{r}(y)} |^{h}∇^{m}φ_{t}(x)|^{2} ≤ c (φ_{1}|_{B_{r}(y)}, n, m, p, ζ, (c_{k}), I_{0}^{m,p}) .

This may be seen as follows (for the rest of this argument, we set ∇ := ^{h}∇). Differentiating gives for V as in (1.1)

(9.2) |∇^{m}V (x, t)| ≤ c_{m} ∑_{k=1}^{m+1} ∑_{{i_{1}+···+i_{k}=m+1 \atop 1≤i_{1}, ..., i_{k}≤m+1}} |∇^{i_{1}}g| ⋯ |∇^{i_{k}}g| ,

|∇^{m}(V (φ(x, t), t))| ≤ c_{m} ∑_{i=1}^{m} ∑_{{j_{1}+···+j_{i}=m \atop 1≤j_{1}, ..., j_{i}≤m}} |(∇^{i}V ) (φ(x, t), t)| · |∇^{j_{1}}φ| ⋯ |∇^{j_{i}}φ| . (9.3)

Then

\frac{∂}{∂t} log (1 + |∇^{m}φ_{t}(x)|^{2}) = \frac{\frac{∂}{∂t} |∇^{m}φ_{t}(x)|^{2}}{(1 + |∇^{m}φ_{t}(x)|^{2})}

≤ c \frac{|∇^{m}φ_{t}(x)| |∇^{m} \frac{∂}{∂t}φ_{t}(x)|}{(1 + |∇^{m}φ_{t}(x)|^{2})}

= c \frac{|∇^{m}φ_{t}(x)| |∇^{m}(V (φ_{t}, t))(x)|}{(1 + |∇^{m}φ_{t}(x)|^{2})}.

Substituting (9.2) and (9.3) into the above, using the estimates for |∇^{k}g| and arguing inductively (for m) leads to the estimate.

Hence φ_{∞} : ℝ^{n} → ℝ^{n} is C^{∞} and φ_{t} → φ_{∞} converges in C_{loc}^{∞} as t → ∞.

Step 3: We wish to show that φ_{∞} is a diffeomorphism. Let F_{t} be the inverse of φ_{t}. Letting l(t) := φ_{t}^{∗}g(t), we know that l solves Ricci flow on ℝ^{n}. But then we get in view of (8.2)

l|\frac{∂}{∂t}l| (x, t) = 2 ·^{l}| Ric(l)|(x, t) = 2 ·^{g}| Ric(g)|(φ_{t}(x), t) ≤ \frac{c}{t^{\frac{3}{2}}} for t ≥ 1,

which is integrable (from 1 to ∞). This implies that l(t) converges locally uniformly to a well defined continuous Riemannian metric l_{∞} as t → ∞ in view of [5, Lemma 14.2]. According to [5], there must exist a c = c(y) with \frac{1}{c}h(x) ≤ l(x, t) ≤ c h(x) for all t ∈ [0, ∞) for all x ∈ B_{1}(y), for all y ∈ ℝ^{n}, as Theorem 4.4 implies this uniform equivalence for t ∈ [0, 1].

Now using the definition of l we get

\frac{1}{c}δ_{αβ} ≤ l_{αβ}(x, t) = \frac{∂φ^{s}}{∂y^{α}} (x, t)\frac{∂φ^{k}}{∂y^{β}} (x, t) g_{sk}(φ_{t}(x), t)

≤ (1 + ε̃)\frac{∂φ^{s}}{∂y^{α}} \frac{∂φ^{k}}{∂y^{β}} (x, t)δ_{sk}

= (1 + ε̃) (Dφ) (Dφ)^{T} (x, t).

In particular, we see that det (Dφ_{t})^{2} (x) ≥ \frac{1}{(1+ε̃)c} > 0, where Df is the Jacobian of

f. Taking the limit as t → ∞ we get det (Dφ_{∞})^{2} (y) ≥ \frac{1}{(1+ε̃)c} > 0. As det (Dφ_{0}) = 1, we see that det (Dφ_{∞})(y) ≥ \frac{1}{\sqrt{(1+ε̃)c(y)}} > 0 for all y ∈ ℝ^{n}. Hence φ_{∞} is an

immersion. In view of the derivative estimates for |∇^{m}φ_{t}|, the function φ_{∞} : ℝ^{n} → ℝ^{n} is a smooth diffeomorphism if it is bijective. Recall that F_{t} ≡ F (·, t) = (φ_{t})^{−1} for t < ∞. The estimates for φ_{t} above ensure that there exists a function F_{∞} : ℝ^{n} → ℝ^{n} such that F_{t} → F_{∞} in C_{loc}^{∞} as t → ∞.

As φ_{t} and F_{t} are smooth diffeomorphisms with estimated derivatives, φ_{t} → φ_{∞} and F_{t} → F_{∞}, we obtain

(F_{∞} ∘ φ_{∞})(x) = lim_{t→∞} (F_{t} ∘ φ_{t}) (x) = x for all x ∈ ℝ^{n}.

Thus φ_{∞} is injective. Similarly, we see that φ_{∞} is surjective. Hence φ_{∞} is a diffeomorphism. □

Proof of Theorem 1.4. In Lemma 9.1 we have constructed diffeomorphisms φ_{t} so that (φ_{t}^{∗}g(t))_{t∈[0,∞)} solves (RF) if (g(t))_{t∈[0,∞)} solves (1.1). Define g̃(t) := φ_{t}^{∗}g(t). According to Theorem 1.3, we get g(t) → h. Theorem 9.2 implies that the diffeomorphisms φ_{t} converge to a diffeomorphism φ_{∞} for t → ∞. Thus g̃(t) → φ_{∞}^{∗}h in C^{∞} as t → ∞. □

We can also show that the diffeomorphisms φ_{t} are close to the identity near infinity, uniformly in t.

Lemma 9.3. Let g ∈ M_{loc}^{∞} (ℝ^{n}, [0, ∞)) be a solution to (1.1) as in Theorem 9.2 and (φ_{t})_{t∈[0,∞]} be the diffeomorphisms of ℝ^{n} constructed before. Then for every η > 0 there exists an R > 0 such that

sup_{ℝ^{n}\B_{R}(0)} |φ_{t}(x) − x| ≤ η

for all t ∈ [0, ∞].

Proof. We first want to show that for any given η′ > 0 there exists an R_{1} > 0 such that g(t) is η′-close to h = δ on ℝ^{n} \ B_{R_{1}}(0) for all t. By Lemma 7.2 we can choose a T > 0 such that g(t) is η′-close to h on all of ℝ^{n} for t ≥ T. Now choose R_{0} > 0 such that g(0) is η′/2-close to h on ℝ^{n} \ B_{R_{0}}(0). Applying the interior closeness estimates, Lemma 5.2, we see that we can find R_{1} > R_{0} such that g(t) is η′-close to h on (ℝ^{n} \ B_{R_{1}}(0)) × [0, T]. Note that this implies

sup_{ℝ^{n}\B_{R_{1}} (0)} |g(t) − δ| ≤ η′

for all t. By Lemma 8.1 we have that

|^{h}∇g| ≤ \frac{c}{t^{\frac{1}{2} + \frac{n}{4p} −ζ}}

for t > 0. Now fix t_{1} > 0 such that

∫_{t_{1}}^{∞} \frac{c}{t^{\frac{1}{2} + \frac{n}{4p} −ζ}} dt ≤ \frac{η}{2} .

By interpolation we deduce from Lemma 5.4 that

|^{h}∇g| ≤ \frac{δ(η′)}{\sqrt{t}} on ℝ^{n} \ B_{R_{1}}(0)

for t > 0, and in particular for 0 < t ≤ t_{1}, where δ(η) → 0 as η → 0. Arguing as in Step 1 of the proof of Theorem 9.2 we obtain the claimed estimate for some R > R_{1}. □

10. Convergence Based on Integral Bounds

Proof of Theorem 1.6. In view of Corollary 5.4 and interpolation inequalities of the form ∥∇v∥_{C^{0}} ≤ c · ∥v∥_{C^{0}} · ∥v∥_{C^{2}}, it suffices to prove that g(t) → h in C^{0} as t → ∞.

Assume the conclusions were false. Then we could find δ > 0 and points

(x_{k}, t_{k}, l_{k}) ∈ ℝ^{n} × (0, ∞) × {1, …, n},

k ∈ ℕ, such that t_{k} → ∞ as k → ∞ and

|λ_{l_{k}}(x_{k}, t_{k}) − 1| ≥ 4δ.

By the construction of the solution g(t) in Section 4 from solutions ^{i}g, we find sequences (i_{j}^{k})_{j∈ℕ} with i_{j}^{k} → ∞ as j → ∞, such that for the eigenvalues (^{i}λ_{l})_{1≤l≤n} of ^{i}g, we have

(10.1)  |^{i_{j}^{k}}λ_{l_{k}}(x_{k}, t_{k}) − 1| ≥ 3δ.

Fix R > 0. We want to establish a similar bound on the spatial ball B_{R}(x_{k}) × {t_{k}}. This is done as follows: We consider only k ≥ k_{0} so that the t_{k} is sufficiently large and hence the right-hand side in the estimate of the first spatial derivative (i = 1) in Corollary 5.4 is bounded above by δ/R. Then we choose R_{k} ≥ R such that γ(2R_{k})^{2} ≥ t_{k}, where γ is as in the Corollary. Now we fix j = j(k) so large that ν_{k} ≡ i_{j(k)}^{k} ≥ 2R_{k} + |x_{k}| + 1. This ensures that the metric ^{ν_{k}}g is defined on the cylinder B_{2R_{k}}(x_{k}) × (0, γ(2R_{k})^{2}). Hence we can apply Corollary 5.4 for that

cylinder and get the gradient estimate |^{h}∇^{ν_{k}}g(·, t_{k})| ≤ δ/R in B_{R}(x_{k}). In view of (10.1), we obtain

(10.2)  inf_{x∈B_{R}(x_{k})} sup_{1≤l≤n} |^{ν_{k}}λ_{l}(x, t_{k}) − 1| ≥ 2δ.

Define

iI_{δ}^{m,p}(t) := \frac{1}{p} ∫_{B_{i}(0)} (^{i}φ_{m} +^{i}ψ_{m} − 2n − δ)_{+}^{p} dx,

where

iφ_{m}(x, t) := ∑_{l=1}^{n} \frac{1}{^{i}λ_{l}^{m}(x, t)} and ^{i}ψ_{m}(x, t) := ∑_{l=1}^{n}^{i}λ_{l}^{m}(x, t).

By construction of the initial metrics ^{i}g_{0}, we have for the eigenvalues (λ_{l}(x))_{1≤l≤n} of the metric g_{0} with respect to h which we label such that λ_{1}(x) ≤ … ≤ λ_{n}(x) and similarly for the eigenvalues ^{i}λ_{1}(x) ≤ … ≤ ^{i}λ_{n}(x) of ^{i}g_{0} for |x| ≤ i the inequalities

\begin{cases}λ_{l}(x) ≤ ^{i}λ_{l}(x) ≤ 1 & if λ_{l}(x) ≤ 1, \\ 1 ≤ ^{i}λ_{l}(x) ≤ λ_{l}(x) & if 1 ≤ λ_{l}(x), \end{cases}

which implies in particular the estimate (^{i}λ_{l}^{m} − 1)^{2} ≤ (λ_{l}^{m} − 1)^{2} for all i, l, m ∈ ℕ. We deduce that

iφ_{m} +^{i}ψ_{m} − 2n = ∑_{l=1}^{n} \frac{1}{^{i}λ_{l}^{m}} (^{i}λ_{l}^{m} − 1)^{2}

≤ ∑_{l=1}^{n} \frac{λ_{l}^{m}}{λ_{l}^{m}} (1 + ε(n))^{m} (λ_{l}^{m} − 1)^{2}

≤ (1 + ε(n))^{2m} ∑_{l=1}^{n} \frac{1}{λ_{l}^{m}} (λ_{l}^{m} − 1)^{2}
= (1 + ε(n))^{2m}(φ_{m} + ψ_{m} − 2n).

Thus

\begin{matrix}^{i}I_{δ}^{m,p}(0) = & \frac{1}{p} ∫_{B_{i}(0)} (^{i}φ_{m} +^{i}ψ_{m} − 2n − δ)_{+}^{p} dx|_{t=0} \\ ≤ & (1 + ε(n))^{2m} \frac{1}{p} ∫_{B_{i}(0)} (φ_{m} + ψ_{m} − 2n − δ)_{+}^{p} dx|_{t=0} \\ ≤ & (1 + ε(n))^{2m} \frac{1}{p} ∫_{ℝ^{n}} (φ_{m} + ψ_{m} − 2n − δ)_{+}^{p} dx|_{t=0} \\ = & (1 + ε(n))^{2m}I_{δ}^{m,p}(0) < ∞ \end{matrix} (10.3)

is uniformly bounded in i.

The proof of Theorem 6.1 extends directly to our situation and ensures that

iI_{δ}^{m,p}(t) ≤ ^{i}I_{δ}^{m,p}(0) for t ≥ 0.

Here, (^{i}φ_{m} +^{i}ψ_{m} − 2n − δ)_{+} is compactly supported in B_{i}(0) due to the boundary values imposed on ^{i}g in Theorem 4.3. Therefore, we do not have to use the local closeness estimates of Lemma 5.2 in order to justify the integration by parts employed in the proof of Theorem 6.1.

Using (10.2) and (10.3), we estimate for ε and m such that (1 + ε(n))^{m} ≤ 2 as follows

(1 + ε(n))^{2m}I_{δ}^{m,p}(0) ≥^{ν_{k}}I_{δ}^{m,p}(0)
≥^{ν_{k}}I_{δ}^{m,p}(t_{k})
= \frac{1}{p} ∫_{B_{ν_{k}}(0)} (^{ν_{k}}φ(x, t_{k}) +^{ν_{k}}ψ(x, t_{k}) − 2n − δ)_{+}^{p} dx

≥ \frac{1}{p} ∫_{x∈B_{R}(x_{k})} (∑_{l=1}^{n} \frac{1}{^{ν_{k}}λ_{l}^{m}(x, t_{k})} (^{ν_{k}}λ_{l}^{m}(x, t_{k}) − 1)^{2} − δ)_{+}^{p} dx

≥ \frac{1}{p} ∫_{B_{R}(x_{k})} (\frac{1}{(1 + ε(n))^{m}} 4δ − δ)^{p}

≥ \frac{1}{p} ∫_{B_{R}(x_{k})} δ^{p}

= \frac{δ^{p}}{p} c(n)R^{n}.

As R ≥ 0 can be chosen arbitrarily large, we arrive at a contradiction. This finishes the proof of Theorem 1.6. □

Appendix A. Conformal Ricci Flow in Two Dimensions

Studying Ricci flow on ℝ^{2} for conformally flat metrics g(x, t) = e^{u(x,t)}δ is equivalent [6, 16] to considering the evolution equation

(A.1)  \frac{∂}{∂t}u = Δ_{g}u = e^{−u}Δ_{δ}u on ℝ^{2} × (0, ∞).

We are especially interested in initial conditions u_{0} : ℝ^{2} → ℝ fulfilling

(A.2)  sup_{ℝ^{2}\B_{r}(0)} |u_{0}| → 0 as r → ∞.

We wish to point out that we do not assume in the following that u_{0} fulfills an ε_{0}-closeness condition for a small constant ε_{0} > 0, corresponding to sup |u_{0}| being small. It suffices to have sup |u_{0}| bounded. Note also that in two dimensions, we do not have to study Ricci harmonic map heat flow first before we can obtain results for Ricci flow.

In this situation, we get the following existence result.

Theorem A.1. Let u_{0} ∈ C^{0} (ℝ^{2}) satisfy ∥u_{0}∥_{L^{∞}} < ∞. Then there exists u ∈ C^{∞}(ℝ^{2} × (0, ∞)) solving (A.1) such that u(·, t) → u_{0} in C_{loc}^{0} (ℝ^{2}) as t ↘ 0. As t → ∞, u(·, t) converges subsequentially in C_{loc}^{∞} to a complete flat metric.

Idea of Proof. Mollifying u_{0}, we get a sequence u_{0}^{i} ∈ C^{∞} (ℝ^{2}), i ∈ ℕ, such that u_{0}^{i} → u_{0} in C_{loc}^{0} as i → ∞, ∥u_{0}^{i}∥_{L^{∞}} ≤ ∥u_{0}∥_{L^{∞}}, and ∥D^{2}u_{0}^{i}∥_{L^{∞}} + ∥Du_{0}^{i}∥_{L^{∞}} ≤ c_{i}. According to [16], there exist smooth solutions u^{i} for t ≥ 0 solving (A.1) with u^{i}(·, 0) = u_{0}^{i} such that |u^{i}(x, t)| ≤ sup |u_{0}|. The techniques of Krylov-Safonov, Schauder, and scaling imply |D^{k}u^{i}| ≤ \frac{c_{k}}{t^{k/2}} . Thus we find a smooth solution u to (A.1). Considering ψ = u^{i}(x, t) − u^{i}(x_{0}), we can argue as in Lemma 5.2 (using ψ^{2} instead of φ_{m} + ψ_{m} − 2n) to see that for any δ > 0 we find i_{0} = i_{0}(δ, u_{0}) and ζ > 0 such that |u^{i}(x, t) − u^{i}(x_{0})| ≤ δ for |x − x_{0}|^{2} + t ≤ ζ and i ≥ i_{0}. Thus u(·, t) → u_{0} in C_{loc}^{0} as t ↘ 0. Note that in the case of (A.2), this convergence is uniform in space. The remaining details are similar to the higher dimensional case. □

Theorem A.2. Let u_{0} be as in Theorem A.1 and assume that (A.2) is fulfilled. Let u be the solution to (A.1) obtained there. Then u(·, t) → 0 in C^{∞} as t → ∞.

Idea of Proof. The proof is similar to the proof of Theorem 1.3. Choose p ≥ sup |u_{0}| + 1. According to the construction of u in the proof of Theorem A.1 and the maximum principle, we have p ≥ sup_{x} |u_{0}(x)| + 1 ≥ sup_{x, t} |u(x, t)| + 1. Instead

of Theorem 6.1, however, we use the following estimate (and a similar argument for the negative part of u)

\frac{d}{dt} ∫_{ℝ^{2}} \frac{1}{p}(u − δ)_{+}^{p} dx = ∫_{\{u>δ\}} (u − δ)^{p−1}e^{−u}Δ_{δ}u dx

= ∫_{{u>δ}} (u − δ)^{p−2}((u − δ) − p + 1)e^{−u}|∇u|^{2} dx ≤ 0.

□

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Oliver Schnürer: Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

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

Felix Schulze: Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

E-mail address: Felix.Schulze@math.fu-berlin.de

Miles Simon: Universität Freiburg, Eckerstraße 1, 79104 Freiburg i. Br., Germany

E-mail address: msimon@gmx.de