Graphical translators for mean curvature flow

Abstract

In this paper we provide a full classification of complete translating graphs in ${\mathbf {R}}^3$. We also construct $(n-1)$-parameter families of new examples of translating graphs in ${\mathbf {R}}^{n+1}$.

Appendix: compactness theorems

Theorem 12.1

For $k=1, 2, \ldots$, let $\Omega _k$ be a convex open subset of ${\mathbf {R}}^n$ and let $u_k:\Omega _k\rightarrow {\mathbf {R}}$ be a smooth translator. Let $M_k$ be the graph of $u_k$. Suppose that W is a connected open subset of ${\mathbf {R}}^n$ such that for each k,

$$\begin{aligned} W\times (-k,k) \end{aligned}$$

does not contain any of the boundary of $M_k$.

Then, after passing to a subsequence, $M_k\cap (W\times {\mathbf {R}})$ converges weakly in $W\times {\mathbf {R}}$ to a translator M that is g-area-minimizing. Furthermore, if S is a connected component of M, then either

• (1) S is the graph of a smooth function over an open subset of W and the convergence to S is smooth, or

• (2) $S=\Sigma \times {\mathbf {R}}$, where $\Sigma$ is a variety in W that is minimal with respect to the Euclidean metric on ${\mathbf {R}}^n$. The singular set of $\Sigma$ has Hausdorff dimension at most $n-7$.

Proof

Since $\Omega _k$ is convex and since $M_k$ and its vertical translates form a g-minimal foliation of $\Omega _k\times {\mathbf {R}}$, standard arguments (cf. [12, §6.2]) show that $M_k$ is g-area-minimizing as an integral current, or even as a mod 2 flat chain.

Thus the standard compactness theorem (cf. [15, §34.5]) gives subsequential convergence (in the local flat topology) to a g-area-minimizing hypersurface M (with no boundary in $W\times I$). Also, standard arguments show that the support of $M_k$ converges to the support of M. Hence we will not make a distinguish here between the flat chain and its support.

For notational simplicity, let us assume that M is connected. Clearly, each vertical line intersects M in a connected set.

Case 1: M contains a vertical segment of some length $\epsilon >0$. Let M(s) be the result of translating M vertically by a distance s, where $0<s<\epsilon$. Then by the strong maximum principle of L. Simon [16], $M=M(s)$. Since this is true for all s with $0<s<\epsilon$, it follows that $M=\Sigma \times {\mathbf {R}}$ for some $\Sigma$. Since $\Sigma \times {\mathbf {R}}$ is g-area-minimizing, its singular set has Hausdorff dimension at most $(n+1)-7$, and therefore the singular set of $\Sigma$ has Hausdorff dimension at most $n-7$. Since $\Sigma \times {\mathbf {R}}$ is g-minimal, $\Sigma$ must be minimal with respect to the Euclidean metric.

Case 2: M contains no vertical segment. Then M is the graph of a continuous function u whose domain is an open subset of W. Let $\overline{{\mathbf {B}}(p,r)}$ be a closed ball in the domain of u. Then $\overline{{\mathbf {B}}(p,r)}$ is contained in the domain of $u_k$ for large k, and $u_k$ converges uniformly to u on $\overline{{\mathbf {B}}(p,r)}$. (The uniform convergence follows from monotonicity.) By Theorem 5.2 of [5] (rediscovered in [4, Theorem 1]), the convergence is smooth on ${\mathbf {B}}(p,r)$. $\square$

Theorem 12.2

For $k=1, 2, \ldots$, let $\Omega _k$ be a convex open subset of ${\mathbf {R}}^n$ such that the $\Omega _k$ converge to an open set $\Omega$. Let $u_k:\overline{\Omega _k}\rightarrow {\mathbf {R}}$ be a translator with boundary values 0, and let $M_k$ be the graph of $u_k$. Then, after passing to a subsequence, the $M_k$ converge smoothly in ${\mathbf {R}}^n\times (0,\infty )$ to a smooth translator M. If S is a connected component of M, then either

• (1) S is the graph of a smooth function whose domain is an open subset of $\Omega$, or

• (2) $S=\Sigma \times [0,\infty )$, where $\Sigma$ is an $(n-1)$-dimensional affine plane in ${\mathbf {R}}^n$.

Furthermore, M is a smooth manifold-with-boundary in a neighborhood of every point of $\partial M$ where $\partial M$ is smooth, and the convergence of $M_k$ to M is smooth up the boundary wherever the convergence of $\partial M_k$ to $\partial M$ is smooth.

Proof

Case 1: S contains a point p in $\partial \Omega \times (0,\infty )$. Let $\Sigma$ be the connected component of $\partial \Omega$ such that $p\in \Sigma \times (0,\infty )$. Then by the strong maximum principle ( [16, 17] or [23, Theorem 7.3]), M contains all of $\Sigma \times (0,\infty )$, and therefore (since $\Omega$ is convex) $\Sigma$ must be a plane. It follows that the convergence to S is smooth in ${\mathbf {R}}^n\times (0,\infty )$.

Case 2: S contains no point in $\partial \Omega \times {\mathbf {R}}$. That is, S is contained in $\Omega \times {\mathbf {R}}$. Indeed, S is contained in $\Omega \times [0,\infty )$, so it cannot be translation-invariant in the vertical direction. Thus by Theorem 12.1, S is a smooth graph over an open subset of $\Omega$, and the convergence to S is smooth.

The assertions about boundary behavior follow, for example, from the Hardt-Simon Boundary Regularity Theorem [6]. (Note that the tangent cone to M at a regular point of $\partial M$ is, after a rotation of ${\mathbf {R}}^n$, a cone in ${\mathbf {R}}^{n-1}\times [0,\infty )\times [0,\infty )$ whose boundary is the plane ${\mathbf {R}}^{n-1}\times \{0\}\times \{0\}$.) $\square$