Subsequent singularities in mean-convex mean curvature flow

Abstract

We use Ilmanen’s elliptic regularization to prove that for an initially smooth mean convex hypersurface in $\mathbf {R}^n$ moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity. In particular, the tangent flows are all shrinking spheres or cylinders. Previously this was known only (1) for $n\le 7$, and (2) for arbitrary $n$ up to the first singular time.

Appendices

Appendix A: translating solitions

Here we prove existence and regularity of the translating solitons $N_\lambda$ that were used in the proofs of Theorems 3 and 6.

Lemma 8

Let $M$ be a connected, oriented hypersurface such that the mean curvature $H$ is everywhere a nonnegative multiple of the unit normal $\nu$ and such that $M$ translates with constant velocity $\mathbf {v}\ne 0$ under mean curvature flow, i.e., such that

$$\begin{aligned} H\cdot \nu = \mathbf {v}\cdot \nu . \end{aligned}$$

Then $H\cdot \nu$ has no local interior minimum unless $H\equiv 0$ (in which case, $M$ lies in $\Sigma \times L$ where $L$ is a line parallel to $\mathbf {v}$ and where $\Sigma$ is a minimal hypersurface in $L^\perp$).

Proof

This is because under mean curvature flow, $H\cdot \nu$ satisfies a nice second order parabolic equation. The result then follows immediately from the strong maximum principle. See for example [8, Theorem 2]. $\square$

Lemma 9

Let $M_1$ and $M_2$ be two disjoint, compact, connected smooth hypersurfaces in $\mathbf {R}^{n+1}$ such that $M_i$ translates by $\mathbf {v}\ne 0$ under mean curvature flow. If the function

$$\begin{aligned} (p_1,p_2)\in M_1\times M_2 \mapsto |p-q| \end{aligned}$$

achieves its minimum value $\mu$ at $(p,q)$ for some interior points $p\in M_1$ and $q\in M_2$, then $M_1$ and $M_2$ lie in parallel hyperplanes, and the vector $\mathbf {v}$ is parallel to those planes.

Proof

Since the result is essentially local, we may assume that $M_1$ and $M_2$ are graphs of functions $f_1$ and $f_2$ over the same domain $D$ in a hyperplane perpendicular to the line $p_1p_2$, with $f_2>f_1$ at all points. Note that $f_2-f_1$ has a local minimum. Thus by the maximum principle, it is constant: $f_2-f_1\equiv \mu$. Consequently, for each point $x$ in the domain of the $f_i$, we have $f_2(x)-f_1(x)=\mu$, the minimum distance between the two graphs. But that implies that $Df_2(x)=Df_1(x)=0$. In other words, $f_1$ and $f_2$ must be constant. Thus $M_1$ and $M_2$ are planar. The fact that $\mathbf {v}$ is parallel to the those planes follows immediately (by Lemma 8, for example). $\square$

Theorem 10

Let $n\ge 2$. Let $W$ be a bounded region in an $\mathbf {R}^n$ with piecewise smooth, mean convex boundary and let $\lambda \ge 0$. Let $S$ be a smooth, closed $(n-1)$-manifold in $(\partial W)\times \mathbf {R}$ that is graph-like in the following sense: each line $\{x\}\times \mathbf {R}$ with $x\in \partial W$ intersects $S$ either in a point or in a line segment.

Then there is a smooth, compact $n$-manifold $N$ in $\overline{W}\times \mathbf {R}$ such that

• (1) $\partial N=S$,

• (2) $N{\setminus } S$ is the graph of a smooth function $f: W\rightarrow \mathbf {R}$,

• (3) $N$ translates with velocity $-\lambda \mathbf {e}_{n+1}$.

Proof

First we recall that there is an entire function $\phi : \mathbf {R}^n\rightarrow \mathbf {R}$ such that $\phi$ is rotationally symmetric (i.e., such that $\phi (x)$ depends only on $|x|$) and such that the graph of $\phi$ translates with velocity $-\lambda \mathbf {e}_{n+1}$. By adding a constant to $\phi$, we can assume that $S$ lies below the graph of $\phi$ Let $b= \min \{y: (x,y)\in S\}$. Let $R$ be the region

$$\begin{aligned} \{ (x,y) \in \overline{W}\times \mathbf {R}: b \le y \le \phi (x)\}. \end{aligned}$$

Now let $N$ be a variety (e.g. an integral current) such that $N$ minimizes the $\lambda$-area among all varieties that have boundary $S$ and that lie in the region $R$.

By the maximum principle, $N{\setminus } S$ lies in the interior of $R$. By [2], $N$ is a smooth manifold with boundary in a neighborhood of $S$. Thus the singular set $X$ is a compact subset of the interior of $R$. Also, the singular set has Hausdorff dimension at most $n-7$.

Let $N(s)$ be the result of translating $N$ upward by distance $s$.

Claim 1: for $s>0$, the interiors of $N$ and $N(s)$ are disjoint.

Proof of Claim 1: Suppose not. Let $s$ be the largest number such that $N$ and $N(s)$ intersect in the interior or are tangent to each other at a common boundary point. By the boundary maximum principle, they cannot be tangent to each other at a common boundary point. Thus they touch at an one or more interior points. Let $Z$ be the set of interior points where they touch. Then $Z$ is a compact subset $W\times \mathbf {R}$. Now it follows that for sufficiently small $\epsilon >0$, the function

$$\begin{aligned} (p,q) \in N\times N(s+\epsilon ) \mapsto |p-q| \end{aligned}$$

will have an interior local minimum $(p,q)$. But Lemma 9 then implies that $N$ and $N(s+\epsilon )$ lie in parallel vertical planes, which is clearly incompatible with the boundary conditions. This completes the Proof of Claim 2.

Claim 1 implies that $N{\setminus } S$ is the graph of a continuous function $f: W\rightarrow \mathbf {R}$.

It remains to show that the singular set $X$ is empty.

Let $K\subset W$ be a compact set with smooth boundary such that the singular set $X$ is contained in the interior of $K\times \mathbf {R}$.

By Lemma 8, $\hbox {Tan}_pN$ cannot be vertical at any regular point. In particular, this means that the function $f$ is smooth on except on the compact set $\pi (X)$ (where $\pi :\mathbf {R}^n\times \mathbf {R}\rightarrow \mathbf {R}^n$ is projection onto the first factor).

Claim 2: $\sup \{ |Df(x)| : x\in K{\setminus } \pi (X)\} \le \sup \{|Df(x)|: x\in \partial K\}$.

For suppose not. Then for every small $\epsilon >0$, the minimum distance between the graph of $f|K$ and the graph of $(f+\epsilon )|K$ is attained at a pair of interior points $p\in \text {graph}(f|K)$ and $q\in \text {graph}((f+\epsilon )|K)$. Note that the tangent cone to $N$ at $p$ is contained in the halfspace $\{v: v\times (q-p)\le 0\}$, which implies that the tangent cone is a plane, which by the standard regularity theory for area minimizing hypersurfaces implies that $p$ is a regular point. Likewise $q$ is a regular point. But now $p$ and $q$ violate Lemma 9. This completes the Proof of Claim 2.

Now Claim 2 implies that $f$ is locally Lipschitz. But that implies (by standard PDE or by standard GMT) that $f$ is smooth on $W$. (For the GMT argument, consider the tangent cone at any point. By Claim 2, the tangent cone is the graph of a lipschitz function. Thus it suffices to show that there is no nonplanar minimal cone that is the graph of lipshitz function $u$. By dimension reducing, we may assume that the cone is smooth away from the origin. Let $\mathbf {v}$ be the unit vector for which $f(\mathbf {v})$ is a maximum. Near $\mathbf {v}$, the graph of $f$ lies below the tangent plane at $(\mathbf {v},f(\mathbf {v}))$. Hence by the strong maximum principle, $f$ is linear in a neighborhood of $\mathbf {v}$. By analytic continuation, $f$ is linear everywhere). $\square$

Appendix B: $\kappa _1/h$ bounded below implies convex type

Theorem 11

Suppose $t \in [0,\infty )\mapsto M(t)$ is a (possibly singular) mean curvature flow of mean convex $(n-1)$-dimensional surfaces in a smooth $n$-dimensional Riemannian manifold. Let $Z=(z,t)$ be a spacetime singular point of the flow with $t>0$, and suppose that

$$\begin{aligned} \kappa _1/h \ge c > -\infty \end{aligned}$$

(*)

in a space-time neighborhood of $Z$.

Then the singular point $z\in M_t$ has convex type.

Proof

In [8], this (and all the results of that paper) are proved under the restriction:

• $(\dag )$ For $n>7$, we work in $\mathbf {R}^n$ and we consider the flow only up to and including the first singular time.

In fact, the only place that restriction is explicitly used is in the proof of [8, Theorem 4]. In the proof of that theorem, the restriction $(\dag )$ is used only to show that $\kappa _1/h$ is bounded below in a (space-time) neighborhood of the singularity in question up to and including the time $t$. Thus, if we assume (as here) the hypothesis (*) that $\kappa _1/h$ is bounded below in a neighborhood of $Z$, then Theorem 4, and therefore all the other theorems of [8], hold at that singularity, without assuming the restriction $(\dag )$. In particular, the singularity is of convex type by [8, Theorem 1] and its corollary.

Incidentally, the assumption (*) that $\kappa _1/h$ is bounded below in a spacetime neighborhood of the singularity (both before and after the singularity) significantly simplifies the proofs in [8] in various places. In particular, one can (throughout that paper) work with the class $\mathcal {F}$ of all blowup flows (limit flows) at $Z$ rather than the class of what are are called there “special limit flows”. For example, [8, Theorem 9] asserts that any each flow in $\mathcal {F}$ is convex for all times $\tau \le 0$. But the class of blowup flows at a space-time point is trivially closed under time translation, so it follows immediately that the restriction $\tau \le 0$ is not necessary. (In [8], an additional argument involving an unpublished reference [W6] was given to remove the restriction $\tau \le 0$). $\square$