# Mean Curvature Flow with Boundary

Brian White  Department of Mathematics
Stanford University
Stanford, CA 94305<EMAIL_ADDRESS>
(Date: January 9, 2019.)

###### Abstract.

We develop a theory of surfaces with boundary moving by mean curvature flow.
In particular, we prove a general existence theorem by elliptic
regularization, and we prove boundary regularity at all positive times under
rather mild hypotheses.

###### 2010 Mathematics Subject Classification:

Primary 53C44; Secondary 49Q20.

##  1\. Introduction

In this paper, we study mean curvature flow for surfaces with boundary: each
point moves so that the normal component of its velocity is equal to the mean
curvature, and the boundary remains fixed. (More generally, the boundary can
be time-dependent, but prescribed.) In particular,

  1. (1)

We define integral Brakke flows with boundary and prove the basic properties.
This is a rather general class that includes network flows. See §5.

  2. (2)

We define the subclass of standard Brakke flows with boundary. These are flows
as in (1) with additional nice properties. In particular, for almost all
times, the moving surface has the prescribed boundary in the sense of mod $2$
homology. This condition excludes, for example, surfaces with triple junctions
(or, more generally, with odd-order junctions). See §10.

  3. (3)

Following Ilmanen [ilmanen-elliptic], we use elliptic regularization to prove
existence of standard Brakke flows with boundary for any prescribed initial
surface. See §11.

  4. (4)

We prove a strong boundary regularity theorem for standard Brakke flows with
boundary. See §14.

As a special case of some of the results (Theorems 29 and 37), we have

######  Theorem 1.

Let $N$ be a smooth, compact, $(m+1)$-dimensional Riemannian manifold with
smooth, strictly mean-convex boundary. Let $M_{0}$ be a smoothly embedded
$m$-dimensional submanifold of $N$ whose boundary is a smooth submanifold
$\Gamma$ of $\partial N$. (More generally, $M_{0}$ can be any $m$-rectifiable
set of finite $m$-dimensional measure whose boundary, in the sense of mod $2$
flat chains, is a smoothly embedded submanifold $\Gamma$ of $\partial N$.)
Then there is a standard Brakke flow

$t\in[0,\infty)\mapsto M(t)$

with boundary $\Gamma$ such that $M(0)=M_{0}$. Furthermore, if $M(\cdot)$ is
any standard Brakke flow with boundary $\Gamma$, then the flow is smooth (with
multiplicity one) in a spacetime neighborhood of each point $(p,t)$ with
$p\in\Gamma$ and $t>0$.

Thus (under the hypotheses of the theorem) we have boundary regularity at all
positive times, even after interior singularities may have occurred.

The regularity in Theorem 1 is uniform as $t\to\infty$. For given any sequence
of times $t_{i}\to\infty$, there is a subsequence $t_{i(j)}$ such that the
time-translated flows $M_{i(j)}(t)=t\mapsto M(t-t_{i(j)})$ converge to a
standard eternal limit flow $M^{\prime}(\cdot)$ by §7 and Theorem 27. The
limit flow is regular at the boundary by Theorem 37, and thus the convergence
$M_{i(j)}(\cdot)\to M^{\prime}(\cdot)$ is smooth near the boundary by the
local regularity theory in [white-local].

The notion of standard Brakke flow with boundary is crucial in Theorem 1; the
regularity assertion of Theorem 1 is false for general integral Brakke flows
with boundary, because interior singularities can move into the boundary.
Consider, for example, a configuration consisting of three curves in the unit
disk in $\mathbf{R}^{2}$, where the three curves meet at equal angles at a
point $P$ in the interior of the disk and where the other endpoints of the
curves are three points $A$, $B$, and $C$ on the unit circle. The
configuration evolves so that the three points on the unit circle are fixed,
and so that interior points move with normal velocity equal to the curvature.
This implies that the triple junction $P(t)$ moves in such a way that the
curves continue to meet at equal angles at the junction. If each interior
angle of the triangle $ABC$ is less than $120^{\circ}$, then the triple
junction remains in the interior, and we have boundary regularity at all
times. However, if one of the angles is greater than $120^{\circ}$, then
$P(t)$ bumps into the corresponding vertex in finite time and thus creates a
boundary singularity.

The flow described in the previous paragraph is an integral Brakke flow with
boundary $\\{A,B,C\\}$. However, it is not a standard Brakke flow with
boundary $\\{A,B,C\\}$, because if we think of the network as a mod $2$ chain,
then the boundary contains $P(t)$ in addition to $A$, $B$, and $C$.

It is natural to wonder whether such a boundary singularity could occur if the
original surface is smooth and embedded. In the case of curves, the answer is
“no”: the flow would remain smooth everywhere for all time by the analog of
Grayson’s Theorem. However, although I do not yet have a proof, I believe that
there is an integral Brakke flow

$t\in[0,\infty)\mapsto M(t)$

with boundary $\Gamma$, where $\Gamma$ consists of smooth embedded curves in
the unit sphere in $\mathbf{R}^{3}$, such that $M(0)$ is a smoothly embedded
surface in the unit ball and such that later the moving surface develops a
triple junction curve that eventually bumps into the boundary. Note that this
could only happen if we had non-uniqueness, since by Theorem 1 there is a
standard Brakke flow $M^{\prime}(\cdot)$ with the same initial surface and the
same boundary, and that flow never develops boundary singularities. Of course
the two flows are equal at least until singularities occur, but they must
differ as soon as $M(\cdot)$ has a triple junction curve.

The regularity part of Theorem 1 is a consequence of the following general
theorem (see Theorem 36):

######  Theorem 2.

Suppose $t\in[0,T]\mapsto M(t)$ is an $m$-dimensional standard mean curvature
flow with boundary $\Gamma$ in a smooth, $(m+1)$-dimensional Riemannian
manifold. If a tangent flow at $(p,t)$ is contained in a wedge, where
$p\in\Gamma$ and $t>0$, then $(p,t)$ is a regular point of the flow
$M(\cdot)$.

Two features of this paper seems to be new even for Brakke flows without
boundary. First, when taking limits of Brakke flows, we get improved
subsequential convergence of the mean curvature for almost all times; see
Remark 23. Second, to prove Huisken’s monotonicity formula, one needs to know
that the mean curvature vector is orthogonal to the variety almost everywhere.
Brakke [brakke]*§5 proved such orthogonality for arbitrary integral varifolds
of bounded first variation. However, the proof is rather long (40 pages). In
this paper, we give a much easier proof that such orthogonality is preserved
when taking weak limits of mean curvature flows, and thus in particular
orthogonality holds in flows coming from elliptic regularization. For this
reason, we have chosen to include orthogonality of mean curvature as part of
the definition of Brakke flow.

For simplicity, in most of the paper we consider flows in which the boundary
is fixed. In §15, we indicate how to modify the theory for moving boundaries.

Although mean curvature flow has been extensively studied, there have been
only a few investigations of mean curvature flow of surfaces with boundary.
The papers [white-topology] and [white-local] dealt with mean curvature flow
of surfaces both with and without boundary. In [stone], Stone proved a theorem
analogous to the boundary regularity part of Theorem 1, but only at the first
singular time and under additional, rather restrictive hypotheses. In
particular, the moving surface was assumed to be mean convex and to satisfy a
Type I estimate. In [ilmanen-white-cones], mean curvature flow with boundary
was used to prove sharp lower density bounds for area-minimizing hypercones.

##  2\. Notation

In this paper, $U$ is a smooth Riemannian manifold (possibly with smooth
boundary). We do not assume that $U$ is complete: it may be an open subset of
a larger Riemannian manifold. We let $G_{m}(U)$ denote the Grassman bundle of
pairs $(x,P)$ where $x\in U$ and $P$ is an $m$-dimensional linear subspace of
$\operatorname{Tan}(U,x)$. We let $\mathcal{X}(U)$ denote the space of
continuous, compactly supported vectorfields on $U$. We let
$\mathcal{X}_{m}(U)$ denote the space of continuous, compactly supported
functions on $G_{m}(U)$ that assign to each $(x,P)$ in $G_{m}(U)$ a vector in
$\operatorname{Tan}(N,x)$.

If $M$ is a Radon Measure on $U$ and if $f$ is a function on $U$, we let

$Mf=\int f\,dM.$

If $\Gamma$ is a $k$-dimensional submanifold of $U$ (or, more generally, a
$k$-rectifiable set of locally finite $k$-dimensional measure), then (by
slight abuse of notation) we will also use $\Gamma$ to denote the associated
Radon measure. Thus

$\int f\,d\Gamma=\int_{\Gamma}f\,d\mathcal{H}^{k}$

and

$\Gamma(K)=\mathcal{H}^{k}(\Gamma\cap K).$

##  3\. $L^{p}$ vectorfields

In the following theorem, $1_{K}$ denotes the characteristic function of the
set $K$. Thus if $M$ is a Radon measure on $U$ and if $p<\infty$, then

$\|Y1_{K}\|_{L^{p}(M)}=\left(\int_{K}|Y|^{p}\,dM\right)^{1/p}.$

Similarly,

$\|Y1_{K}\|_{L^{\infty}(M)}$

is the essential supremum of $|Y|$ on the set $K$ with respect to the measure
$M$.

######  Theorem 3.

Let $V_{i}$ and $V$ be rectifiable $m$-varifolds in $U$ such that
$V_{i}\rightharpoonup V$. Let $M_{i}$ and $M$ be the associated Radon measures
on $U$. Suppose that $p\in(1,\infty]$ and that $Y_{i}$ is a Borel vectorfield
on $U$ such that

$c_{K}:=\sup_{i}\|Y_{i}1_{K}\|_{L^{p}(M_{i})}<\infty$

for every $K\subset\subset U$. Then, after passing to a subsequence, there is
a Borel vectorfield $Y$ on $U$ in $L^{p}_{\textnormal{loc}}(M)$ such that

$\int X(x,\operatorname{Tan}(M_{i}(x))\cdot Y_{i}(x)\,dM_{i}(x)\to\int
X(x,\operatorname{Tan}(M,x))\cdot Y(x)\,dM(x)$

for every $X\in\mathcal{X}_{m}(U)$.

###### Proof.

Define

$\displaystyle L_{i}:\mathcal{X}_{m}(U)\to\mathbf{R},$ $\displaystyle
L_{i}(X)=\int X(x,\operatorname{Tan}(M_{i}(x))\cdot Y_{i}(x)\,dM_{i}(x).$

If $K\subset\subset U$ and if $X\in\mathcal{X}_{m}(U)$ is supported in
$\\{(x,P):x\in K\\}$, then by Hölder’s Inequality,

(1) $|L_{i}(X)|\leq
c_{K}\left(\int|X(x,\operatorname{Tan}(M_{i},x))|^{q}\,dM_{i}(x)\right)^{1/q},$

where $q=p/(p-1)$, or, equivalently,

(2) $|L_{i}(X)|\leq c_{K}\left(\int|X(x,P)|^{q}\,dV_{i}(x,P)\right)^{1/q}.$

From (1), we see that

(3) $|L_{i}(X)|\leq c_{K}d_{K}{}^{1/q}\sup|X|$

where $d_{K}:=\sup_{i}M_{i}(K)$. Note that $d_{K}<\infty$ by weak convergence
of $M_{i}$ to $M$.

By (3) and Banach-Alaoglu, we can assume, after passing to a subsequence, that
there is an $L:\mathcal{X}_{m}(U)\to\mathbf{R}$ such that

$L_{i}(X)\to L(X)\quad\text{for every $X\in\mathcal{X}_{m}(U)$}.$

Letting $i\to\infty$ in (2) gives

(4) $|L(X)|\leq c_{K}\left(\int|X(x,P)|^{q}\,dV(x,P)\right)^{1/q}.$

By the Riesz Representation Theorem, there is an $M$-measurable vectorfield
$\tilde{Y}(x,P)$ such that

(5) $L(X)=\int X(x,P)\cdot\tilde{Y}(x,P)\,dV(x,P)$

for every $X\in\mathcal{X}_{m}(U)$. Since $V$ is rectifiable and since $M$ is
the associated Radon measure on $U$, we can rewrite (5) as

$L(X)=\int
X(x,\operatorname{Tan}(M,x))\cdot\tilde{Y}(x,\operatorname{Tan}(M,x))\,dM(x).$

Thus if we set $Y(x)=\tilde{Y}(x,\operatorname{Tan}(M,x))$, then we have

$L(X)=\int X(x,\operatorname{Tan}(M,x))\cdot Y(x)\,dM(x),$

as desired. ∎

######  Corollary 4.

In Theorem 3, if $Y_{i}(x)$ is perpendicular to $\operatorname{Tan}(M_{i},x)$
for $M_{i}$-almost every $x$, then $Y(x)$ is perpendicular to
$\operatorname{Tan}(M,x)$ for $M$-almost every $x$.

###### Proof.

Suppose $X\in\mathcal{X}_{m}(U)$. Let

$\tilde{X}(x,P)=\Pi_{P}X(x,P).$

Then $\tilde{X}$ is also in $\mathcal{X}_{m}(U)$. Hence

$\int\tilde{X}(x,\operatorname{Tan}(M_{i},x)\cdot
Y_{i}(x)\,dM_{i}(x)\to\int\tilde{X}(x,\operatorname{Tan}(M_{i},x))\cdot
Y(x)\,dM(x),$

i.e,

$\int\Pi_{\operatorname{Tan}(M_{i}(x)}X(x)\cdot
Y(x)\,dM_{i}(x)\to\int\Pi_{\operatorname{Tan}(M,x)}X(x)\cdot Y(x)\,dM(x)$

The left hand side is $0$, so

$\int\Pi_{\operatorname{Tan}(M,x)}X(x)\cdot Y(x)\,dM(x)=0$

or, equivalently,

$\int X(x)\cdot\Pi_{\operatorname{Tan}(M,x)}Y(x)\,dM(x)=0$

for all $X\in\mathcal{X}(U)$. Since $\mathcal{X}(U)$ is dense in
$L^{q}_{\textnormal{loc}}(M)$ (cf. [ilmanen-elliptic]*§7.4), it follows that

$\Pi_{\operatorname{Tan}(M,x)}Y(x)=0$

for $M$-almost every $x$. ∎

######  Theorem 5.

Suppose $\beta_{i}$ $(i=1,2,\dots)$ and $\beta$ are Radon measures on $U$ such
that $\beta_{i}$ converges to $\beta$. Suppose that $p\in(1,\infty]$ and that
$Y_{i}$ is a Borel vectorfield on $U$ such that

$\sup_{i}\|1_{K}Y_{i}\|_{L^{p}(\beta_{i})}<\infty$

for every $K\subset\subset U$. Then (after passing to a subsequence) there is
a Borel vectorfield $Y$ on $U$ in $L^{p}_{\textnormal{loc}}(M)$ such that

$\int X\cdot Y_{i}\,d\beta_{i}\to\int X\cdot Y\,d\beta$

for all $X$ in $\mathcal{X}(U)$.

The proof is essentially the same as the proof of Theorem 3, except that we
work in $U$ rather than in $G_{m}(U)$.

##  4\. A Varifold Closure Theorem

Let $U$ be an open subset of a smooth Riemannian manifold, let
$\mathcal{M}(U)$ be the set of all Radon measures on $U$, and let
$\mathcal{M}_{k}(U)$ be the set of Radon measures associated to
$k$-dimensional rectifiable varifolds in $U$. Equivalently,
$\mathcal{M}_{k}(U)$ is the set of Radon measures $M$ such that

  1. (i)

$M(U\setminus S)=0$ for some countable union $S$ of $k$-dimensional $C^{1}$
submanifolds of $U$, and

  2. (ii)

$M$ is absolutely continuous with respect to $\mathcal{H}^{k}$.

Let $I\mathcal{M}_{k}(U)$ be the set of $M\in\mathcal{M}_{k}(U)$ such that
$\Theta(M,x)$ is an integer for $M$-almost every $x$.

If $M\in\mathcal{M}_{k}(U)$, we let $\operatorname{Var}(M)$ be the associated
$k$-dimensional varifold in $U$. Thus $M\in I\mathcal{M}_{k}(U)$ if and only
if $\operatorname{Var}(M)$ is an integral varifold.

Now suppose that $M\in\mathcal{M}_{k}(U)$ and that $\operatorname{Var}(M)$ has
bounded first variation. Then there exist an $M$-locally integrable
vectorfield $H(\cdot)=H(M,\cdot)$, a Radon measure $\beta(M)$ that is singular
with respect to $M$, and a $\beta(M)$-locally integrable unit vectorfield
$\eta(\cdot)=\eta(M,\cdot)$ with the following property: if $X$ is any
compactly supported, $C^{1}$ vectorfield on $U$, then

(6) $\int_{M}\operatorname{Div}_{M}X\,dM=-\int H\cdot X\,dM+\int
X\cdot\eta\,d\beta.$

######  Definition 6.

If $\Gamma$ is a properly embedded $(m-1)$-dimensional submanifold of $U$,
then $\mathcal{V}_{m}(U,\Gamma)$ is the space of $M\in I\mathcal{M}_{m}(U)$
such that $M$ has bounded first variation and such such that

  1. (1)

$\beta(M)\leq\mathcal{H}^{m-1}\llcorner\Gamma$.

  2. (2)

$H(\cdot)$ and $\operatorname{Tan}(M,\cdot)$ are perpendicular $M$-almost
everywhere.

(As mentioned in the introduction, the orthogonality condition (2) in
Definition 6 is superfluous according to a theorem of Brakke [brakke]*§5, but
the proof of that theorem is rather difficult. Including Condition (2) in the
definition makes that theorem unnecessary for us.)

Let

(7) $\nu(M,x)=\lim_{r\to
0}\frac{1}{\omega_{m-1}r^{m-1}}\int_{\mathbf{B}(x,r)}\eta(\cdot)\,d\beta$

where the limit exists, and let $\nu(M,x)=0$ where the limit does not exist.
Note that the limit exists $\mathcal{H}^{m-1}$ almost everywhere. Note also
that we can rewrite (6) as

(8) $\int\operatorname{Div}_{M}X\,dM=-\int H\cdot X\,dM+\int_{\Gamma}\nu\cdot
X\,d\mathcal{H}^{m-1}$

or (using the notational conventions described in Section 2) as

$\int\operatorname{Div}_{M}X\,dM=-\int H\cdot X\,dM+\int\nu\cdot X\,d\Gamma.$

######  Remark 7.

The condition that $\beta\leq\mathcal{H}^{m}\llcorner\Gamma$ is equivalent to
the condition that $|\nu(x)|\leq 1$ for $\mathcal{H}^{m-1}$ almost every
$x\in\Gamma$.

######  Remark 8.

If $M\in\mathcal{V}_{m}(U,\Gamma)$, then $\nu(M,\cdot)$ is perpendicular to
$\Gamma$ at almost every point of $\Gamma$ by [allard-boundary]*§3.1.

In the following theorem, we write $H_{i}(\cdot)$ and $H(\cdot)$ for
$H(M_{i},\cdot)$ and $H(M,\cdot)$, and $\nu_{i}(\cdot)$ and $\nu(\cdot)$ for
$\nu(M_{i},\cdot)$ and $\nu(M,\cdot)$.

######  Theorem 9 (Varifold Closure Theorem).

Suppose for $i=1,2,\dots$ that $M_{i}\in\mathcal{V}_{m}(U,\Gamma_{i})$, where
the $\Gamma_{i}$ are smooth $(m-1)$-dimensional submanifolds of $U$ that
converge in $C^{1}$ to a smooth manifold $\Gamma$. Suppose that the $M_{i}$
converge to a Radon measure $M$ and that

(9) $d_{K}:=\sup_{i}\int_{K}|H_{i}|^{2}\,dM_{i}<\infty$

for every $K\subset\subset U$. Then

  1. (1)

For every $K\subset\subset U$,

$\sup_{i}\left(\int_{K}|H_{i}|\,dM_{i}+\beta(M_{i})(K)\right)<\infty.$

  2. (2)

$M\in I\mathcal{M}_{m}(U)$ and $\operatorname{Var}(M_{i})$ converges to
$\operatorname{Var}(M)$. Thus if $f:G_{m}(U)\to\mathbf{R}$ is continuous and
compactly supported, then

$\int f(x,\operatorname{Tan}(M_{i},x))\,dM_{i}(x)\to\int
f(x,\operatorname{Tan}(M,x))\,dM(x)$

  3. (3)

If $X\in\mathcal{X}_{m}(U)$, then

$\int X(x,\operatorname{Tan}(M_{i},x))\cdot H_{i}(x)\,dM_{i}\to\int
X(x,\operatorname{Tan}(M,x)))\cdot H(x)\,dM(x).$

  4. (4)

If $Z\in\mathcal{X}_{m-1}(U)$, then

$\int
Z(x,\operatorname{Tan}(\Gamma_{i},x))\cdot\nu_{i}(x)\,d\Gamma_{i}(x)\to\int
Z(x,\operatorname{Tan}(\Gamma_{i},x)\cdot\nu(x)\,d\Gamma(x).$

  5. (5)

$M\in\mathcal{V}_{m}(U,\Gamma)$.

###### Proof.

Since the $M_{i}$ converge to $M$,

(10) $c_{K}:=\sup_{i}M_{i}(K)<\infty$

for every $K\subset\subset U$. Thus

$\displaystyle\int_{K}|H_{i}|\,dM_{i}$
$\displaystyle\leq(M_{i}(K))^{1/2}\left(\int_{K}|H_{i}|^{2}\,dM_{i}\right)^{1/2}$
$\displaystyle\leq(c_{K}d_{K})^{1/2}$

Also,

$\sup_{i}\beta(M_{i})(K)\leq\sup_{i}\mathcal{H}^{m-1}(\Gamma_{i}\cap
K)<\infty$

since $\Gamma_{i}$ converges in $C^{1}$ to $\Gamma$. This proves Assertion
(1).

By Assertion (1) and by Allard’s Closure Theorem for Integral Varifolds
([allard-first-variation]*Theorem 6.4 or [simon-gmt]*§42.8 or [simon-new-
gmt]*chapter 8, §5.9), the varifolds $\operatorname{Var}(M_{i})$ converge
(after passing to a subsequence) to an integral varifold $V$ of bounded first
variation. Note that $\mu_{V}=M$, so the limit $V=\operatorname{Var}(M)$ does
not depend on the choice of subsequence. Thus the original sequence
$\operatorname{Var}(M_{i})$ converges to $\operatorname{Var}(M)$. Thus we have
proved Assertion (2) of the theorem.

By Theorem 3, every sequence of $i$ tending to infinity has a subsequence
$i(j)$ for which there exist an $M$-measurable vectorfield $\tilde{H}$ and a
$\Gamma$-measurable vectorfield $\tilde{\nu}$ such that

(11) $\displaystyle\int X(x,\operatorname{Tan}(M_{i(j)},x))\cdot H_{i(j)}$
$\displaystyle(x)\,dM_{i(j)}(x)$ $\displaystyle\to\int
X(x,\operatorname{Tan}(M,x))\cdot\tilde{H}(x)\,dM(x)$

for every $X\in\mathcal{X}_{m}(U)$ and

(12) $\displaystyle\int
Z(x,\operatorname{Tan}(\Gamma_{i(j)},x))\cdot\nu_{i(j)}($ $\displaystyle
x)\,d\Gamma_{i(j)}(x)$ $\displaystyle\to\int
Z(x,\operatorname{Tan}(\Gamma,x))\cdot\tilde{\nu}(x)\,d\Gamma(x)$

for every $Z\in\mathcal{X}_{m-1}(U)$. Furthermore, by Corollary 4, the
perpendicularity almost everywhere of $H(M_{i},\cdot)$ and
$\operatorname{Tan}(M_{i},\cdot)$ implies the perpendicularity almost
everywhere of $\tilde{H}(\cdot)$ and $\operatorname{Tan}(M,\cdot)$. Also, from
(12) (and Remark 7) we see that $|\tilde{\nu}(\cdot)|\leq 1$ almost everywhere
with respect to $\Gamma$.

For every $C^{1}$, compactly supported vectorfield $X$ on $U$, we have

$\int\operatorname{Div}_{M_{i(j)}}X\,dM_{i(j)}=-\int H_{i(j)}\cdot
X\,dM_{i(j)}(\cdot)+\int\nu_{i(j)}\cdot X\,d\Gamma_{i(j)}(\cdot).$

By the convergence $\operatorname{Var}(M_{i(j)})$ to $\operatorname{Var}(M)$
and by (11) and (12), it follows that

$\int\operatorname{Div}_{M}X\,dM=-\int\tilde{H}\cdot
X\,dM_{i}+\int\tilde{\nu}\cdot X\,d\Gamma.$

Consequently, $M\in\mathcal{V}_{m}(U,\Gamma)$, $\tilde{H}(\cdot)=H(M,\cdot)$
and $\tilde{\nu}(\cdot)=\nu(M,\cdot)$. We passed to a subsequence $i(j)$, but
since the limits $H(M,\cdot)$ and $\nu(M,\cdot)$ are independent of the choice
of subsequence, in fact (11) and (12) hold for the original sequence. ∎

##  5\. Brakke Flows with Boundary

######  Definition 10.

An $m$-dimensional integral Brakke flow with boundary in $U$ is a pair
$(M(\cdot),\Gamma)$ where $\Gamma$ is a smooth, properly embedded
$(m-1)$-dimensional submanifold of $U$ and where

$t\in I\mapsto M(t)$

is a Borel map from an interval $I$ to the space $\mathcal{M}(U)$ of Radon
measures in $U$ such that

  1. (1)

For almost every $t\in I$, $M(t)$ is in $\mathcal{V}_{m}(U,\Gamma)$ (see
Definition 6).

  2. (2)

If $[a,b]\subset I$ and if $K\subset U$ is compact, then

$\int_{a}^{b}\int_{K}(1+|H|^{2})\,dM(t)\,dt<\infty.$

  3. (3)

If $[a,b]\subset I$ and if $u$ is a nonnegative, compactly supported, $C^{2}$
function on $U\times[a,b]$, then

$(Mu)(a)-(Mu)(b)\geq\int_{a}^{b}\int\left(u|H|^{2}-H\cdot\nabla
u-\frac{\partial u}{\partial t}\right)\,dM(t)\,dt.$

We also say that “$M(\cdot)$ is an integral Brakke flow with boundary
$\Gamma$”.

By (2), the integral in (3) is finite.

(The condition that $t\mapsto M(\cdot)$ is a Borel map is equivalent to the
condition that $t\mapsto M(t)f$ is a Borel map for every continuous, compactly
supported function $f$ on $U$.)

######  Proposition 11.

If $t\in I\mapsto M(t)$ is a Brakke flow with boundary $\Gamma$, then the
defining inequality (3) in Definition 10 holds for every nonnegative,
compactly supported, Lipschitz function $u$ on $U$ that is $C^{1}$ on
$\\{u>0\\}$.

###### Proof.

Approximate $u$ by $C^{2}$ functions $u_{n}$ and use the Dominated Convergence
Theorem. ∎

######  Lemma 12.

Suppose that $M$ is a rectifiable varifold of bounded first variation and that
$u$ is a nonnegative, compactly supported, Lipschitz function such that
$u|\\{u>0\\}$ is $C^{2}$ and such that

$\sup_{u(x)>0}|\nabla^{2}u(x)|<\infty$

Then

$\int\operatorname{Div}_{M}\nabla u\,dM\leq-\int H(M,\cdot)\cdot\nabla
u\,dM+\int\eta(M,\cdot)\cdot\nabla u\,d\beta(M).$

###### Proof.

Let $\phi:\mathbf{R}\to\mathbf{R}$ be a smooth increasing function such that
$\phi(x)=0$ for $x<1$, $\phi(x)=x-1$ for $x\geq 3$, and such that
$\phi^{\prime\prime}\geq 0$ everywhere. Let $\kappa>0$, and apply the
Divergence Theorem to $\kappa^{-1}\phi(\kappa u)$:

$\displaystyle\int\operatorname{Div}_{M}\nabla(\kappa^{-1}\phi(\kappa u)\,dM$
$\displaystyle=-\int H\cdot\nabla(\kappa^{-1}\phi(\kappa
u))\,dM+\int\eta\cdot\nabla(\kappa^{-1}\phi(\kappa u))\,d\beta$

Now $\nabla(\kappa^{-1}\phi(\kappa u))=\phi^{\prime}(\kappa u)\nabla u$, so

$\displaystyle\operatorname{Div}_{M}\nabla(\kappa^{-1}\phi(\kappa u))$
$\displaystyle=\operatorname{Div}_{M}(\phi^{\prime}(\kappa u)\nabla u)$
$\displaystyle=\phi^{\prime\prime}(\kappa
u)|\nabla_{M}u|^{2}\kappa+\phi^{\prime}(\kappa u)\operatorname{Div}_{M}\nabla
u$ $\displaystyle\geq\phi^{\prime}(\kappa u)\operatorname{Div}_{M}\nabla u.$

Thus

$\displaystyle\int\phi^{\prime}(\kappa u)\operatorname{Div}_{M}\nabla u\,dM$
$\displaystyle\leq-\int\phi^{\prime}(\kappa u)H\cdot\nabla
u\,dM+\int\phi^{\prime}(\kappa u)\eta\cdot\nabla u\,d\beta$

Now let $\kappa\to\infty$ and use the Dominated Convergence Theorem. ∎

If $S$ is an $n\times n$ symmetric matrix with eigenvalues
$\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{n}$, let
$\operatorname{Trace}_{k}(S)=\sum_{i=1}^{k}\lambda_{i}$. Thus if $M$ is an
$m$-dimensional submanifold (or, more generally, if $M$ is in
$\mathcal{M}_{m}(U)$), then

$\operatorname{Div}_{M}\nabla u\geq\operatorname{Trace}_{m}(\nabla^{2}u).$

######  Corollary 13.

If $t\in[a,b]\mapsto M(t)$ is a Brakke Flow with boundary $\Gamma$, if

$f:U\times[a,b]\to\mathbf{R}$

is a nonnegative, $C^{2}$ function with $\overline{\\{f>0\\}}$ compact, and if
$u:=1_{f\geq 0}f$, then

$\displaystyle(Mu)(a)-(Mu)(b)$
$\displaystyle\qquad\geq\int_{a}^{b}\int\left(u|H|^{2}+\operatorname{Div}_{M}\nabla
u-\frac{\partial u}{\partial
t}\right)\,dM(t)\,dt-\int_{a}^{b}\int\nu\cdot\nabla u\,d\Gamma\,dt$
$\displaystyle\qquad\geq\int_{a}^{b}\int\left(u|H|^{2}+\operatorname{Trace}_{m}(\nabla^{2}u)-\frac{\partial
u}{\partial t}\right)\,dM(t)\,dt-\int_{a}^{b}\int\nu\cdot\nabla
u\,d\Gamma\,dt.$

###### Proof.

This follows immediately from Proposition 11 and Lemma 12. ∎

As a special case of Corollary 13, we have

######  Theorem 14.

Let $t\in[0,T]\mapsto M(t)$ be a Brakke Flow with boundary $\Gamma$. Suppose
that

$\displaystyle\text{$\overline{\mathbf{B}(x,R)}$ is a compact subset of $U$},$
$\operatorname{dist}(\cdot,x)^{2}$ is smooth on $\overline{\mathbf{B}(x,R)}$,
and
$\operatorname{Trace}_{m}(-\nabla^{2}(\operatorname{dist}(\cdot,x)^{2}))\geq-4m$
on $\mathbf{B}(x,R)$.

Let $u=(R^{2}-\operatorname{dist}(\cdot,x)^{2}-4mt)^{+}$. Then for
$t\in[0,T]$,

$(Mu)(t)\leq(Mu)(0)+tR\mathcal{H}^{m-1}(\Gamma\cap\mathbf{B}(x,R)).$

Note that
$\operatorname{Trace}_{m}(-\nabla^{2}(\operatorname{dist}(\cdot,x)^{2})=-2m$
at $x$, so the hypotheses in Theorem 14 are satisfied if $R$ is sufficiently
small.

######  Theorem 15.

Let $t\in I\mapsto M(t)$ be a Brakke Flow with boundary $\Gamma$. Let $u$ be a
nonnegative, compactly supported, $C^{2}$ function on $U$. Then for
$[a,b]\subset I$,

$\displaystyle\frac{1}{2}\int_{a}^{b}\int u|H|^{2}\,dM(t)\,dt$
$\displaystyle\leq M(a)u-M(b)u$
$\displaystyle\quad+(b-a)(\max|\nabla^{2}u|)K_{[a,b]},$

where

$K_{[a,b]}:=\sup_{t\in[a,b]}M(t)(\operatorname{spt}u).$

###### Proof.

By [ilmanen-elliptic]*Lemma 6.6,

$|\nabla u|^{2}\leq 2\,|u|\max{|\nabla^{2}u|}.$

Thus wherever $u>0$,

$\displaystyle u|H|^{2}-H\cdot\nabla u$
$\displaystyle=\frac{1}{2}u|H|^{2}+\frac{1}{2}|u^{1/2}H-u^{-1/2}(\nabla
u)|^{2}-\frac{1}{2}\frac{|\nabla u|^{2}}{|u|}$
$\displaystyle\geq\frac{1}{2}u|H|^{2}-\max|\nabla^{2}u|$

Consequently,

$\displaystyle M(a)u-M(b)u$
$\displaystyle\geq\int_{a}^{b}\int(u|H|^{2}-H\cdot\nabla u)\,dM_{i}(t)\,dt$
$\displaystyle\geq\frac{1}{2}\int_{a}^{b}\int
u|H|^{2}\,dM_{i}(t)\,dt-\max|\nabla^{2}u|\int_{a}^{b}M_{i}(\operatorname{spt}u)\,dt$
$\displaystyle\geq\frac{1}{2}\int_{a}^{b}\int
u|H_{i}|^{2}\,dM_{i}(t)\,dt-(b-a)\max|\nabla^{2}u|K_{[a,b]}.$

∎

######  Corollary 16.

If $K_{I}<\infty$, then

$t\in I\mapsto M(t)u-(\max|\nabla^{2}u|)K_{I}t$

is a non-increasing function of $t$.

##  6\. Monotonicity with Boundary in a Manifold

Now consider mean curvature flow in a smooth Riemannian manifold $N$. We embed
$N$ isometrically in a Euclidean space $\mathbf{R}^{d}$. By spacetime
translation, it suffices to consider monotonicity about the origin in
spacetime. By parabolic scaling, we can assume that $N$ is properly embedded
in an open subset of $\mathbf{R}^{d}$ that contains $\mathbf{B}^{d}(0,1)$. If
$M$ is an $m$-dimensional submanifold of $N$, we let $H$ be the mean curvature
as a submanifold of $\mathbf{R}^{d}$, and we let $H_{N}$ and $H_{N^{\perp}}$
be the projections of $H$ to $\operatorname{Tan}(N,\cdot)$ and to
$\operatorname{Tan}(N^{\perp},\cdot)$. Thus $H_{N}$ is the mean curvature of
$M$ as a submanifold of $N$.

For $x\in\mathbf{R}^{d}$ and $t<0$, let

$\rho(x,t)=\frac{1}{(4\pi|t|)^{m/2}}\exp\left(-\frac{|x|^{2}}{4|t|}\right)$

and

$\hat{\rho}(x,t)=\phi(|x|)\rho(x,t),$

where $\phi$ is a a smooth function compactly supported in $[0,1)$ such that
$\phi=1$ on $[0,1/2]$, and $\phi^{\prime}\leq 0$.

A straightforward calculation (see [kasai-tonegawa]*§6.1) shows that

(13) $K:=\sup\left|\frac{\partial\hat{\rho}}{\partial
t}+\operatorname{Div}_{M}\nabla\hat{\rho}+\frac{|\nabla^{\perp}\rho|^{2}}{\rho}\right|<\infty$

and that if $\Gamma$ is a smooth, properly embedded, $(m-1)$-dimensional
submanifold of $\overline{\mathbf{B}(0,1)}$, then

(14)
$C_{\Gamma}:=\sup_{t<0}\int|(\nabla\hat{\rho}(\cdot,t))_{\Gamma^{\perp}}|\,d\Gamma<\infty.$

######  Theorem 17 (Huisken Monotonicity).

Let $U$ be an open subset of $\mathbf{R}^{d}$ that contains
$\mathbf{B}^{d}(0,1)$ and $N$ be a smooth, properly embedded submanifold of
$U$. Let $\Gamma$ be a smooth, properly embedded $(m-1)$-dimensional
submanifold of $U$. Let $t\in I\mapsto M(t)$ be a Brakke flow in $N$ with
boundary $\Gamma$. Suppose that

(15) $M(t)\mathbf{B}(0,1)\leq C$

for $t\in I$ and that the norm of the second fundamental form of $U$ is
bounded by $A$. Then for $a,b\in I$ with $a\leq b<0$,

(16) $\displaystyle(M\hat{\rho})(a)-(M\hat{\rho})(b)$
$\displaystyle\geq\int_{a}^{b}\int\left|H_{N}-\frac{(\nabla^{\perp}\hat{\rho})_{N}}{\hat{\rho}}\right|^{2}\hat{\rho}\,dM(t)\,dt$
$\displaystyle\qquad+\int_{a}^{b}\int\nu_{M}\cdot\nabla\hat{\rho}\,d\Gamma\,dt$
$\displaystyle\qquad-mA^{2}\int_{a}^{b}\int\hat{\rho}\,dM(t)\,dt$
$\displaystyle\qquad-CK(b-a)$

where $C$ and $K$ are as in (13) and (15). Furthermore,

$e^{-mA^{2}t}\left((M\hat{\rho})(t)+\int_{\tau=T_{0}}^{t}\int\nu_{M}\cdot\nabla\hat{\rho}\,d\Gamma\,d\tau-
CKt\right)$

is a decreasing function of $t$ for $t<0$ in $I$.

###### Proof.

(17) $\displaystyle(M\hat{\rho})(a)-(M\hat{\rho})(b)$
$\displaystyle\quad\geq\int_{a}^{b}\int\left(\hat{\rho}|H_{N}|^{2}-H_{N}\cdot\nabla\hat{\rho}-\frac{\partial\hat{\rho}}{\partial
t}\right)\,dM(t)\,dt$
$\displaystyle\quad\geq\int_{a}^{b}\int\left(\hat{\rho}|H_{N}|^{2}-H_{N}\cdot\nabla\hat{\rho}+\operatorname{Div}_{M}\nabla\hat{\rho}+\frac{|\nabla^{\perp}\hat{\rho}|^{2}}{\hat{\rho}}-K1_{\mathbf{B}}\right)\,dM(t)\,dt$
$\displaystyle\quad=\int_{a}^{b}\int\left(\hat{\rho}|H_{N}|^{2}-H_{N}\cdot\nabla\hat{\rho}-H\cdot\nabla\hat{\rho}+\frac{|\nabla^{\perp}\hat{\rho}|^{2}}{\hat{\rho}}-K\,1_{\mathbf{B}}\right)\,dM(t)\,dt$
$\displaystyle\qquad\qquad+\int_{a}^{b}\int\nu_{M}\cdot\nabla\hat{\rho}\,d\Gamma\,dt$
$\displaystyle\quad\geq\int_{a}^{b}\int_{\mathbf{B}}\left(\hat{\rho}|H_{N}|^{2}-H_{N}\cdot\nabla\hat{\rho}-H\cdot\nabla\hat{\rho}+\frac{|\nabla^{\perp}\hat{\rho}|^{2}}{\hat{\rho}}\right)\,dM(t)\,dt$
$\displaystyle\qquad\qquad+\int_{a}^{b}\int\nu_{M}\cdot\nabla\hat{\rho}\,d\Gamma\,dt-
KC(b-a).$

We rewrite the penultimate the integrand in (17) as follows, using the
orthogonality of the mean curvature:

$\displaystyle\hat{\rho}|H_{N}|^{2}-H_{N}\cdot\nabla\hat{\rho}-H\cdot\nabla\hat{\rho}+\frac{|\nabla^{\perp}\hat{\rho}|^{2}}{\hat{\rho}}$
$\displaystyle\quad=\hat{\rho}|H_{N}|^{2}-2H_{N}\cdot\nabla\hat{\rho}-H_{N^{\perp}}\cdot\nabla\hat{\rho}+\frac{|\nabla^{\perp}\hat{\rho}|^{2}}{\hat{\rho}}$
$\displaystyle\quad=\hat{\rho}|H_{N}|^{2}-2H_{N}\cdot\nabla^{\perp}\hat{\rho}-H_{N^{\perp}}\cdot\nabla\hat{\rho}+\frac{|\nabla^{\perp}\hat{\rho}|^{2}}{\hat{\rho}}$
$\displaystyle\quad=\hat{\rho}|H_{N}|^{2}-2H_{N}\cdot(\nabla^{\perp}\hat{\rho})_{N}+\frac{|(\nabla^{\perp}\hat{\rho})_{N}|^{2}}{\hat{\rho}}-H_{N^{\perp}}\cdot(\nabla\hat{\rho})_{N^{\perp}}+\frac{|(\nabla\hat{\rho})_{N^{\perp}}|^{2}}{\hat{\rho}}$
$\displaystyle\quad=\left|H_{N}-\frac{(\nabla^{\perp}\hat{\rho})_{N}}{\hat{\rho}}\right|^{2}\hat{\rho}+\left|\frac{1}{2}H_{N^{\perp}}-\frac{(\nabla\hat{\rho})_{N^{\perp}}}{\hat{\rho}}\right|^{2}\hat{\rho}-\frac{1}{4}|H_{N^{\perp}}|^{2}\hat{\rho}$
$\displaystyle\quad\geq\left|H_{N}-\frac{(\nabla^{\perp}\hat{\rho})_{N}}{\hat{\rho}}\right|^{2}\hat{\rho}-mA^{2}\hat{\rho}$

since $|H_{N^{\perp}}|^{2}\leq mA^{2}$. Substituting this into (17) gives
(16).

Now let

(18)
$f(t)=(M\hat{\rho})(t)+\int_{\tau=T_{0}}^{t}\nu_{M}\cdot\nabla\hat{\rho}\,d\Gamma\,d\tau-
CKt.$

By (16), we have

$f^{\prime}(t)\leq mA^{2}f(t)$

in the distributional sense, which immediately implies that

$\frac{d}{dt}\left(e^{-mA^{2}t}f(t)\right)\leq 0.$

∎

######  Corollary 18.

The quantity $(M\hat{\rho})(t)$ has a finite limit as $t\to 0$.

###### Proof.

Since $e^{-mA^{2}t}f(t)$ is in a non-increasing function of $t$ (where $f$ is
given by (18)), $\lim_{t\uparrow}f(t)$ exists and is in $[-\infty,\infty)$. By
(14),

$\lim_{t\uparrow
0}\int_{\tau=T_{0}}^{t}\int\nu_{M}\cdot\nabla\hat{\rho}\,d\Gamma\,d\tau$

exists and is finite. Thus $\lim_{t\uparrow 0}(M\hat{\rho})(t)$ exists and is
$<\infty$. Since $(M\hat{\rho})(t)\geq 0$, the limit is $\geq 0$, and thus is
a finite, nonnegative number. ∎

######  Definition 19.

The Gaussian density of $M(\cdot)$ at $(0,0)$ is

$\Theta(M(\cdot),(0,0))=\lim_{t\uparrow 0}(M\hat{\rho})(t)$

It is straightforward to prove that the Gauss density does not depend on the
isometric embedding of $N$ into $\mathbf{R}^{d}$ or on the choice of the
cutoff function $\phi$.

##  7\. Compactness Theorems

######  Theorem 20.

Suppose for $i=1,2,\dots$ that

$t\in[0,T]\mapsto M_{i}(t)$

is an integral Brakke flow in $U$ with boundary $\Gamma_{i}$. Suppose also
that the $\Gamma_{i}$ converge in $C^{1}$ to a smooth, properly embedded
$(m-1)$-dimensional submanifold $\Gamma$ of $U$, and that

$c_{K}:=\sup_{i}\sup_{t\in[0,T]}M_{i}(t)K<\infty$

for each compact subset $K$ of $U$. Then there is a subsequence $i(j)$ such
that for each $t\in[0,T]$, $M_{i(j)}(t)$ converges to a Radon measure $M(t)$.

###### Proof.

Let $\mathcal{F}$ be a countable collection of $C^{2}$, nonnegative, compactly
supported functions on $U$ such that the linear span of $\mathcal{F}$ is dense
in the space of all continuous, compactly supported functions. By Corollary
16, for each $u\in\mathcal{F}$, the function

(19) $t\in[0,T]\mapsto M(t)u-c_{\operatorname{spt}u}(\max|\nabla^{2}u|)t$

is non-increasing. Each such function is also bounded. Hence by passing to a
subsequence, we can assume that each of the functions (19) converges to a
limit function. Theorem 20 follows immediately from the Riesz Representation
Theorem. ∎

######  Theorem 21.

Suppose

  1. (1)

$\Gamma_{i}$ $(i\in\mathbf{N})$ and $\Gamma$ are smooth, properly embedded,
$(m-1)$-dimensional submanifolds of $U$, and the $\Gamma_{i}$ converge
smoothly to $\Gamma$.

  2. (2)

For $i\in\mathbf{N}$, $t\in[0,T]\mapsto M_{i}(t)$ is an $m$-dimensional
integral Brakke flow with boundary $\Gamma_{i}$.

  3. (3)

For each $t$, $M_{i}(t)$ converges to a Radon measure $M(t)$.

Then

$c_{K}:=\sup_{i}\sup_{t\in[0,T]}M_{i}(t)K<\infty.$

###### Proof.

This follows immediately from Theorem 14. ∎

In the following theorem, we write $H_{i}(x,t)$ for $H(M_{i}(t),x)$ and
$\nu_{i}(x,t)$ for $\nu(M_{i}(t),x)$.

######  Theorem 22.

Suppose

  1. (1)

$\Gamma_{i}$ $(i\in\mathbf{N})$ and $\Gamma$ are smooth, $(m-1)$-dimensional
submanifolds of $U$, and the $\Gamma_{i}$ converge smoothly to $\Gamma$.

  2. (2)

For $i\in\mathbf{N}$, $t\in[0,T]\mapsto M_{i}(t)$ is an $m$-dimensional
integral Brakke flow with boundary $\Gamma_{i}$.

  3. (3)

For each $t$, $M_{i}(t)$ converges to a Radon measure $M(t)$.

Then $t\mapsto M(t)$ is an integral Brakke Flow with boundary $\Gamma$, and
there is a continuous, everywhere positive function $\phi:U\to\mathbf{R}$ with
the following property. For almost every $t$, there is a subsequence $i(j)$
such that:

(20) $\sup_{j}\int\phi|H(M_{i(j)}(t),\cdot)|^{2}\,dM_{i(j)}(t)<\infty,$ (21)
$\operatorname{Var}(M_{i(j)}(t))\to\operatorname{Var}(M(t)),$ (22)
$\displaystyle\int H_{i(j)}(x)\cdot X(x,$
$\displaystyle\operatorname{Tan}(M_{i(j)},x))\,dM_{i}(t)$
$\displaystyle\to\int H(x)\cdot X(x,\operatorname{Tan}(M,x))\,dM(t),$ (23)
$\displaystyle\int\nu_{i(j)}(x)\cdot Y(x,$
$\displaystyle\operatorname{Tan}(\Gamma_{i(j)},x))\,d\Gamma_{i(j)}(x)$
$\displaystyle\to\int\nu(x)\cdot
Y(x,\operatorname{Tan}(\Gamma,x))\,d\Gamma(x)$

for every $X\in\mathcal{X}_{m}(U)$ and for every $Y\in\mathcal{X}_{m-1}(U)$.

######  Remark 23.

Even for Brakke flows without boundary, the fact that $X$ in (22) can depend
on $x$ and $\operatorname{Tan}(M_{i(j)},x)$ (rather than just on $x$) seems to
be new.

###### Proof.

By Theorem 21,

$c_{K}:=\sup_{i}\sup_{t\in[0,T]}M_{i}(t)K<\infty.$

By Theorem 15,

$d_{K}:=\sup_{i}\int_{0}^{T}\int_{K}|H_{i}|^{2}\,dM_{i}(t)\,dt<\infty.$

It follows that there is a continuous, everywhere positive function
$\phi:U\to\mathbf{R}$ such that

(24) $C:=\sup_{i}\int_{0}^{T}\int\phi|H_{i}|^{2}\,dM_{i}(t)\,dt<\infty.$

By Fatou’s Lemma,

$\int_{0}^{T}\left(\liminf_{i}\int\phi|H_{i}|^{2}\,dM_{i}(t)\right)\,dt<\infty,$

so for almost every $t$,

$\liminf_{i}\int\phi|H_{i}|^{2}\,dM_{i}(t)<\infty.$

For every such $t$, there is a subsequence $i(j)$ such that

$\sup_{j}\int\phi|H_{i(j)}|^{2}\,dM_{i(j)}(t)<\infty.$

By the Varifold Closure Theorem 9, $M(t)\in\mathcal{V}_{m}(U,\Gamma)$, and
(21), (22), and (23) hold.

Now let $u:U\times[0,T]\to\mathbf{R}$ be a nonnegative, compactly supported,
$C^{2}$ function. For each $i$,

$\displaystyle(M_{i}u)(a)-(M_{i}u)(b)$
$\displaystyle\quad\geq\int_{a}^{b}\int\left(u|H_{i}|^{2}-H_{i}\cdot\nabla
u-\frac{\partial u}{\partial t}\right)\,dM_{i}(t)\,dt$
$\displaystyle\quad=\int_{a}^{b}\int\left(\left|u^{1/2}H_{i}-\frac{1}{2}u^{-1/2}\nabla
u\right|^{2}-\frac{1}{4}\frac{|\nabla u|^{2}}{u}-\frac{\partial u}{\partial
t}\right)\,dM_{i}(t)\,dt$

Therefore by (24),

$\displaystyle(M_{i}u)(a)-(M_{i}u)(b)+\epsilon\,C$
$\displaystyle\quad\geq\int_{a}^{b}\int\left(\left|u^{1/2}H_{i}-\frac{1}{2}u^{-1/2}\nabla
u\right|^{2}+\epsilon\phi|H_{i}|^{2}-\frac{1}{4}\frac{|\nabla
u|^{2}}{u}-\frac{\partial u}{\partial t}\right)\,dM_{i}(t)\,dt.$

Letting $i\to\infty$ gives, by Fatou’s Lemma,

(25) $\displaystyle(Mu)(a)-(Mu)(b)+\epsilon\,C$
$\displaystyle\quad\geq\int_{a}^{b}\lambda_{\epsilon}(t)\,dt-\int_{a}^{b}\int\left(\frac{1}{4}\frac{|\nabla
u|^{2}}{u}+\frac{\partial u}{\partial t}\right)\,dM(t)\,dt,$

where

$\lambda_{\epsilon}(t)=\liminf_{i}\int\left(\left|u^{1/2}H_{i}-\frac{1}{2}u^{-1/2}\nabla
u\right|^{2}+\epsilon\phi|H_{i}|^{2}\right)\,dM_{i}(t).$

For each $t$ with $\lambda_{\epsilon}(t)<\infty$, there is a subsequence
$i(j)$ such that

$\int\left(\left|u^{1/2}H_{i(j)}-\frac{1}{2}u^{-1/2}\nabla
u\right|^{2}+\epsilon\phi|H_{i(j)}|^{2}\right)\,M_{i(j)}(t)\to\lambda_{\epsilon}(t).$

For such $t$, we have (as above),
$\operatorname{Var}(M_{i(j)}t))\to\operatorname{Var}(M(t))$ and

$\int H_{i(j)}\cdot X\,dM_{i(j)}\to\int H\cdot X\,dM(t),$

for all $X\in\mathcal{X}(U)$. Consequently,

$\int\left(u^{1/2}H_{i(j)}-\frac{1}{2}u^{-1/2}\nabla u\right)\cdot
X\,dM_{i(j)}(t)\to\int\left(u^{1/2}H-\frac{1}{2}u^{-1/2}\nabla u\right)\cdot
X\,dM(t).$

Since this holds for all $X\in\mathcal{X}(U)$,

$\displaystyle\int\left|u^{1/2}H-\frac{1}{2}u^{-1/2}\nabla
u\right|^{2}\,dM(t)$
$\displaystyle\leq\liminf\int\left|u^{1/2}H_{i(j)}-\frac{1}{2}u^{-1/2}\nabla
u\right|^{2}\,dM_{i(j)}(t)$ $\displaystyle\leq\lambda_{\epsilon}(t)$

Substituting this into (25) and letting $\epsilon\to 0$ gives

$\displaystyle(Mu)(a)-(Mu)(b)$ $\displaystyle\geq\int_{a}^{b}\
\int\left|u^{1/2}H-\frac{1}{2}u^{-1/2}\nabla u\right|^{2}\,dM(t)\,dt$
$\displaystyle\qquad-\int_{a}^{b}\int\left(\frac{1}{4}\frac{|\nabla
u|^{2}}{u}+\frac{\partial u}{\partial t}\right)\,dM(t)\,dt$
$\displaystyle=\int_{a}^{b}\int\left(u|H|^{2}-H\cdot\nabla u-\frac{\partial
u}{\partial t}\right)\,dM(t)\,dt$

∎

##  8\. Tangent Flows

Consider an integral Brakke flow $t\in I\mapsto M(t)$ in $N$ with boundary
$\Gamma$. As in §6, we isometrically embed $N$ in a Euclidean space
$\mathbf{R}^{d}$. We now discuss tangent flows at a spacetime point
$(p_{0},t_{0})$. By making a spacetime translation, it suffices to consider
the case $(p_{0},t_{0})=(0,0)$.

For $\lambda>0$, let $M^{\lambda}$ be the result of applying the parabolic
dilation

$\mathcal{D}_{\lambda}:(x,t)\mapsto(\lambda x,\lambda^{2}t)$

to the flow

$t\in I\cap(-\infty,0)\mapsto M(t).$

Just as for mean curvature flow without boundary, monotonicity together with
the compactness and closure theorems in §7 implies existence of tangent flows:
for every sequence $\lambda(i)\to\infty$, there is a subsequence
$\lambda(i(j))$ such that the flows $M^{\lambda(i(j))}(\cdot)$ converge to a
flow $M^{\prime}(\cdot):t\in(-\infty,0]\mapsto M^{\prime}(t)$. If
$0\notin\Gamma$, it is an integral Brakke flow in the Euclidean space
$\operatorname{Tan}(N,0)$. If $0\in\Gamma$, it is an integral Brakke flow in
$\operatorname{Tan}(N,0)$ with boundary $\operatorname{Tan}(\Gamma,0)$. In
either case, the tangent flow $M^{\prime}(\cdot)$ is self-similar: it is
invariant under parabolic dilations $\mathcal{D}_{\lambda}$ with $\lambda>0$.

######  Definition 24.

We say that an integral Brakke flow $t\in I\mapsto M(t)$ with boundary
$\Gamma$ is unit-regular provided the following holds:

> For each $p\in N$ and $t\in I$, if one of the tangent flows at $(p,t)$ is a
> multiplicity-$1$ plane or halfplane, then the flow $M(\cdot)$ is fully
> smooth in a spacetime neighborhood of $(p,t)$.

Equivalently,

> For each $p\in N$ and $t\in I$, if $p\notin\Gamma$ and
> $\Theta(M(\cdot),(p,t))=1$ or if $p\in\Gamma$ and
> $\Theta(M(\cdot),(p,t))=1/2$, then the flow $M(\cdot)$ is fully smooth in a
> spacetime neighborhood of $(p,t)$.

Here “fully smooth” means “smooth and with no sudden vanishing”.

######  Remark 25.

If $p\notin\Gamma$ and if $\Theta(M(\cdot),(p,t))=1$, then $(p,t)$ is a
$C^{1,\alpha}$-regular point of the flow by Brakke’s Regularity Theorem
[brakke] if the ambient space is Euclidean or by the Kasai-Tonegawa [kasai-
tonegawa] generalization of that theorem for general ambient manifolds, and
consequently is a $C^{\infty}$-regular point by [tonegawa-higher]. Presumably
the analogous theorems are true for $(p,t)$ with $p\in\Gamma$ and
$\Theta(M(\cdot),(p,t))=1/2$. If so, then every integral Brakke flow with
boundary would have the smoothness (but not necessarily the full smoothness)
described in the definition of unit-regularity. However, none of those facts
are required for this paper; the simpler local regularity theorems in [white-
local] suffice.

##  9\. Mod $2$ Flat Chains

Let $\mathcal{L}_{\textnormal{$m$-rec}}(U,\mathbf{Z}^{+})$ denote the space of
functions on $U$ that take values in the nonnegative integers, that are
locally $L^{1}$ with respect to Hausdorff $m$-dimensional measure on $U$, and
that vanish except on a countable union of m-dimensional $C^{1}$-submanifolds
of $U$. We identify functions that agree except on a set of Hausdorff
$m$-dimensional measure zero. Let
$\mathcal{L}_{\textnormal{$m$-rec}}(U,\mathbf{Z}_{2})$ be the corresponding
space with the nonnegative integers $\mathbf{Z}^{+}$ replaced by
$\mathbf{Z}_{2}$, the integers mod $2$. The space $I\mathcal{M}_{m}(U)$
(defined in §2) is naturally isomorphic to
$\mathcal{L}_{\textnormal{$m$-rec}}(U,\mathbf{Z}^{+})$: given any $M\in
I\mathcal{M}_{m}(U)$, the corresponding function in
$\mathcal{L}_{\textnormal{$m$-rec}}(U,\mathbf{Z}^{+})$ is the density function
$\Theta(M,\cdot)$ given by

$\Theta(M,x)=\lim_{r\to 0}\frac{M\mathbf{B}(x,r)}{\omega_{m}r^{m}},$

where $\omega_{m}$ is the volume of the unit ball in $\mathbf{R}^{m}$. In
particular, this limit exists and is a nonnegative integer for
$\mathcal{H}^{m}$-almost every $x\in U$. Similarly, the space of
$m$-dimensional rectifiable mod $2$ flat chains111As in [simon-gmt] and in
[white-duke], we do not require flat chains to have compact support. In
Federer’s terminology [federer-book], they would be called “locally flat
chains”. See the discussion in [white-duke]*§2.1. in $U$ is naturally
isomorphic to $\mathcal{L}_{\textnormal{$m$-rec}}(U,\mathbf{Z}_{2})$: given
any such flat chain $A$, the corresponding function is the density function
$\Theta(A,\cdot)$ given by

$\Theta(A,x):=\lim_{r\to 0}\frac{\mu_{A}\mathbf{B}(x,r)}{\omega_{m}r^{m}}$

where $\mu_{A}$ is the Radon measure on $U$ determined by $A$. In particular,
this limit exists and is $0$ or $1$ for $\mathcal{H}^{m}$-almost every $x\in
U$.

The surjective homomorphism

$\displaystyle[\cdot]:\mathbf{Z}^{+}\to\mathbf{Z}_{2},$ $\displaystyle
k\mapsto[k]$

determines a homomorphism from
$\mathcal{L}_{\textnormal{$m$-rec}}(U,\mathbf{Z}^{+})$ to
$\mathcal{L}_{\textnormal{$m$-rec}}(U,\mathbf{Z}_{2})$ and thus also a
homomorphism from the additive semigroup $\mathcal{M}_{m}(U)$ to the additive
group of $m$-dimensional rectifiable mod $2$ flat chains in $U$. If
$M\in\mathcal{M}_{m}(U)$, we let $[M]$ denote the corresponding rectifiable
mod $2$ flat chain. Thus $[M]$ is the unique rectifiable mod $2$ flat chain in
$U$ such that

$\Theta(M,x)=[\Theta(M,x)]$

for $\mathcal{H}^{m}$-almost every $x\in U$.

The following is Theorem 3.3 in [white-duke]:

######  Theorem 26.

Suppose that $M_{i}$ $(i=1,2,\dots)$ and $M$ are Radon measures in
$\mathcal{M}_{m}(U)$ with the following properties:

  1. (1)

$\operatorname{Var}(M_{i})\rightharpoonup\operatorname{Var}(M)$.

  2. (2)

Each $M_{i}$ has bounded first variation, and

$q_{K}:=\sup_{i}\left(\int_{K}|H(M_{i},\cdot)\,dM_{i}+\beta(M_{i})(K)\right)<\infty.$

  3. (3)

The $\partial[M_{i}]$ converge (in the flat topology) to a mod $2$ flat chain
$C$.

Then the $[M_{i}]$ converge (in the flat topology) to $[M]$. In particular,
$\partial[M_{i}]=C$.

##  10\. Standard Brakke Flows and The Closure Theorem

Suppose $(M_{i}(\cdot),\Gamma_{i})$ ($i\in\mathbf{N}$) and $(M(\cdot),\Gamma)$
are integral Brakke flows with boundary in $U$ defined on a time interval $I$.
We say that the $(M_{i}(\cdot),\Gamma_{i})$ converge to $(M(\cdot),\Gamma)$ if
$M_{i}(t)\rightharpoonup M(t)$ for each $t\in I$ and the $\Gamma_{i}$ converge
smoothly to $\Gamma$.

######  Theorem 27 (Closure Theorem).

Suppose $(M_{i}(\cdot),\Gamma_{i})$ converges to $(M(\cdot),\Gamma)$, where
each $M_{i}:t\in[0,T]\to M_{i}(t)$ is an integral Brakke flows with boundary
$\Gamma_{i}$.

  1. (1)

If the flows $M_{i}(\cdot)$ are unit-regular, then so is the flow $M(\cdot)$.

  2. (2)

If $\partial[M_{i}(t)]=[\Gamma_{i}(t)]$ for almost every $t$, then
$\partial[M(t)]=[\Gamma(t)]$ for almost every $t$.

###### Proof.

See [schulze-white-triple]*Theorem 4.2 for the proof of Assertion (1). (The
proof in [schulze-white-triple] is for Brakke flows without boundary, but the
same proof works for flows with boundary.)

We now prove Assertion (2). By Theorem 22, for almost every $t$, there is a
subsequence $M_{i(j)}(t)$ such that

$\operatorname{Var}(M_{i(j)}(t))\rightharpoonup\operatorname{Var}(M(t))$

and such that

$\sup_{j}\int_{K}|H_{i(j)}|^{2}\,dM_{i(j)}(t)<\infty$

for every $K\subset\subset U$. Also,

$\sup_{j}\beta(M_{i(j)})(t)K\leq\sup_{j}\mathcal{H}^{m-1}(\Gamma_{i(j)}(t)\cap
K)<\infty$

since $\Gamma_{i}\to\Gamma$ smoothly. Thus by Theorem 26,
$[M_{i(j)}(t)]\to[M(t)]$ and $\partial[M(t)]=[\Gamma]$. ∎

######  Definition 28.

An integral Brakke flow $M(\cdot)$ with boundary $\Gamma$ is called standard
if it has properties described in Theorem 27:

  1. (1)

the flow is unit-regular, and

  2. (2)

$\partial[M(t)]=[\Gamma]$ for almost every $t$.

##  11\. Existence

######  Theorem 29.

Let $N$ be a smooth Riemannian manifold. If $N$ has nonempty boundary, we
assume that the boundary is smooth and $m$-convex. Let $\Gamma$ be smooth,
properly embedded $(m-1)$-dimensional manifold in $N$. Let $M_{0}$ be a
smoothly embedded $m$-dimensional manifold in $N$ with boundary $\Gamma$ and
with finite area, or, more generally, let $M_{0}$ be an $m$-rectifiable set of
finite $m$-dimensional measure such that $\partial[M_{0}]=[\Gamma]$. Then
there exists a standard Brakke flow

$t\in[0,\infty)\mapsto M(t)$

with boundary $\Gamma$ such that

$M(t)\rightharpoonup\mathcal{H}^{m}\llcorner M_{0}$

as $t\to 0$.

If $p\notin\Gamma$ and if $M_{0}$ is smooth in a neighborhood of $p$, then the
flow is smooth in a spacetime neighborhood of $(p,0)$. If $p\in\Gamma$ and if
$M_{0}$ is $C^{1,\alpha}$ in a neighborhood of $p$, then the flow is
parabolically $C^{1}$ in a spacetime neighborhood of $(p,0)$.

###### Proof.

Let $\Gamma^{*}=(\Gamma\times\\{0\\})\cup(\Gamma\times[0,\infty)$. Let
$\lambda>0$. Consider a mod $2$ flat chain $\Sigma^{\lambda}$ in
$N\times\mathbf{R}$ that minimizes

$\int_{(x,z)\in N\times\mathbf{R}}e^{-\lambda z}d\mu_{\Sigma^{\lambda}}(x,z)$

subject to $\partial\Sigma^{\lambda}=[\Gamma^{*}]$. Let $M^{\lambda}$ be the
associated Radon measure; thus $\Sigma^{\lambda}=[M^{\lambda}]$.

Note that $\Sigma^{\lambda}$ is mass-minimizing with respect to the Ilmanen
metric $e^{-2\lambda z/m}(g+dz^{2})$ (where $g$ is the metric on $N$.) Thus
$\operatorname{spt}(M^{\lambda})\setminus\Gamma^{*}$ is smooth (with
multiplicity $1$) away from a closed set of Hausdorff dimension $\leq m-1$
[federer-short].

For $t\geq 0$, let $M^{\lambda}(t)$ be the portion of $M^{\lambda}-\lambda
t(0,1)$ in $\tilde{N}=N\times(0,\infty)$. Then

$t\in[0,\infty)\mapsto M^{\lambda}(t)$

is an integral Brakke flow with boundary
$\tilde{\Gamma}:=\Gamma\times(0,\infty)$ in $\tilde{N}$. In fact, it is
standard:

  1. (1)

Because $M^{\lambda}$ is smooth almost everywhere, the mean curvature vector
is orthogonal to the surface almost everywhere.

  2. (2)

Unit regularity follows from Allard’s Regularity Theorem and Boundary
Regularity Theorem applied to $M^{\lambda}$.

  3. (3)

The mod $2$ boundary condition holds by construction.

By [ilmanen-elliptic]*5.1, 3.2(ii), the areas of the
$\mathcal{M}^{\lambda}(t)$ have a uniform upper bound on area as
$\lambda\to\infty$:

$\operatorname{area}(\mathcal{M}^{\lambda}_{0}\cap\\{a<z<a+b\\})\leq(b+\lambda^{-1})\operatorname{area}(M_{0})$

for any $a,b>0$, and thus

$\operatorname{area}(\mathcal{M}^{\lambda}(t)\llcorner\\{0<z<b\\})\leq(b+\lambda^{-1})\operatorname{area}(M_{0}).$

Consequently (by Theorems 20, 22, and 27), the flows
$t\in[0,\infty)\mapsto\mathcal{M}^{\lambda}(t)$ converge as $\lambda\to\infty$
(after passing to a subsequence) to a standard Brakke flow
$\mathcal{M}(\cdot)$ in $\tilde{N}$ with boundary $\tilde{\Gamma}$.

Furthermore, as in [ilmanen-elliptic],

$\mathcal{M}(0)=M_{0}\times(0,\infty)$

and

$\mathcal{M}(t)=M(t)\times(0,\infty)$

(except possibly for countably many $t$), where $M(t)$ is a Radon measure in
$N$.

Since $\mathcal{M}(\cdot)$ is a standard Brakke flow with boundary
$\Gamma^{*}$ in $N^{*}$, it follows that $t\in[0,\infty)\mapsto M(t)$ is a
standard Brakke flow in $N$ with boundary $\Gamma$ and with

$M(0)=M_{0}.$

(See [ilmanen-elliptic]*8.9.)

If $M_{0}$ is $C^{1,\alpha}$ in a neighborhood of a point $p$, then the flow
is parabolically $C^{1}$ in a spacetime neighborhood of $(p,0)$ by [white-
local]. If $M_{0}$ is smooth in a neighborhood of a point $p\in
N\setminus\Gamma$, then the flow is smooth in a spacetime neighborhood of
$(p,0)$ by [schulze-white-triple]*Corollary A.3. ∎

##  12\. Self-Similar Flows

######  Theorem 30 (Shrinker Theorem).

Let $\Gamma$ be an $(m-1)$-dimensional linear subspace of $\mathbf{R}^{m+1}$.
Suppose that

$t\in(-\infty,0)\mapsto S(t)$

is an $m$-dimensional integral Brakke flow in
$\mathbf{R}^{m+1}\setminus\Gamma$ that is self-similar (i.e, invariant under
parabolic dilations $\mathcal{D}_{\lambda}:(x,t)\mapsto(\lambda
x,\lambda^{2}t)$ with $\lambda>0$). Let $M:=S(-1)$, and suppose that
$\operatorname{spt}(M)\setminus\Gamma$ is disjoint from some $m$-dimensional
halfplane $P$ with boundary $\Gamma$. Then $M$ is a sum of half-planes (each
with boundary $\Gamma$) with multiplicities.

###### Proof.

Let $\mathcal{P}$ be the set of halfplanes with boundary $\Gamma$ that are
disjoint from $(\operatorname{spt}M)\setminus\Gamma$. We claim that
$\mathcal{P}$ is open. To see this, suppose $P\in\mathcal{P}$. By rotating, we
can assume that $P$ is the halfplane $\\{x_{2}=0,\,x_{1}\geq 0\\}$. Let
$M^{+}$ and $S^{+}(t)$ be the portions of $M$ and $S(t)$ in the region
$\\{x_{2}\geq 0,\,x_{1}>0\\}$.

Let $f:\mathbf{B}^{m}(0,1)\cap\\{x_{1}>0\\}\to\mathbf{R}$ be a smooth,
compactly supported, nonnegative function that is $>0$ at some points. Extend
$f$ to $\mathbf{B}^{m}(0,1)$ so that it is odd in $x_{1}$:

$f(-x_{1},x_{2},\dots,x_{m})=-f(x_{1},x_{2},\dots,x_{m}).$

Now let

$u:\mathbf{B}^{m}(0,1)\times[-1,\infty)\to\mathbf{R}$

be the solution of the nonparametric MCF equation with

$\displaystyle u(\cdot,-1)=f,$ $\displaystyle
u(\cdot,t)|\partial\mathbf{B}^{m}\equiv 0.$

By the boundary maximum principle, $c:=\frac{\partial}{\partial
x_{1}}u(0,0)>0$. Let $G(t,\epsilon)$ be the graph of $f(\cdot,t)-\epsilon$.
Let $\epsilon>0$. By the maximum principle (Theorem 44), the graph of
$f(\cdot,t)-\epsilon$ lies below $\operatorname{spt}(S^{+}(t))$ for all
$t\in[-1,0]$. Hence the graph $G(t)$ of $f(\cdot,t)$ lies below
$\operatorname{spt}(S^{+}(t))$. Equivalently, $G^{*}(t):=|t|^{-1/2}G(t)$ lies
below $\operatorname{spt}(M)$ for all $t\in[-1,0)$. At $t\to 0$, $G^{*}(t)$
converges to the plane $\\{x:x_{2}=cx_{1}\\}$. Thus we see that the halfplane
$P_{\lambda}:=\\{x:x_{2}=\lambda x_{1},\,x_{1}>0\\}$ is in $\mathcal{P}$ for
all $\lambda\in[0,c)$. Likewise there is a $c^{\prime}<0$ such that the
halfplane $P_{\lambda}$ is disjoint from $M$ for all
$\lambda\in(c^{\prime},0]$. This completes the proof of openess of
$\mathcal{P}$.

Now let $P$ be a plane in the boundary of $\mathcal{P}$. Then
$\operatorname{spt}(M)$ touches $P$, so, by the strong maximum principle, $M$
contains $P$. Now repeat the process with $S(t)$ replaced by

$t\in(-\infty,0)\mapsto S(t)-(\mathcal{H}^{m}\llcorner P).$

The process must stop in finitely many steps, since otherwise $M$ would
contain infinitely many halfplanes and thus would not have locally finite area
in $\mathbf{R}^{m+1}\setminus\Gamma$. ∎

######  Definition 31.

Consider two distinct $m$-dimensional linear subspaces $P$ and $P^{\prime}$ of
$\mathbf{R}^{m+1}$. The closure of a component of
$\mathbf{R}^{m+1}\setminus(P\cup P^{\prime})$ is called a wedge, and $P\cap
P^{\prime}$ is the edge of the wedge.

######  Corollary 32.

Suppose $t\in(-\infty,0)\mapsto M(t)$ is a nontrivial $m$-dimensional self-
similar, integral Brakke flow (without boundary) in $\mathbf{R}^{m+1}$. Then
$\operatorname{spt}M(-1)$ is not contained in any wedge.

###### Proof.

If $\operatorname{spt}M(-1)$ were contained in such a wedge $W$, then by
Theorem 35 it would be a union of half-planes in $W$, which is impossible. ∎

######  Corollary 33.

Let $V$ be a stationary integral $m$-varifold in
$\mathbf{R}^{m+1}\setminus\Gamma$ that is invariant under positive dilations
about $0$. Suppose there is an open halfplane with boundary $\Gamma$ that is
disjoint from the support of $V$. Then $V$ is a sum of halfplanes with
multiplicities.

This is the special case of Theorem 30 when the shrinker is a minimal cone.

######  Remark 34.

Theorem 30 and Corollary 32 remain true (with the same proofs) if the
hypothesis that $M(\cdot)$ is an integral Brakke flow is replaced by the
hypothesis that $M(\cdot)$ is a Brakke flow such that the density of $M(-1)$
is $\geq a>0$ almost everywhere (with respect to $M(-1)$). Likewise Corollary
33 remains true for any stationary varifold $V$ such that $\Theta(V,\cdot)\geq
a>0$ holds $\mu_{V}$ almost everywhere.

##  13\. The Wedge Theorem and Boundary Regularity

######  Theorem 35 (Wedge Theorem).

Suppose $W$ is a wedge (see Definition 31) in $\mathbf{R}^{m+1}$ with edge
$\Gamma$. Suppose

$t\in(-\infty,0)\mapsto S(t)$

is a self-similar, standard Brakke flow in $W$ with boundary $\Gamma$. Then
$S(\cdot)$ is a non-moving halfplane with multiplicity $1$.

###### Proof.

Let $M=S(-1)$. By Theorem 30, $M=\sum_{i=1}^{k}P_{i}$ where each $P_{i}$ is a
multiplicity-one halfplane plane with boundary $\Gamma$. Since
$\partial[M]=[\Gamma]$, $k$ is odd.

For each $i$, let $\nu_{i}$ be the unit vector in the plane of $P_{i}$ that is
normal to $\Gamma$ and that points out from $P_{i}$. For any smooth, compactly
supported vectorfield $X$,

$\int\operatorname{Div}_{P_{i}}X\,dP_{i}=\int_{\Gamma}X\cdot\nu_{i}\,d\mathcal{H}^{m-1},$

so

$\displaystyle\int\operatorname{Div}_{M}X\,dM$
$\displaystyle=\int_{\Gamma}\left(\sum_{i=1}^{k}\nu_{i}\right)\cdot
X\,d\mathcal{H}^{m-1}.$

Thus

$\nu(M,\cdot)=\sum_{i=1}^{k}\nu_{k}.$

By definition of mean curvature flow with boundary, $|\nu(M,\cdot)|\leq 1$.
Now we use the following elementary fact: if
$\nu^{\prime}=\sum_{i=1}^{k}\nu_{i}$ where the
$\nu_{i}=(\cos\theta_{i},\sin\theta_{i})$ are unit vectors with
$|\theta_{i}|\leq\theta<\pi/2$, then

(26) $|\nu^{\prime}|\geq\begin{cases}k\cos\theta&\text{if $k$ is even},\\\
\sqrt{1+(k^{2}-1)\cos^{2}\theta}&\text{if $k$ is odd}.\end{cases}$

(The inequalities (26) can be proved as follows. Given $k$ and $\theta$, it is
easy to show that at the mimimum of $|\nu^{\prime}|$, each $\theta_{i}$ is
$\pm\theta$. If $k$ is even, the minimum is attained by having half of the
$\theta_{i}$ equal to $\theta$ and the other half equal to $-\theta$. If
$k=2j+1$ is odd, the minimum is attained when $j$ of the $\theta_{i}$ are
equal to $\theta$ and $j+1$ are equal to $-\theta$.)

In our case, the number of planes (counting multiplicity) in $M^{\prime}$ is
odd, so

$1\geq|\nu^{\prime}|\geq\sqrt{1+(k^{2}-1)\cos^{2}\theta}.$

Therefore $k=1$. ∎

As an immediate consequence of Theorem 35, we have

######  Theorem 36.

Suppose $t\in[0,T]\mapsto M(t)$ is an $m$-dimensional standard mean curvature
flow with boundary $\Gamma$ in a smooth, $(m+1)$-dimensional Riemannian
manifold. If a tangent flow at $(p,t)$ is contained in a wedge, where
$p\in\Gamma$ and $t>0$, then $(p,t)$ is a regular point of the flow.

##  14\. A Boundary Regularity Theorem

######  Theorem 37.

Suppose $N$ is a smooth, $(m+1)$-dimensional Riemannian manifold with smooth,
weakly mean-convex boundary. Suppose $\Gamma$ is a smooth, properly embedded
$(m-1)$-dimensional submanifold of $N$. Suppose

$t\in[0,T]\mapsto M(t)$

is a standard Brakke flow in $N$ with boundary $\Gamma$. If

  1. (1)

$\partial N$ is strictly mean convex, or if

  2. (2)

$\operatorname{spt}(M(0))\cap\partial N=\Gamma\cap\partial N$,

then for every $p$ in $\Gamma\cap\partial N$ and for every $t\in(0,T]$, the
spacetime point $(p,t)$ is a regular point of the flow.

###### Proof.

Note that if $\partial N$ is strictly mean convex, then by the maximum
principle ([ilmanen-elliptic]*10.5 or [hershkovits-white-avoid]*theorem 27),

$\operatorname{spt}(M(t))\cap\partial N=\Gamma\cap\partial N$

for all $t>0$. In other words, as soon as $t>0$, Hypothesis (2) holds. Thus it
suffices to prove Theorem 37 under Hypothesis (2).

Since the result is local, it suffices to work in a small neighborhood of the
point $p$. Such a neighborhood is diffeomorphic to a halfspace, so we may
assume that

$N=\\{x\in\mathbf{R}^{m+1}:x_{m+1}\geq 0\\}$

with some smooth Riemannian metric $g$. We may also assume that $p$ is the
origin and that the metric is Euclidean at the origin (i.e., that
$g_{ij}(0)=\delta_{ij}$). In the rest of the proof,
$\operatorname{dist}(\cdot,\cdot)$ refers to $g$-distance, but
$\mathbf{B}^{m+1}(a,r)$ is the Euclidean ball of radius $r$ about $a$, and if
$f$ is a function from a domain in $\mathbf{R}^{m}$ to $[0,\infty)$ (so that
the graph lies in $N$), then expressions such as $\nabla f$ and
$\|f\|_{C^{2}}$ are with respect to the Euclidean metric.

######  Lemma 38.

There is an $\epsilon>0$ with the following property. If $a\in\partial
N\cong\mathbf{R}^{m}$, $|a|\leq\epsilon$, $0<r\leq\epsilon$, and

$f:\mathbf{B}^{m}(a,r)\to[0,\infty)$

is a function with $\|f\|_{C^{3}}\leq\epsilon$, then there is a $C^{2}$
function

$F:\mathbf{B}^{m}(a,r)\times[0,\infty)\to\mathbf{R}$

such that

$\displaystyle F(\cdot,0)=f(\cdot),$ $\displaystyle
F(\cdot,t)|\partial\mathbf{B}^{m}(a,r)=f|\partial\mathbf{B}^{m}(a,r),$
$\displaystyle\sup\ |\nabla F|\leq 1,$

and such that the graph of $F(\cdot,t)$ moves by mean curvature flow with
respect to the metric on $N$.

###### Proof of lemma.

If not, there would be a sequence of solutions

$F_{i}:\mathbf{B}^{m}(a_{i},r_{i})\times[0,T_{i}]\to[0,\infty)$

of the nonparametric $g$-mean-curvature flow equation with $|a_{i}|\to 0$ and
$r_{i}\to 0$ such that

(27) $\|F_{i}(\cdot,0)\|_{C^{3}}\to 0$

and such that

$\max|\nabla F_{i}|=|\nabla F_{i}(x_{i},T_{i})|=1$

for some $x_{i}$. Let $D_{i}(t)$ be the graph of $F_{i}(\cdot,t)$. By (27),

$\frac{g\operatorname{area}(D_{i}(0))}{\omega_{m}r_{i}^{m}}\to 1$

and thus

(28)
$\sup_{t\in[0,T_{i}]}\frac{g\operatorname{area}(D_{i}(0))}{\omega_{m}r_{i}^{m}}=\frac{g\operatorname{area}(D_{i}(0))}{\omega_{m}r_{i}^{m}}\to
1$

as $i\to\infty$. Translating in space by $-a_{i}$ and in time by $-T_{i}$,
dilating parabolically by $1/r_{i}$, and passing to a subsequential limit
gives a smooth solution

$F:\mathbf{B}^{m}(0,1)\times(-\infty,0]\to\mathbf{R}$

of the Euclidean nonparametric mean curvature flow equation such that

$F(\cdot,t)|\partial\mathbf{B}^{m}(0,1)\equiv 0$

and such that

$\sup|\nabla F|=\sup|\nabla F(\cdot,0)|=1.$

But by (28), the area of the graph of $F(\cdot,t)$ is less than of equal to
area of $\mathbf{B}^{m}(0,1)$, and thus $F\equiv 0$, a contradiction. This
proves the lemma. ∎

We now complete the proof of Theorem 37. Choose
$a\in\operatorname{Tan}(\Gamma,0)^{\perp}\cap\partial N$ with $0<|a|<\epsilon$
(where $\epsilon$ is as in Lemma 38.) We choose $a$ sufficiently close to $0$
that the Euclidean ball $\mathbf{B}^{m+1}(a,|a|)$ intersects $\Gamma$ in the
single point $0$. Let $2R=|a|$.

Let $f:\mathbf{R}^{m}\to[0,\infty)$ be a smooth function such that: $f(a)>0$,
$f$ is supported in the interior of $\mathbf{B}^{m}(a,R)$,
$\|f\|_{C^{3}}<\epsilon$, and

(29) $(\operatorname{spt}M(0))\setminus\Gamma$ lies in the set
$\\{x:x_{m+1}>f(x_{1},\dots,x_{m})\\}$.

For $R/2\leq r\leq R$, let

$F_{r}:\mathbf{B}^{m}(a,r)\times[0,T]\to\mathbf{R}$

be the solution of the nonparametric mean curvature flow equation (with
respect to the metric $g$) such that

$F_{r}(\cdot)=0$ on $\partial\mathbf{B}^{m}(a,r)$, and $F(\cdot,0)=f(\cdot)$
on $\mathbf{B}^{m}(a,r)$.

By choice of $a$,

$|\nabla F_{r}(x,t)|\leq 1$

for all $x\in\mathbf{B}^{m}(a,r)$ and $t\in[0,T]$. It follows that the graph
$D_{r}(t)$ of $F_{r}(\cdot,t)$ is contained in the ball
$\mathbf{B}^{m+1}(a,r)$, and thus

(30) $D_{r}(t)\cap\Gamma=\emptyset\quad\text{for all $r\in[R,2R)$ and
$t\in[0,T]$}.$

By the strong maximum principle,

$F_{r}(x,t)>0$ for all $x$ in the interior of $\mathbf{B}(a,r)$ and all
$t\in[0,T]$

and

(31) $|\nabla F_{r}(x,t)|\neq 0$ for all $x\in\partial\mathbf{B}^{m}(a,r)$ and
$t\in(0,T]$.

Let

$Q_{r}(t)=\\{x:0<x^{m+1}<F_{r}(x_{1},\dots,x_{m},t)\\}.$

We claim that

(32) If $R\leq r<2R$, then $\overline{Q_{r}(t)}$ is disjoint from
$\operatorname{spt}M(t)$ for $t\in[0,T]$.

For suppose not. At the first time $t$ of contact, let $q$ be a point in
$\operatorname{spt}M(t)\cap\overline{Q_{r}(t)}$. Then $q$ is in the graph
$D_{r}(t)$ of $F(\cdot,r)$. If $q$ were in $\partial D_{r}(t)$, then the
tangent flow to $M(\cdot)$ at $(q,t)$ would be contained in wedge (by (31)),
which is impossible (see Corollary 32). Thus

(33) $\partial D(\tau)\cap M(\tau)=\emptyset\quad\text{for all
$\tau\in[0,t]$},$

and $q\in D(t)\setminus\partial D(t)$. But this (together with (29) and (30))
violates the maximum principle (Theorem 44). This proves (32).

From (32), we see that

(34) $\operatorname{spt}M(t)\cap Q_{2R}(t)=\emptyset$

since $\cup_{r<2R}Q_{r}(t)=Q_{2R}(t)$.

Now let $t\in(0,T]$. By (34) and (31), any tangent flow to $M(\cdot)$ at
$(p,t)$ must be contained in a wedge. Thus $(p,t)$ is a regular point of the
flow $M(\cdot)$ by Theorem 36. ∎

##  15\. Moving Boundaries

Let $I$ be an interval in $\mathbf{R}$. A moving $(m-1)$-dimensional boundary
in $U\times I$ is a smooth, properly embedded, $m$-dimensional submanifold
(without boundary) $\Gamma$ of $U\times I$ such that the time function
$(x,t)\in\Gamma\mapsto t$ has no critical points on $\Gamma$. For $t\in I$, we
let $\Gamma(t)=\\{x:(x,t)\in\Gamma\\}$. For $(x,t)\in\Gamma$, we let
$\dot{\Gamma}(x,t)$ be normal velocity of $\Gamma(t)$ at $x$: it is the unique
vector $v\in\operatorname{Tan}(\Gamma(t),x)^{\perp}$ such that $(v,1)$ is
tangent to $\Gamma$ at $(x,t)$.

######  Definition 39.

Let $\Gamma\subset U\times I$ be a moving $(m-1)$-dimensional boundary. A
Brakke flow with (moving) boundary $\Gamma$ is a Borel map

$t\in I\mapsto M(t)\in\mathcal{M}(U)$

such that

  1. (1)

For almost every $t\in I$, $M(t)$ is in $\mathcal{V}_{m}(U,\Gamma(t))$.

  2. (2)

If $[a,b]\subset I$ and $K\subset U$ is compact, then

$\int_{a}^{b}\int_{K}(1+|H|^{2})\,dM(t)\,dt<\infty.$

  3. (3)

If $[a,b]\subset I$ and if $u$ is a nonnegative, compactly supported $C^{2}$
function on $U\times[a,b]$, then

$\displaystyle(Mu)(a)-(Mu)(b)$
$\displaystyle\quad\geq\int_{a}^{b}\int\left(u|H|^{2}-H\cdot\nabla
u-\frac{\partial u}{\partial t}\right)\,dM(t)\,dt-\int_{a}^{b}\int
u\nu\cdot\dot{\Gamma}\,d\Gamma(t)\,dt.$

######  Theorem 40.

Suppose $t\in I\mapsto M(t)$ is a Brakke flow with moving boundary $\Gamma$.

  1. (1)

The defining inequality (3) in Definition 39 holds for every nonnegative,
compactly supported, Lipschitz function $u$ on $U$ that is $C^{1}$ on
$\\{u>0\\}$.

  2. (2)

If

$f:U\times[a,b]\to\mathbf{R}$

is a nonnegative, $C^{2}$ function with $\overline{\\{f>0\\}}$ compact, and if
$u:=1_{f\geq 0}f$, then

$\displaystyle(Mu)(a)-(Mu)(b)$
$\displaystyle\quad\geq\int\left(u|H|^{2}+\operatorname{Div}_{M}\nabla
u-\frac{\partial u}{\partial t}\right)\,dM(t)\,dt+\int\nu\cdot(\nabla
u-\dot{\Gamma})\,d\Gamma$
$\displaystyle\quad\geq\int\left(u|H|^{2}+\operatorname{Trace}_{m}(\nabla^{2}u)-\frac{\partial
u}{\partial t}\right)\,dM(t)\,dt+\int\nu\cdot(\nabla
u-\dot{\Gamma})\,d\Gamma.$

  3. (3)

Suppose that

$\displaystyle\text{$\overline{\mathbf{B}(x,R)}$ is a compact subset of $U$},$
$\operatorname{dist}(\cdot,x)^{2}$ is smooth on $\overline{\mathbf{B}(x,R)}$,
$\operatorname{Trace}_{m}(-\nabla^{2}(\operatorname{dist}(\cdot,x)^{2}))\geq-4m$
on $\mathbf{B}(x,R)$, and $\displaystyle\text{$|\dot{\Gamma}|\leq\delta$ on
$\Gamma\cap(\mathbf{B}(x,R)\times[0,T])$}.$

Let $u=(R^{2}-\operatorname{dist}(\cdot,x)^{2}-4mt)^{+}$. Then for
$t\in[0,T]$,

$(Mu)(t)\leq(Mu)(0)+t(R+\delta
R^{2})\mathcal{H}^{m-1}(\Gamma\cap\mathbf{B}(x,R)).$

  4. (4)

Let $u$ be a $C^{2}$, nonnegative, compactly supported function on $U$.

$\displaystyle\frac{1}{2}\int_{a}^{b}\int u|H|^{2}\,dM(t)\,dt$
$\displaystyle\leq M(a)u-M(b)u$
$\displaystyle\quad+(b-a)(\max|\nabla^{2}u|)K_{[a,b]}$
$\displaystyle\quad+(b-a)L_{[a,b]}$

for $[a,b]\subset I$, where

$\displaystyle K_{[a,b]}$
$\displaystyle:=\sup_{t\in[a,b]}M(t)(\operatorname{spt}u),$ $\displaystyle
L_{[a,b]}$ $\displaystyle:=\sup_{t\in[a,b]}\int u|\dot{\Gamma}|\,d\Gamma(t).$

The proofs are almost identical to the proofs of Proposition 11, Corollary 13,
Theorem 14, and Theorem 15.

######  Theorem 41.

Let $U$ be an open subset of $\mathbf{R}^{d}$ containing
$\mathbf{B}^{d}(0,1)$. Let $N$ be a smooth, properly embedded submanifold of
$U$. Let $\Gamma$ be an $(m-1)$-dimensional moving boundary in
$N\times[T_{0},0]$. Let $t\in[T_{0},0]\mapsto M(t)$ be an integral Brakke flow
in $N$ with moving boundary $\Gamma$. Suppose that

$M(t)\mathbf{B}(0,1)\leq C$

for $t\in I$ and that the norm of the second fundamental form of $N$ is
bounded by $A$. Then for $a,b\in[T_{0},0]$ with $a\leq b<0$, Then for $a,b\in
I$ with $a\leq b<0$,

(35) $\displaystyle(M\hat{\rho})(a)-(M\hat{\rho})(b)$
$\displaystyle\geq\int_{a}^{b}\int\left|H_{N}-\frac{(\nabla^{\perp}\hat{\rho})_{N}}{\hat{\rho}}\right|^{2}\hat{\rho}\,dM(t)\,dt$
$\displaystyle\qquad+\int_{a}^{b}\int\nu_{M}\cdot(\nabla\hat{\rho}-\rho\dot{\Gamma})\,d\Gamma\,dt$
$\displaystyle\qquad-mA^{2}\int_{a}^{b}\int\hat{\rho}\,dM(t)\,dt$
$\displaystyle\qquad-CK(b-a)$

where $C$ and $K$ are as in (13) and (15). Furthermore,

(36)
$e^{-mA^{2}t}\left((M\hat{\rho})(t)+\int_{\tau=T_{0}}^{t}\int\nu_{M}\cdot(\nabla\hat{\rho}-\hat{\rho}\,\dot{\Gamma})\,d\Gamma\,d\tau-
CKt\right)$

is a decreasing function of $t$ for $t<0$ in $I$.

###### Proof.

The proof is exactly like the proof of the proof of the Monotonicity Theorem
17, except that, starting with the right hand side of the first inequality in
(17), there is one additional extra term:

$-\int_{a}^{b}\int\hat{\rho}\,\dot{\Gamma}\cdot\nu\,d\Gamma\,dt.$

∎

######  Corollary 42.

As $t\uparrow 0$, $(M\hat{\rho})(t)$ converges to a finite limit.

###### Proof.

A straightforward computation shows that

(37)
$\int_{T_{0}}^{0}\int|(\nabla\hat{\rho}-\hat{\rho}\,\dot{\Gamma})_{\Gamma^{\perp}}|\,d\Gamma\,dt<\infty.$

The corollary now follows immediately from the monotonicity of (36). ∎

The compactness and closure theorems, existence of tangent flows, and the
definition of standard flows are the exact analogs are the corresponding
theorems and definition for fixed boundaries (§7, §8, §10) so we will not
state them. For those theorems, the extra term arising from the motion of
boundary is easy to control, so only trivial modifications of the proofs are
required.

Just as for fixed boundaries, we have (as an immediate consequence of the
Wedge Theorem 35),

######  Theorem 43.

Suppose that $M:[0,T]\mapsto M(t)$ is an $m$-dimensional standard Brakke flow
with moving boundary $\Gamma$ in an $(m+1)$-dimensional Riemannian manifold.
If $t>0$, if $p\in\Gamma(t)$, and if a tangent flow at $(p,t)$ is contained in
a slab, then $(p,t)$ is a regular point of the flow $M(\cdot)$.

##  16\. Appendix: A Strong Maximum Principle

######  Theorem 44.

Suppose that $t\in[0,T]\mapsto M(t)$ is an integral Brakke flow with moving
boundary $\Gamma$ in a smooth Riemannian manifold $N$. Suppose that
$t\in[0,T]\mapsto D(t)$ is a smooth, one-parameter family of compact, smoothly
embedded, $m$-dimensional manifolds with boundary in $N$ that are moving by
mean curvature. (In other words, the normal velocity at each point is equal to
the mean curvature vector at that point.) If

$\displaystyle\text{$D(0)$ is disjoint from $\operatorname{spt}(M(0))$},$
$\partial D(t)$ is disjoint from $\operatorname{spt}M(t)$ for all $t\in[0,T]$,
and $\Gamma(t)$ is disjoint from $D(t)$ for all $t\in[0,T]$,

then $D(t)$ is disjoint from $\operatorname{spt}(M(t))$ for all $t\in[0,T]$.

###### Proof.

Choose $\delta>0$ small enough so that

$Q:=\cup_{t\in[0,T]}\\{x\in N:\operatorname{dist}(x,D(t))\leq\delta\\}$

is compact, and so that for $t\in[0,T]$,

(38) $\operatorname{dist}(p,q)>\delta$

for all $p\in\Gamma(t)$ and $q\in D(t)$, and for all
$p\in\operatorname{spt}M(t)$ and $q\in\partial D(t)$.

Let $\lambda<0$ be a strict lower bound for the Ricci curvature of $N$ in the
set $Q$. We claim that

$e^{-\lambda t}\operatorname{dist}(D(t),\operatorname{spt}M(t))>\delta$

for all $t\in[0,T]$. For if not, there would be first time $t>0$ such that

(39) $e^{-\lambda t}\operatorname{dist}(D(t),\operatorname{spt}M(t))=\delta.$

At that time, there would be an arc-length parametrized geodesic

$\gamma:[0,L]\to N$

such that

$\displaystyle\gamma(0)\in D(t),$
$\displaystyle\gamma(L)\in\operatorname{spt}M(t),\,\text{and}$ $\displaystyle
L=\operatorname{dist}(D(t),\operatorname{spt}M(t))$

Note that $\gamma(0)\notin\partial D(t)$ and $\gamma(L)\notin\Gamma(t)$ by
(38) and (39). (Recall that $\lambda<0$). By [hershkovits-white-avoid]*Lemma
11 and [hershkovits-white-avoid]*Theorem 28, at the point $\gamma(0)\in D(t)$,
the surface $D(\cdot)$ moves in the direction $\gamma^{\prime}(0)$ faster than
the mean curvature in that direction, a contradiction. ∎

## References