# Epsilon-regularity for the Brakke flow with boundary

Carlo Gasparetto Mathematics Area, SISSA, Trieste (Italy).
mailto:cgaspare<EMAIL_ADDRESS>
###### Abstract

We prove that, if a Brakke flow with boundary is close enough to a stationary
half-plane with density one, then it is $C^{1,\alpha}$. Our approach is based
on viscosity techniques introduced by Savin in the context of elliptic
equations. The same techniques can be used to give a proof of Brakke’s
(interior) regularity theorem which is alternative to the original one. We
also prove a constancy theorem for the Brakke flow.

##  1 Introduction

In this paper, we state and prove a Brakke-type theorem for the mean curvature
flow with boundary, that is a flow of $m$-dimensional surfaces in
${\mathbb{R}}^{d}$ so that at every point the normal component of the velocity
is equal to the mean curvature and the boundary is fixed. A weak notion of
such a flow has been recently introduced in [20] by using integral varifolds,
as devised by Brakke [5]. The objects in question are called integral Brakke
flows with boundary.

In short, given a $(m-1)$-dimensional submanifold $\Gamma$, an integral Brakke
flow with boundary $\Gamma$ is a collection $\\{V_{t}\\}_{t\in I}$ of
$m$-dimensional integral varifolds with the constraint that the first
variation of $V_{t}$ is a measure whose singular part with respect to
$||V_{t}||$ behaves like $\mathcal{H}^{m-1}\llcorner\Gamma$ and the varifolds
satisfy an evolution equation that encodes the information on the velocity. A
precise definition will be given in Section 2.

The main result of this paper is that, if a Brakke flow in a ball of radius
$1$ is close enough (in some appropriate topology) to a unit-density half
plane (which is a stationary solution to the mean curvature flow with a
prescribed straight boundary), then the Brakke flow becomes smooth up to the
boundary in a smaller ball and after some fixed waiting time. Namely, we prove
the following

######  Theorem 1.1 ($\varepsilon$-regularity).

Let $\Gamma$ be a $C^{1,\alpha}$, $(m-1)$-dimensional submanifold of $B_{1}$
and let $\\{V_{t}\\}_{t\in[-\Lambda,0]}$ be an integral Brakke flow with
boundary $\Gamma$ in $B_{1}\times[-\Lambda,0]$. Assume the following:

  1. 1.

$0\in\operatorname{supp}||V_{0}||$;

  2. 2.

the mass of $V_{t}$ is close to the measure of a $m$-dimensional half disk,
namely

$\left\|V_{t}\right\|\leq\frac{3}{4}\omega_{m}\mbox{ for every
}t\in[-\Lambda,0],$

where $\omega_{m}$ is the Lebesgue measure of a ball of radius $1$ in
${\mathbb{R}}^{m}$.

  3. 3.

there exists a half-plane $S^{+}$ such that, for every $t\in[-\Lambda,0]$:

$\operatorname{supp}\left\|V_{t}\right\|\subset\\{x\in{\mathbb{R}}^{d}\colon\operatorname{dist}(x,S^{+})\leq\varepsilon\\}.$

If $\varepsilon$ and $\Lambda$ are small enough, then there exist small
constants $\eta,\beta$ and a family $\\{N_{t}\\}_{t\in(-\eta^{2},0]}$ of
$C^{1,\beta}$ surfaces with boundary $\Gamma$ such that

$\left\|V_{t}\right\|\llcorner B_{\eta}=\mathcal{H}^{m}\llcorner N_{t}$

for every $t\in(-\eta^{2},0]$.

We briefly comment on the assumptions. The key assumptions are Item 2 and Item
3, which describe how the Brakke flow is close to being a half-plane (with a
straight boundary). Item 1, on the other hand, prevents a “pathological”
behavior of Brakke flows, which is the possibility of a sudden loss of mass
(see, for example, [15, Section 2.3]). All the assumptions will be made more
rigorous in Section 4.

As a corollary of Theorem 1.1 and of the maximum principle that we will prove
in Subsection 3.2, $\\{N_{t}\\}$ is a viscosity solution to the mean curvature
flow. Therefore more regularity may be proved, a posteriori.

A central point in our work is that, under appropriate assumptions, the
support of an integral Brakke flow with boundary satisfies a maximum
principle. In order to fix ideas, assume that the support of the flow is the
graph of some function $u:{\mathbb{R}}^{m}\to{\mathbb{R}}^{d-m}$. Then it can
be proved that $|u|$ is a viscosity subsolution (in a suitable sense which we
will describe at a later stage) to

$\partial_{t}\varphi-\mathcal{M}^{+}(D^{2}\varphi)\leq 0,$

where $\mathcal{M}^{+}$ is a Pucci maximal operator. We may therefore exploit
this property to adopt a technique developed by Savin in [12] in the framework
of elliptic equations and later adapted by Wang in [17] to parabolic
equations, which we now summarize in our case. The key step in proving Theorem
1.1 is proving the following improvement of flatness:

######  Proposition 1.2 (Improvement of flatness).

Under the assumption of Theorem 1.1, there exist $\eta>0$ and a half plane
$T^{+}$ close to $S^{+}$ such that, for every $t\in(-\eta^{2},0]$,

$\operatorname{supp}\left\|V_{t}\right\|\cap
B_{\eta}\subset\bigg{\\{}x\in{\mathbb{R}}^{d}\colon\operatorname{dist}(x,T^{+})\leq\frac{\varepsilon}{2}\eta\bigg{\\}}.$

In summary, if the Brakke flow is “$\varepsilon$-flat” at scale $1$, then it
becomes “$\eta\varepsilon/2$-flat” at scale $\eta$, for some $\eta$ small and
universal; from this, proving $C^{1,\alpha}$-regularity is classical.

The proof of Proposition 1.2 is based on a contradiction and compactness
argument. Assume one can find a sequence of flatter and flatter Brakke flows
for which the conclusion of Proposition 1.2 does not hold. Then appropriate
rescalings of the supports of such flows converge in a suitable sense to the
graph of a solution to the heat equation. The desired improvement of flatness
is a straightforward consequence of classical Schauder estimates. The above
convergence is achieved via a Harnack-type inequality, in the spirit of [17],
and a barrier argument that describes the behavior of the Brakke flow near the
boundary.

Theorem 1.1 answers a question left open in [20, Remark 11.2], that is whether
an integral Brakke flow with boundary that has a tangent flow which is a unit-
density half-plane is smooth in a backward neighborhood. The reader should
also compare our results with the regularity theorems proved in [20]. The
latter are proved under the additional assumption that the flow is standard:
namely the flow has to be smooth at every point where a tangent flow is a
unit-density half-plane (see [20, Definition 11.1]). Since we only prove
backward regularity, our result does not guarantee (as it should not be
expected) that an integral Brakke flow with boundary satisfying the
assumptions of Theorem 1.1 is actually standard.

The first $\varepsilon$-regularity theorem for the mean curvature flow
(without boundary) was proved in [5] and then refined in [9], where the
authors extended the result to mean curvature flow in general ambient
manifolds. Both those proof are variational and rely on $L^{2}$ energy
estimates, somehow in the spirit of [1]. We think that a variational proof of
Theorem 1.1 may be performed, by adapting the arguments in [2] and in [4] to
account for the presence of the boundary. As mentioned, our proof is based on
an argument first developed in [12] for elliptic equations and then adapted to
parabolic equations in [17]. This method was used in [11] to prove an Allard-
type theorem for minimal surfaces. Although an adaptation of the same
techniques to the mean curvature flow seems quite natural, to the best of the
author’s knowledge this paper is the first instance in which these techniques
are used for the mean curvature flow.

The regularity of mean curvature flow with boundary has been briefly
investigated also in [21, 18]. One should also see [7], where the author
defines a Brakke flow with a free boundary condition. Another definition of
Brakke flow with fixed boundary has been investigated in [14].

###  1.1 Structure of the paper

In Section 2, we collect some notations that will be used throughout the paper
and some well known facts about rectifiable measures. We then recall the
definition of Integral Brakke flow with boundary, as stated in [20].

Section 3 is dedicated to collecting some known results about Integral Brakke
flows and to adapting them to the case of an integral Brakke flow with fixed
boundary. In Subsection 3.3, we state and prove a constancy theorem for the
Brakke flow both with and without boundary, which, to the best of the author’s
knowledge, has not appeared in the literature yet. We think this result is of
independent interest.

The core of the paper is Section 4, where we state and prove the improvement
of flatness described in Proposition 1.2, which will later yield the desired
$C^{1,\beta}$ regularity. The proof of this result is described in Subsection
4.2. The aforementioned barrier argument and Harnack-type inequality, which
are crucial for obtaining the desired compactness, are discussed in Section 5
and Section 6, respectively.

Finally, the proof of Theorem 1.1 is given in Section 7, where we iterate the
improvement of flatness to obtain the desired regularity.

### Acknowledgements

The author is extremely grateful to Guido De Philippis for his crucial help in
writing this paper and to Luigi De Masi for countless discussions and
comments. This material is based upon work partially supported by INDAM-
GNAMPA.

##  2 Preliminaries, notation and definitions

Throughout the paper, we consider fixed two positive integers $m$ and $d$ such
that $m\leq d$. All the constants taken in consideration in the present work
depend, in general, on these two parameters, although we will mostly avoid
stating such dependency.

For the present section, we introduce two generic positive integers $k\leq n$
to define some objects in full generality.

###  2.1 Space-time

By ${\mathbb{R}}^{n,1}$ we denote the space $\\{(x,t)\colon
x\in{\mathbb{R}}^{n}\mbox{ and }t\in{\mathbb{R}}\\}$. We use upper-case
letters to denote points in ${\mathbb{R}}^{n,1}$, for example $X=(x,t)$.

For any couple $X=(x,t)$ and $Y=(y,s)$ of points in ${\mathbb{R}}^{n,1}$, we
let

$\rho(X,Y)=|x-y|+|t-s|^{1/2};$

$\rho$ is a metric on ${\mathbb{R}}^{n,1}$ (see, for example [10, Exercise
8.5.1]) and the topology that $\rho$ induces on ${\mathbb{R}}^{n,1}$ coincides
with the euclidean topology of ${\mathbb{R}}^{n+1}$. In particular, if
$d_{H}(E,F)$ is the Hausdorff distance between $E$ and $F$ with respect to
$\rho$ and $K$ is a compact subset of ${\mathbb{R}}^{n,1}$, then the space of
non-empty closed subsets of $K$ is a compact metric space, when endowed with
the metric $d_{H}$.

If $x\in{\mathbb{R}}^{n}$ and $r>0$, we set
$B^{n}_{r}(x)=\\{y\in{\mathbb{R}}^{n}\colon|y-x|<r\\}$. When the dimension of
the space is clear, we omit its indication and simply write $B_{r}(x)$. We
also omit the indication of the center of the ball, whenever it coincides with
$0$, so that $B_{r}=B_{r}(0)$. If $(x,t)\in{\mathbb{R}}^{n,1}$, we define the
parabolic cylinder

$Q^{n}_{r}(x,t)=B^{n}_{r}(x)\times(t-r^{2},t],$

where the apex $n$ indicates the dimension of the space component; as above,
its indication will be omitted when no confusion shall arise. Lastly, we let
$Q_{r}=Q_{r}(0,0)$.

$\partial_{p}(U\times(a,b))$ denotes the parabolic boundary of the cylinder
$U\times(a,b)$, where $U\subset{\mathbb{R}}^{n}$:

$\partial_{p}(U\times(a,b)):=\big{(}\overline{U}\times\\{a\\}\big{)}\cup\big{(}\partial
U\times(a,b)\big{)}.$

We define the measures $\mathcal{L}^{n,1}$ and $\mathcal{H}^{s,1}$ (for any
$0\leq s\leq n$) on ${\mathbb{R}}^{n,1}$ by

$\mathcal{L}^{n,1}(E\times
F)=\mathcal{L}^{n}(E)\times\mathcal{L}^{1}(F),\qquad\mathcal{H}^{s,1}(E\times
F)=\mathcal{H}^{s}(E)\times\mathcal{L}^{1}(F)$

for $E\subset{\mathbb{R}}^{n}$ and $F\subset{\mathbb{R}}$, where
$\mathcal{L}^{n}$ is the Lebesgue measure in ${\mathbb{R}}^{n}$ and
$\mathcal{H}^{s}$ is the $s$-dimensional Hausdorff measure in
${\mathbb{R}}^{n}$.

For any function $f:{\mathbb{R}}^{n,1}\to{\mathbb{R}}^{k}$, we denote by
$\nabla f(x,t)$ the gradient of the function $f(\cdot,t)$ computed at $x$ and
by $\partial_{t}f(x,t)$ the derivative of $f(x,\cdot)$ computed at $t$,
whenever they are defined.

Lastly, for a set $E\subset{\mathbb{R}}^{n}$ and $x\in{\mathbb{R}}^{n}$, we
let $\chi_{E}(x)=0$ if $x\notin E$ and $\chi_{E}(x)=1$ if $x\in E$.

###  2.2 Linear functions and subspaces of the euclidean space

We let $\\{\mathbf{e}_{1},\dots,\mathbf{e}_{n}\\}$ be the canonical
orthonormal basis of ${\mathbb{R}}^{n}$.

We define the Grassmannian $\mathrm{Gr}(k,n)$ as the space of (unoriented)
$k$-dimensional linear subspaces of ${\mathbb{R}}^{n}$; we identify
$S\in\mathrm{Gr}(k,n)$ with the endomorphism
$S\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ representing the orthogonal
projection onto $S$. When no confusion shall arise and an orthonormal basis
$\\{\zeta_{1},\dots,\zeta_{k}\\}$ of $S$ is fixed, we identify $S$ with
${\mathbb{R}}^{k}$ via the canonical bijection

$\iota:S\ni
x\mapsto(x\cdot\zeta_{1},\dots,x\cdot\zeta_{k})\in{\mathbb{R}}^{k}:$

therefore by $Sx$ we denote both the point $Sx\in S\subset{\mathbb{R}}^{n}$
and its image via $\iota$. In particular, when
$S=\operatorname{span}\\{\mathbf{e}_{1},\dots,\mathbf{e}_{k}\\}$ and
$X=(x,t)\in{\mathbb{R}}^{n,1}$, we will often use the notation
$x^{\prime}=Sx=(x\cdot\mathbf{e}_{1},\dots,x\cdot\mathbf{e}_{k})\in{\mathbb{R}}^{k}$.

We also let $S:{\mathbb{R}}^{n,1}\to{\mathbb{R}}^{k,1}$ be the map
$S(x,t)=(Sx,t)$ and, in the case
$S=\operatorname{span}\\{\mathbf{e}_{1},\dots,\mathbf{e}_{k}\\}$, for
$X=(x,t)\in{\mathbb{R}}^{n,1}$ we let $X^{\prime}=(x^{\prime},t)$.

Lastly, if $S$ and $T$ are two endomorphisms of ${\mathbb{R}}^{n}$, we define
the scalar product between $S$ and $T$ by

$S\colon T=\sum_{i,j=1}^{n}S_{ij}T_{ij},$

where $(S_{ij})$ is the representation of $S$ as a $n\times n$ matrix such
that

$S_{ij}=(S\mathbf{e}_{i})\cdot\mathbf{e}_{j}.$

We also let $|S|=\sqrt{S\colon S}$.

###  2.3 Hölder regularity

We point out some facts and definitions on Hölder regularity for several
objects. In what follows, $\kappa\in(0,1)$ is a fixed parameter.

  * •

Functions on ${\mathbb{R}}^{n}$. A function $u:{\mathbb{R}}^{n}\supset
U\to{\mathbb{R}}^{k}$ is said to be in $C^{1,\kappa}(U;{\mathbb{R}}^{k})$ if
$u\in C^{1}(U;{\mathbb{R}}^{k})$ and

$[\nabla u]_{C^{0,\kappa}(U)}=\sup_{\begin{subarray}{c}x,y\in U\\\ x\neq
y\end{subarray}}\frac{|\nabla u(x)-\nabla u(y)|}{|x-y|^{\kappa}}<\infty.$

We also let

$\left\|u\right\|_{C^{1,\kappa}(U)}=\left\|u\right\|_{L^{\infty}(U)}+[\nabla
u]_{C^{0,\kappa}(U)}.$

  * •

Functions on ${\mathbb{R}}^{n,1}$. Let $\Omega\subset{\mathbb{R}}^{n,1}$. We
say that $u:\Omega\to{\mathbb{R}}^{k}$ is in
$C^{1,\kappa}(\Omega;{\mathbb{R}}^{k})$ if it is everywhere differentiable
with respect to the space variable and the quantity

$\left\|u\right\|_{C^{1,\kappa}(\Omega)}:=\sup_{\Omega}|u|+\sup_{\begin{subarray}{c}X,Y\in\Omega\\\
X\neq Y\end{subarray}}\frac{|\nabla u(X)-\nabla
u(Y)|}{\rho(X,Y)^{\kappa}}+\sup_{\begin{subarray}{c}(x,t),(x,s)\in\Omega\\\
s\neq t\end{subarray}}\frac{|u(x,t)-u(x,s)|}{|t-s|^{(1+\kappa)/2}}$

is finite.

  * •

Submanifolds. We say that a $k$-dimensional, properly embedded submanifold
$\Gamma$ of some open set $U\subset{\mathbb{R}}^{n}$ is $C^{1,\kappa}$ if
there exists some $\kappa>0$ such that, for every $x,y\in\Gamma$, it holds

$[\Gamma]_{C^{1,\kappa}(U)}:=\sup_{\begin{subarray}{c}x,y\in\Gamma\\\ x\neq
y\end{subarray}}\frac{|T_{x}\Gamma-T_{y}\Gamma|}{|x-y|^{\kappa}}<\infty,$

where $T_{\cdot}\Gamma\in\mathrm{Gr}(k,n)$ is the tangent space to $\Gamma$
and $|T_{x}\Gamma-T_{y}\Gamma|$ should be intended as in Subsection 2.2.

Before stating the next definition, we introduce the following notation
denoting a closed half-space in ${\mathbb{R}}^{m}$:

${\mathbb{R}}_{+}^{m}=\\{x^{\prime}\in{\mathbb{R}}^{m}\colon
x^{\prime}_{m}\geq 0\\}.$

######  Definition 2.1 ($C^{1,\kappa}$-regular single sheet flow with
boundary).

Let $U\subset{\mathbb{R}}^{d}$ be an open connected set,
$I\subset{\mathbb{R}}$ be a non-empty interval and let $\Gamma$ be a
$(m-1)$-dimensional, properly embedded submanifold of $U$. We say that
$N=\bigcup_{t\in I}N_{t}\times\\{t\\}\subset U\times I$ is a
$C^{1,\kappa}$-regular single sheet flow with boundary $\Gamma$ in $U\times I$
if there exist $x^{\prime}\in{\mathbb{R}}_{+}^{m}$, $r>0$ and a family of
functions

$f_{t}:\overline{B_{r}(x^{\prime})}\cap{\mathbb{R}}_{+}^{m}\to\overline{U}\mbox{
for every }t\in I$

such that:

  * •

for every $t\in I$, $N_{t}$ is homeomorphic to
$\overline{B_{r}(x^{\prime})}\cap{\mathbb{R}}_{+}^{m}$ via $f_{t}$;

  * •

for every $t\in I$, $f_{t}(\partial
B_{r}(x^{\prime})\cap{\mathbb{R}}_{+}^{m})\subset\partial U$;

  * •

$f_{t}(B_{r}(x^{\prime})\cap\partial{\mathbb{R}}^{m}_{+})=\Gamma$ for every
$t\in I$;

  * •

the map $(t,y)\mapsto f_{t}(y)$ is $C^{1,\kappa}$ in
$(\overline{B_{r}(x^{\prime})}\cap{\mathbb{R}}_{+}^{m})\times I$.

In particular, if $N$ is a $C^{1,\kappa}$-regular single sheet flow with
boundary, then $N_{t}$ is a $C^{1,\kappa}$, $m$-dimensional submanifold of $U$
with boundary $\Gamma$. Notice that it well may be that
$B_{r}(x^{\prime})\cap({\mathbb{R}}^{m}_{+})^{c}=\varnothing$. Clearly, in
this case, $\Gamma=\varnothing$.

###  2.4 Integral Varifolds

We adopt most of the terminology from [20]. Let $U\subset{\mathbb{R}}^{d}$ be
an open set and let $\mathcal{M}(U)$ be the set of non-negative Radon measures
on $U$; if $\varphi$ is continuous and compactly supported on $U$, we let
$M(\varphi)=\int\varphi(x)\,dM(x)$. Let $\mathcal{M}_{m}(U)$ be the set of
$m$-dimensional rectifiable non-negative Radon measures on $U$. Namely,
$M\in\mathcal{M}_{m}(U)$ if and only if there exist a $m$-dimensional
rectifiable set $E$ and a non-negative function $\theta\in
L^{1}_{loc}(\mathcal{H}^{m}\llcorner E)$ such that

$M(\varphi)=\int_{E}\theta(x)\varphi(x)\,d\mathcal{H}^{m}(x)\qquad\mbox{for
all }\varphi\in C_{c}(U).$

We also let $\mathcal{IM}_{m}(U)$ be the set of those $M\in\mathcal{M}_{m}(U)$
such that their density $\theta(x)$ is a non-negative integer at $M$-a.e. $x$.
If $M\in\mathcal{M}_{m}(U)$, then for $M$-a.e. $x$ the approximate tangent
space $T_{x}M\in\mathrm{Gr}(m,d)$ is well defined (see, for instance, [13,
Chapter 3]). A $m$-dimensional varifold on $U$ is a Radon measure on
$U\times\mathrm{Gr}(m,d)$ (see [13, Chapter 8]). In particular, to each
$M\in\mathcal{M}_{m}(U)$ we may associate a $m$-dimensional varifold
$\mathrm{Var}(M)$ by

$\mathrm{Var}(M)(\varphi)=\int\varphi(x,T_{x}M)\,dM(x)\qquad\mbox{for all
}\varphi\in C_{c}(U\times\mathrm{Gr}(m,d)).$

Such an object is called a rectifiable varifold (see [13, Chapter 4]);
$\mathrm{Var}(M)$ is said to be integral if and only if
$M\in\mathcal{IM}_{m}(U)$. If $M\in\mathcal{M}_{m}(U)$, we say that
$\mathrm{Var}(M)$ has bounded first variation if there exists $C>0$ such that,
for every smooth vector field $F:U\to{\mathbb{R}}^{d}$ with compact support in
$U$, it holds

$\int T_{x}M\colon\nabla F(x)\,dM(x)\leq C\left\|\nabla F\right\|_{\infty}.$

If $\mathrm{Var}(M)$ has bounded first variation, then there exists a
$M$-locally integrable vector field $H_{M}$, a Radon measure $\beta_{M}$ that
is singular with respect to $M$ and a $\beta_{M}$-locally integrable unit
vector field $\zeta_{M}$ such that, for every $F\in
C^{1}_{c}(U;{\mathbb{R}}^{d})$, it holds

$\int T_{x}M:\nabla F(x)\,dM(x)=-\int H_{M}\cdot F\,dM+\int
F\cdot\zeta_{M}\,d\beta_{M}.$ (2.1)

In the following, we will often denote

$\operatorname{div}_{S}F(x)=S\colon\nabla F(x).$

When $M\in\mathcal{M}_{m}(U)$, we also let

$\operatorname{div}_{M}F(x):=\operatorname{div}_{T_{x}M}F(x)=T_{x}M\colon\nabla
F(x),$

whenever it is well defined.

######  Definition 2.2.

Let $\Gamma$ be a properly embedded $(m-1)$-dimensional submanifold of
$U\subset{\mathbb{R}}^{d}$. We let $\mathcal{V}_{m}(U,\Gamma)$ be the space of
those $M\in\mathcal{IM}_{m}(U)$ such that $\mathrm{Var}(M)$ has bounded first
variation and the following hold true:

  1. 1.

$\beta_{M}(E)\leq\mathcal{H}^{m-1}(E\cap\Gamma)$ for every $E\subset U$;

  2. 2.

$H_{M}(x)$ and $T_{x}M$ are perpendicular at $M$-a.e. $x$.

As mentioned in the remark following [20, Definition 6], Item 2 is actually
redundant, as it can be derived from [5, §5].

As in [20], for $M\in\mathcal{V}_{m}(U,\Gamma)$ we let

$\nu_{M}(x)=\lim_{r\searrow
0}\frac{1}{\omega_{m-1}r^{m-1}}\int_{B_{r}(x)}\zeta_{M}\,d\beta_{M}$ (2.2)

where the limit exists, and $\nu_{M}(x)=0$ otherwise. Notice that the
requirement $\beta_{M}\leq\mathcal{H}^{m-1}\llcorner\Gamma$ in Definition 2.2
yields $|\nu_{M}|\leq 1$ $\mathcal{H}^{m-1}\llcorner\Gamma$-a.e.. Moreover, by
[2, §3.1], $\nu_{M}(y)\perp\Gamma$ for $\mathcal{H}^{m-1}$-a.e. $y\in\Gamma$.

In the following, whenever $\Gamma$ is a $(m-1)$-dimensional submanifold of
${\mathbb{R}}^{d}$, by a small abuse of notation we denote by $\Gamma$ the
Hausdorff measure $\mathcal{H}^{m-1}\llcorner\Gamma$, if no confusion shall
arise.

###  2.5 Integral Brakke flows with boundary

Let $U\subset{\mathbb{R}}^{d}$ be an open set, $I\subset{\mathbb{R}}$ be a
non-empty interval and let $\Gamma$ be a properly embedded $(m-1)$-dimensional
submanifold of $U$.

######  Definition 2.3 (Integral Brakke flow).

A $m$-dimensional integral Brakke flow with boundary $\Gamma$ in $U\times I$
is a collection $\mathbf{M}=\\{M_{t}\colon t\in I\\}\subset\mathcal{M}(U)$
such that the following hold true:

  1. 1.

for almost every $t$, $M_{t}\in\mathcal{V}_{m}(U,\Gamma)$;

  2. 2.

if $I^{\prime}\subset\subset I$ and $U^{\prime}\subset\subset U$, then
$\int_{I^{\prime}}\int_{U^{\prime}}(1+|H_{M_{t}}|^{2})\,dM_{t}\,dt<+\infty$,

  3. 3.

if $[a,b]\subset I$ and $u$ is a non-negative, compactly supported, $C^{1}$
function on $U\times I$, then

$\int u(\cdot,a)\,dM_{a}-\int
u(\cdot,b)\,dM_{b}\geq\int_{a}^{b}\int\left(u|H_{M_{t}}|^{2}-H_{M_{t}}\cdot\nabla
u-\partial_{t}u\right)\,dM_{t}\,dt.$ (2.3)

We denote by $\mathcal{BF}_{m}(U\times I,\Gamma)$ the set of all
$m$-dimensional integral Brakke flows in $U\times I$ with boundary $\Gamma$.

When $\Gamma=\varnothing$, we drop its indication and simply write
$\mathcal{BF}_{m}(U\times I)$; notice that in this case $\beta_{M_{t}}=0$ for
a.e. $t$, and the definition agrees with the one of integral Brakke flow
(without boundary) given, for instance, in [15].

Given $\mathbf{M}\in\mathcal{BF}_{m}(U\times I,\Gamma)$, we define its space-
time mass measure $M$ by

$\int\varphi(x,t)\,dM(x,t)=\int\int\varphi(x,t)\,dM_{t}\,dt$

for every $\varphi\in C_{c}(U\times I)$. We define the space-time track of
$\mathbf{M}$ to be the closed set

$\Sigma_{\mathbf{M}}=\operatorname{Clos}\bigg{(}\bigcup_{t\in
I}\operatorname{supp}M_{t}\times\\{t\\}\bigg{)}$

and we let $\Sigma_{\mathbf{M}}(t)$ be the slice at time $t$ of
$\Sigma_{\mathbf{M}}$, namely
$\Sigma_{\mathbf{M}}(t)=\\{x\in\mathbb{R}^{n}\colon(x,t)\in\Sigma_{\mathbf{M}}\\}$.
It is straightforward to check that
$\operatorname{supp}M\subset\Sigma_{\mathbf{M}}$. Under reasonable
assumptions, the opposite inclusion holds true as well: we further discuss
this point in Lemma 3.5. Whenever no confusion may arise, we write $\Sigma$
and $\Sigma_{t}$ in place of $\Sigma_{\mathbf{M}}$ and
$\Sigma_{\mathbf{M}}(t)$, respectively.

######  Remark 2.4 (Scaling properties).

A Brakke flow $\mathbf{M}\in\mathcal{BF}_{m}(U\times I,\Gamma)$ may be
translated and parabolically dilated while preserving the requirements in
Definition 2.3. For $x_{0}\in\mathbb{R}^{n}$ and $r>0$, let
$T_{x_{0},r}(y)=(y-x_{0})/r$. By $(T_{x_{0},r})_{\sharp}\mu$ we denote the
push-forward of $\mu\in\mathcal{M}(\mathbb{R}^{n})$ through $T_{x_{0},r}$.
Then $\mathbf{M}^{\prime}=\\{M^{\prime}_{s}\\}$ given by

$M_{s}^{\prime}=r^{-m}(T_{x_{0},r})_{\sharp}M_{t_{0}+r^{2}s}$

is a Brakke flow in $\frac{U-x_{0}}{r}\times\frac{I-t_{0}}{r^{2}}$ with
boundary $\frac{1}{r}(\Gamma-x_{0})$. In this case, we will write

$\mathbf{M}^{\prime}=\mathcal{D}_{r}(\mathbf{M}-X_{0})$

where, as usual, $X_{0}=(x_{0},t_{0})$.

##  3 Properties of Integral Brakke flows with boundary

We collect some known results about Integral Brakke flows, which we will use
throughout the rest of the paper.

###  3.1 Monotonicity properties

We denote by $\Psi:{\mathbb{R}}^{d}\times(-\infty,0)\to{\mathbb{R}}$ the
$m$-dimensional backward heat kernel

$\Psi(x,t)=\frac{1}{(4\pi(-t))^{m/2}}\exp\bigg{(}-\frac{|x|^{2}}{4(-t)}\bigg{)}.$

We also pick a smooth cut-off function $\phi\in C^{\infty}_{c}([0,2))$ such
that $\phi\equiv 1$ in $[0,1]$, $|\phi^{\prime}|\leq 2$ and $0\leq\phi\leq 1$
everywhere, which from now on we consider fixed. $\phi$ being chosen, for
$R>0$ we set

$\Psi_{R}(x,t)=\Psi(x,t)\phi\bigg{(}\frac{|x|}{R}\bigg{)}.$

######  Proposition 3.1 (Huisken monotonicity formula).

There exists a universal constant $C>0$ such that, if
$\mathbf{M}\in\mathcal{BF}_{m}(U\times(-T,0),\Gamma)$ and $B_{2R}\subset U$,
then for every $-T<s\leq t<0$ it holds

$\displaystyle\int\Psi_{R}(x,t)\,dM_{t}-\int\Psi_{R}(x,s)\,dM_{s}$
$\displaystyle\qquad\leq\int_{s}^{t}\int\nu_{M_{\tau}}\cdot\nabla\Psi_{R}(\cdot,\tau)\,d\Gamma\,d\tau$
(3.1)
$\displaystyle\qquad\quad+C\frac{t-s}{R^{2}}\sup_{\tau\in[s,t]}\frac{M_{\tau}(B_{2R})}{R^{m}},$
(3.2)

where $\nu_{M_{\tau}}$ is defined in (2.2).

Proof. See [20, Theorem 6.1]. ∎

In several points of the present work, we are going to need some precise
bounds on (3.1) and (3.2). While in most cases we will assume a uniform bound
of the form

$\sup_{t}\sup_{B_{r}(x)}\frac{M_{t}(B_{r}(x))}{r^{m}}\leq E_{1}<\infty$

which takes care of (3.2), estimating (3.1) requires some more attention. What
we prove in the following lemma is that, at a small enough scale, (3.1) is
close to $\frac{1}{2}$ if $0\notin\Gamma$, otherwise it is very small.

######  Lemma 3.2.

For every $\delta>0$, there exist small positive constants $\Lambda$ and $c$
with the following property. Let $U\subset{\mathbb{R}}^{d}$ be open and let
$\Gamma$ be a $C^{1,\alpha}$ submanifold of $U$. Then, for every $R\leq
c/[\Gamma]_{C^{1,\alpha}(U)}$ and for every $(x,t)\in U\times{\mathbb{R}}$
such that $B_{2R}(x)\subset U$, it holds

$\int_{t-\Lambda
R^{2}}^{t}\int|T_{y}\Gamma^{\perp}\nabla\psi_{R}(y-x,s-t)|\,d\Gamma(y)\,ds\leq\frac{1}{2}\chi_{\Gamma^{c}}(x)+\delta.$

The proof of Lemma 3.2 is somehow cumbersome and is therefore postponed to
Appendix A.

Exploiting the above result, we may prove a sort of clearing-out lemma, in the
spirit of [15, Proposition 3.6]. Namely, we prove that, provided we have some
control on (3.2) and (3.1), if a point $(x,t)$ is in the space-time track of
$\mathbf{M}$, then $M_{s}$ cannot be too small in a backward neighborhood of
$(x,t)$.

Before proceeding with this result, we introduce the following terminology:

######  Definition 3.3 (Maximal density ratio).

A Brakke flow $\mathbf{M}$ (possibly with boundary) in $U\times I$ is said to
have _bounded maximal density ratio_ in $U^{\prime}\times I^{\prime}$, where
$U^{\prime}\subset U$ and $I^{\prime}\subset I$, if

$\operatorname{mdr}(\mathbf{M},U^{\prime}\times
I^{\prime}):=\sup_{B_{r}(x)\subset U^{\prime}}\sup_{t\in
I^{\prime}}\frac{M_{t}(B_{r}(x))}{r^{m}}<\infty.$

######  Proposition 3.4 (Clearing-out lemma).

For every $K<\infty$ there exist positive constants $c_{1},c_{2}$ with the
following property. Let $\Gamma$ be a $C^{1,\alpha}$ submanifold of $U$ and
let $\mathbf{M}\in\mathcal{BF}_{m}(U\times(a,b),\Gamma)$ be such that

$\operatorname{mdr}(\mathbf{M},U\times(a,b))\leq K.$

If $(x,t)\in\Sigma_{\mathbf{M}}$, and $R$ is small enough depending on
$\Gamma$, then

$M_{t-c_{1}R^{2}}(B_{4R}(x))\geq c_{2}R^{m}.$

Proof. The proof of the case without a boundary can be found, for example, in
[15, Proposition 3.6]. For the sake of completeness, we sketch the proof along
the same line in the case of an Integral Brakke flow with boundary.

Corresponding to $\delta=1/4$, choose $\Lambda$ and $c$ as in Lemma 3.2. Let
$(x,t)\in\Sigma_{t}$ and let $R\leq c/[\Gamma]_{C^{1,\alpha}(U)}$.

We first assume that $x\in\operatorname{supp}M_{t}$ and that
$M_{t}\in\mathcal{V}_{m}(U,\Gamma)$, so that in particular
$M_{t}=\theta(\cdot)\mathcal{H}^{m}\llcorner E$ for some $m$-rectifiable set
$E$. Then there exists $y\in B_{R}(x)$ such that

$1\leq\theta(y)=\lim_{\tau\nearrow 0}M_{t}(\Psi_{R}(\cdot-y,\tau)).$ (3.3)

Therefore, by Proposition 3.1, for any $t_{1}<t$, it holds

$M_{t_{1}}(\Psi_{R}(\cdot-y,t_{1}-t))\geq\theta(y)-CK\frac{t-t_{1}}{R^{2}}-\int_{t_{1}}^{t}\int\nu_{M_{s}}\cdot\Psi_{R}(\cdot-y,s-t)\,d\Gamma\,ds.$

We now choose $c_{1}$ so small that both $CKc_{1}\leq\frac{1}{8}$ and
$c_{1}\leq\Lambda$ and we set $t_{1}=t-c_{1}R^{2}$. Then, using Lemma 3.2, we
obtain

$M_{t_{1}}(\Psi_{R}(\cdot-y,-c_{1}R^{2}))\geq\theta(y)-\frac{1}{8}-\bigg{(}\frac{1}{2}+\frac{1}{4}\bigg{)}\geq\frac{1}{8},$

where the second inequality is given by (3.3). Notice that, for every
$z\in{\mathbb{R}}^{d}$, simple computations yield

$\Psi_{R}(z-y,-c_{1}R^{2})\leq CR^{-m}\chi_{B_{2R}(y)}(z)\leq
CR^{-m}\chi_{B_{3R}(x)}(z)$

for some $C>0$ universal. Hence, by integrating the above inequality in
$M_{t-c_{1}R^{2}}$, we obtain

$M_{t-c_{1}R^{2}}(B_{3R}(x))\geq\frac{R^{m}}{C}M_{t-c_{1}R^{2}}(\Psi_{R}(\cdot-y,-c_{1}R^{2}))\geq\frac{R^{m}}{8C},$

as desired.

If $x\notin\operatorname{supp}M_{t}$ or
$M_{t}\notin\mathcal{V}_{m}(U,\Gamma)$, then one can find a sequence of points
$(x_{i},t_{i})$ such that $M_{t_{i}}\in\mathcal{V}_{m}(U,\Gamma)$,
$x_{i}\in\operatorname{supp}M_{t_{i}}$ and such that $(x_{i},t_{i})\to(x,t)$.
It is then sufficient to choose $R_{i}$ so that
$t_{i}-c_{1}R_{i}^{2}=t-c_{1}R^{2}$ to obtain, for $i$ large enough,

$M_{t-c_{1}R^{2}}(B_{4R}(x))\geq M_{t-c_{1}R^{2}}(B_{3R}(x_{i}))\geq
c_{2}R^{m}.$

∎

We now state two important consequences of Proposition 3.4.

######  Lemma 3.5.

Let $\mathbf{M}\in\mathcal{BF}_{m}(U\times I,\Gamma)$ have bounded maximal
density ratio in $U\times I$ and let $\Gamma\in C^{1,\alpha}(U)$. Then

$\Sigma_{\mathbf{M}}=\operatorname{supp}M.$

Proof. The inclusion $\operatorname{supp}M\subset\Sigma$ is trivial. For the
opposite inclusion, notice that, for a point $(x,t)\in\Sigma$ and for every
$r>0$ small enough, Proposition 3.4 gives

$M_{t-cr^{2}}(B_{r}(x))\geq cr^{m}.$

It is now sufficient to integrate this inequality in $r$ to obtain that for
every $r>0$ small enough, there is a set of the form

$A_{r}=\big{\\{}(y,s)\colon|y-x|\leq\theta\sqrt{t-s}\leq r\big{\\}}$

for some positive $\theta,c$ depending only on
$\operatorname{mdr}(\mathbf{M})$ such that $M(A_{r})>0$, hence
$(x,t)\in\operatorname{supp}M$, as claimed. ∎

######  Lemma 3.6.

Let $\mathbf{M}\in\mathcal{BF}_{m}(U\times I,\Gamma)$ have bounded maximal
density ratio and let $\Gamma\in C^{1,\alpha}(U)$. Then

$M\geq\mathcal{H}^{m,1}\llcorner\Sigma_{\mathbf{M}}.$

For the proof of the above lemma, we refer the reader to Appendix B.

###  3.2 Maximum principle

In the present subsection, we assume that $U\subset\mathbb{R}^{d}$ is open and
$I\subset\mathbb{R}$ is an interval of the form $(a,b]$. We also let $\Gamma$
be a $(m-1)$-dimensional, $C^{1,\alpha}$ submanifold of $U$.

The main result of the present section is the following maximum principle.

######  Proposition 3.7 (Maximum principle).

Let $\mathbf{M}\in\mathcal{BF}_{m}(U\times I,\Gamma)$ have bounded maximal
density ratio.

If there exist $u\in C^{2}(U\times I)$ and a point
$(x_{0},t_{0})\in\Sigma\setminus\partial_{p}(U\times I)$ with
$x_{0}\notin\Gamma$ such that $u|_{\Sigma\cap\\{t\leq t_{0}\\}}$ has a local
maximum at $(x_{0},t_{0})$ and $\nabla u(x_{0},t_{0})\neq 0$, then

$\partial_{t}u(x_{0},t_{0})-\inf_{\begin{subarray}{c}T\in\mathrm{Gr}(m,d)\\\
T\perp\nabla u(x_{0},t_{0})\end{subarray}}T\colon D^{2}u(x_{0},t_{0})\geq 0.$

Proof.  This proposition is a corollary of the results in [8, Section 13], see
also [3]; for the reader’s convenience, we give a self-contained proof, in the
spirit of, for example, [19].

We may assume, without loss of generality, that $(x_{0},t_{0})=(0,0)$ and that
$u|_{\Sigma\cap\\{t\leq 0\\}}$ has a strict local maximum at $(0,0)$ (if not,
replace $u$ by $u-|x|^{4}-|t|^{2}$).

Step 1. We first prove that

$\partial_{t}u(0,0)-\operatorname{trace}_{m}D^{2}u(0,0)\geq 0,$

where $\operatorname{trace}_{m}D^{2}u$ is the sum of the $m$ smallest
eigenvalues of $D^{2}u$. Assume the result does not hold. In particular, since
$u\in C^{2}$, we can choose $\rho>0$ and $\varepsilon>0$ small so that:

  1. (i)

$\partial_{t}u-\operatorname{trace}_{m}D^{2}u<-\varepsilon<0$ in $Q_{\rho}$;

  2. (ii)

$B_{\rho}\cap\Gamma=\varnothing$;

  3. (iii)

$u>\varepsilon>0$ in $\Sigma\cap Q_{\rho/2}$ and $u<0$ in $\Sigma\cap\\{t\leq
0\\}\setminus Q_{\rho}$;

where the last point holds up to adding a constant to $u$. We now let
$\varphi(x,t)=(u^{+}(x,t))^{4}$, where $u^{+}=\max\\{u,0\\}$, and we use
$\varphi$ as a test function for (2.3). Since $\varphi(\cdot,-\rho^{2})=0$ by
assumption, we have

$\displaystyle 0$ $\displaystyle\leq\int\varphi(\cdot,0)\,dM_{0}$
$\displaystyle=\int\varphi(\cdot,0)\,dM_{0}-\int\varphi(\cdot,-\rho^{2})\,dM_{-\rho^{2}}$
$\displaystyle\leq\int_{-\rho^{2}}^{0}\int\bigg{(}-|H|^{2}\varphi+H\cdot\nabla\varphi+\partial_{t}\varphi\bigg{)}\,dM_{t}\,dt$

where the last inequality is given by (2.3) and we have set
$H(\cdot,t)=H_{M_{t}}(\cdot)$ for a.e. $t$. We now use the fact that
$\operatorname{supp}\varphi\subset\Gamma^{c}$, thus

$\int
H\cdot\nabla\varphi\,dM_{t}=-\int\operatorname{div}_{M_{t}}\nabla\varphi\,dM_{t}$

for a.e. $t$. Since the term $|H|^{2}\varphi$ is non-negative, we obtain from
the above chain of inequalities:

$0\leq\int_{-\rho^{2}}^{0}\int\bigg{(}-\operatorname{div}_{M_{t}}\nabla\varphi+\partial_{t}\varphi\bigg{)}\,dM_{t}\,dt.$

Some straightforward computations show that

$\operatorname{div}_{M_{t}}\nabla\varphi=4(u^{+})^{3}\operatorname{div}_{M_{t}}\nabla
u\geq 4(u^{+})^{3}\operatorname{trace}_{m}D^{2}u$

and $\partial_{t}\varphi=4(u^{+})^{3}\partial_{t}u$. Therefore

$\displaystyle 0\leq\int_{-\rho^{2}}^{0}\int
4(u^{+})^{3}\big{(}\partial_{t}u-\operatorname{trace}_{m}D^{2}u\big{)}\,dM_{t}\,dt\leq-4\varepsilon^{4}M(Q_{\rho/2})$

where the last inequality is given by Item i and Item iii above. In
particular, it must be

$M(Q_{\rho/2})=0;$

however, by Lemma 3.5, $(0,0)\in\Sigma=\operatorname{supp}M$, thus we reach a
contradiction.

Step 2. We now prove the general result. It is sufficient to show that one can
find a $m$-dimensional subspace $\bar{T}$ of ${\mathbb{R}}^{d}$ such that

$\partial_{t}u(0,0)-\bar{T}\colon D^{2}u(0,0)\geq 0$

and $\bar{T}\nabla u(0,0)=0$. Without loss of generality, assume that
$u(0,0)=0$. Let $\psi_{j}(z)=z+\frac{j}{2}z^{2}$ and let

$u_{j}(X)=\psi_{j}(u(X)).$

Then, for every $j$, $u_{j}|_{\Sigma\cap\\{t\leq 0\\}}$ has a local maximum at
$(0,0)$. Therefore, by Step 1, there is a $m$-dimensional subspace $T_{j}$ of
${\mathbb{R}}^{d}$ such that, at $(0,0)$,

$0\leq\partial_{t}u_{j}-T_{j}\colon D^{2}u_{j}=(\partial_{t}u-T_{j}\colon
D^{2}u)+jT_{j}\colon(\nabla u\otimes\nabla u).$

Up to a subsequence, which we do not relabel, we have that $T_{j}\to\bar{T}$
for some $m$-dimensional subspace $\bar{T}$, and

$\bar{T}\colon(\nabla u\otimes\nabla
u)\leq\liminf_{j}\frac{1}{j}(\partial_{t}u-T_{j}\colon D^{2}u)=0,$

thus $\bar{T}\perp\nabla u$. On the other hand, since $jT_{j}\colon(\nabla
u\otimes\nabla u)\geq 0$, we have

$\bar{T}\colon D^{2}u\leq\liminf_{j}T_{j}\colon
D^{2}u\leq\liminf_{j}(\partial_{t}u-jT_{j}\colon(\nabla u\otimes\nabla
u))\leq\partial_{t}u,$

as desired. ∎

Given a upper-semicontinuous function
$u:{\mathbb{R}}^{m,1}\to[0,1]\cup\\{-\infty\\}$ and a smooth function
$\varphi:{\mathbb{R}}^{m,1}\to{\mathbb{R}}$, we say that $\varphi$ touches $u$
from above at $(x_{0}^{\prime},t_{0})\in{\mathbb{R}}^{m,1}$ if there exists
$r>0$ such that

$\begin{cases}\varphi(x^{\prime},t)\geq u(x^{\prime},t)\quad\mbox{for every
}(x^{\prime},t)\in Q_{r}^{m}(x_{0}^{\prime},t_{0}),\\\
\varphi(x_{0}^{\prime},t_{0})=u(x^{\prime}_{0},t_{0}).\end{cases}$

We recall the definition of Pucci’s maximal operator (see, for instance, [6,
Section 2.2]). For a symmetric matrix $N\in{\mathbb{R}}^{d\times d}$, we let

$\mathcal{M}^{+}(N):=\mathcal{M}^{+}\Big{(}N,\frac{1}{2},2\Big{)}=\frac{1}{2}\sum_{\lambda_{i}<0}\lambda_{i}+2\sum_{\lambda_{i}>0}\lambda_{i},$
(3.4)

where $\lambda_{i}=\lambda_{i}(N)$ are the eigenvalues of $N$. The following
result is a consequence of Proposition 3.7.

######  Corollary 3.8.

Let $\mathbf{M}\in\mathcal{BF}_{m}({\mathbb{R}}^{d,1})$ have bounded maximal
density ratio. For every $(x^{\prime},t)\in{\mathbb{R}}^{m,1}$, let

$u(x^{\prime},t)=\sup\\{|z|\colon z\in{\mathbb{R}}^{d-m}\mbox{ and
}(x^{\prime},z)\in\Sigma_{\mathbf{M}}(t)\\}$

with the convention $\sup\varnothing=-\infty$ and assume that $u\leq 1$
everywhere. There is $\delta>0$ universal such that, whenever a smooth
function $\varphi:{\mathbb{R}}^{m,1}\to{\mathbb{R}}$ touches $u$ from above at
$X_{0}^{\prime}=(x_{0}^{\prime},t_{0})$ and
$\max\\{|D^{2}\varphi(X_{0}^{\prime})|,|\nabla\varphi(X_{0}^{\prime})|\\}\leq\delta$,
it holds

$\partial_{t}\varphi(X_{0}^{\prime})-\mathcal{M}^{+}(D^{2}\varphi(X_{0}^{\prime}))\leq
0.$

Proof. We assume $x_{0}^{\prime}=0$ and $t_{0}=0$. Notice that, since $\Sigma$
is closed and $u(0,0)=\varphi(0,0)$, the supremum in the definition of $u$ is
attained and, without loss of generality, we may assume that the contact point
is $x_{0}=\varphi(0,0)\mathbf{e}_{d}\in\Sigma_{0}$. We let
$S=\operatorname{span}\\{\mathbf{e}_{1},\dots,\mathbf{e}_{m}\\}$ and
$S^{\prime}=\operatorname{span}\\{\mathbf{e}_{m+1},\dots,\mathbf{e}_{d-1}\\}$,
so that
${\mathbb{R}}^{d}=S+S^{\prime}+\operatorname{span}\\{\mathbf{e}_{d}\\}$.
Consider the function

$H(x,t)=\frac{1}{4}\big{|}S^{\prime}x\big{|}^{2}+x\cdot\mathbf{e}_{d}-\varphi(Sx,t).$

By assumption, in a neighborhood of $(x_{0},0)$ it holds

$|S^{\perp}x|\leq\varphi(Sx,t)\leq 2$

for every $(x,t)\in\Sigma$; therefore it can be checked that
$H|_{\Sigma\cap\\{t\leq 0\\}}\leq 0$ in the same neighborhood. Since
$H(x_{0},0)=0$, $H|_{\Sigma\cap\\{t\leq 0\\}}$ has a local maximum at
$(x_{0},0)$. Hence, by Proposition 3.7, it holds

$\partial_{t}H(x_{0},0)-\inf_{T\perp\nabla H(x_{0},0)}T\colon
D^{2}H(x_{0},0)\geq 0.$ (3.5)

We now estimate the two summands in the above inequality. In order to do so,
we first remark that

$\nabla H(x_{0},0)=\begin{pmatrix}-\nabla\varphi(0,0)\\\ 0\\\
1\end{pmatrix}\qquad D^{2}H(x_{0},0)=\begin{pmatrix}-D^{2}\varphi(0,0)&0&0\\\
0&\frac{1}{2}I_{S^{\prime}}&0\\\ 0&0&0\end{pmatrix}.$

Consider $T\in\mathrm{Gr}(m,d)$ and an orthonormal basis
$\zeta_{1},\dots,\zeta_{m}$ of $T$. Then

$\displaystyle T\colon D^{2}H$ $\displaystyle=\sum_{i=1}^{m}\left\langle
D^{2}H\zeta_{i};\zeta_{i}\right\rangle$
$\displaystyle=\sum_{i=1}^{m}\bigg{(}-\left\langle
D^{2}\varphi(Sx,t)S\zeta_{i};S\zeta_{i}\right\rangle+\frac{1}{2}\big{|}S^{\prime}\zeta_{i}\big{|}^{2}\bigg{)}$
$\displaystyle\geq-\sum_{i=1}^{m}\left\langle
D^{2}\varphi(Sx,t)S\zeta_{i};S\zeta_{i}\right\rangle.$

In particular, $S\colon D^{2}H(x,t)=-\Delta\varphi(Sx,t)$ and

$\displaystyle T\colon D^{2}H$ $\displaystyle=S\colon D^{2}H+(T-S)\colon
D^{2}H$ $\displaystyle\geq-\Delta\varphi-|T-S||D^{2}\varphi|,$

Now, if $|T-S|\leq c_{1}$, then the above inequality yields that, for some
small $c_{1}$ universal,

$T\colon D^{2}H(x_{0},0)\geq-\mathcal{M}^{+}(D^{2}\varphi(0,0)).$

On the other hand, if $|T-S|\geq c_{1}$, then we may choose an orthonormal
basis $\zeta_{1},\dots,\zeta_{m}$ of $T$ such that $|S^{\perp}\zeta_{1}|\geq
c_{2}$ for some $c_{2}$ universal. Since we are also assuming $T\perp\nabla
H(x_{0},0)$, we have

$0=\zeta_{1}\cdot\nabla
H(x_{0},0)=-S\zeta_{1}\cdot\nabla\varphi(0,0)+\zeta_{1}\cdot\mathbf{e}_{d}$

thus, in particular,
$|\zeta_{1}\cdot\mathbf{e}_{d}|\leq|\nabla\varphi|\leq\delta$ and

$|S^{\prime}\zeta_{1}|\geq|S^{\perp}\zeta_{1}|-|\zeta_{1}\cdot\mathbf{e}_{d}|\geq
c_{2}-\delta\geq\frac{c_{2}}{2},$

provided $\delta\leq c_{2}/2$. Therefore

$\displaystyle T\colon D^{2}H(x_{0},0)\geq-\sum_{i=1}^{m}\left\langle
D^{2}\varphi(0,0)\zeta_{i};\zeta_{i}\right\rangle+\frac{1}{2}|S^{\prime}\zeta_{1}|^{2}\geq-\Delta\varphi(0,0)+\frac{c_{2}^{2}}{8}\geq-C\delta+\frac{c_{2}^{2}}{8}$

for some $C$ universal, since $|D^{2}\varphi(0,0)|\leq\delta$ by assumption.
We may choose $\delta$ smaller, if needed, so that

$-\mathcal{M}^{+}(D^{2}\varphi(0,0))\leq
C\delta\leq-C\delta+\frac{c_{2}^{2}}{8}.$

Therefore, whether $|T-S|\leq c_{1}$ or not, it holds

$T\colon D^{2}H(x_{0},0)\geq-\mathcal{M}^{+}(D^{2}\varphi(0,0)).$

We conclude the proof by remarking that

$\partial_{t}H(x_{0},0)=-\partial_{t}\varphi(0,0),$

thus (3.5) gives the desired result.

∎

######  Remark 3.9.

With some more accurate computations, one may show that, actually, at the
contact point $\varphi$ satisfies the following inequality:

$\partial_{t}\varphi-\sqrt{1+|\nabla\varphi|^{2}}\operatorname{div}\bigg{(}\frac{\nabla\varphi}{\sqrt{1+|\nabla\varphi|^{2}}}\bigg{)}\leq
0.$

However, the weaker result proved in Corollary 3.8 will be sufficient for the
rest of the paper.

###  3.3 Constancy theorem

######  Theorem 3.10 (Constancy theorem).

Let $\Gamma$ be a (possibly empty) $(m-1)$-dimensional properly embedded
submanifold of $B_{1}$ and let $\mathbf{M}\in\mathcal{BF}_{m}(Q_{1},\Gamma)$
have bounded maximal density ratio in $Q_{1}$. Assume that there is a
$C^{1,\beta}$-regular single sheet flow $N$ with boundary $\Gamma$ in $Q_{1}$
such that

$\Sigma_{\mathbf{M}}\subset N$

and that $(0,0)\in\Sigma_{\mathbf{M}}$.

  1. 1.

If $\Gamma\neq\varnothing$, then

$M=\mathcal{H}^{m,1}\llcorner N;$

  2. 2.

if $\Gamma=\varnothing$, then there is
$\theta:(-1,0]\to{\mathbb{N}}\setminus\\{0\\}$ such that

$M_{t}=\theta(t)\mathcal{H}^{m}\llcorner N_{t}$

for almost every $t\in(-1,0]$.

Proof. First of all, given $N$ as in the statement, let us fix the ball
$B_{0}:=B_{r}(x^{\prime})\subset{\mathbb{R}}^{m}_{+}$ and the family of
homeomorphisms $\\{f_{t}\\}_{t\in(1,0]}$ as in Definition 2.1.

Step 1. First, we prove that $\Sigma=N$. Assume, by contradiction, that there
is $X_{0}=(x_{0},t_{0})\in N\cap\Sigma^{c}$. By Proposition 3.4, there is also
a point $X_{1}=(x_{1},t_{1})\in\Sigma\setminus\Gamma$ so close to $(0,0)$ that
$t_{0}<t_{1}<0$. Since $(y,t)\mapsto f_{t}(y)$ is $C^{1,\beta}$, there exist
$r>0$ and a continuous curve $[t_{0},t_{1}]\ni t\mapsto
p(t)\in{\mathbb{R}}^{d}$ such that

$\begin{cases}p(t)\in N_{t}\mbox{ for every }t;\\\ p(t_{0})=x_{0}\mbox{ and
}p(t_{1})=x_{1};\\\ B_{r}(p(t))\subset B_{1}\setminus\Gamma\mbox{ and
}B_{r}(x_{0})\subset\Sigma_{t_{0}}^{c}.\end{cases}$

To prove that such a curve and radius exists, take an affine curve
$q:[t_{0},t_{1}]\to B_{0}$ such that $q(t_{0})=f_{t_{0}}^{-1}(x_{0})$ and
$q(t_{1})=f_{t_{1}}^{-1}(x_{1})$. Then set $p(t)=f_{t}(q(t))$.

Since $X_{1}\in\Sigma$, the continuity of $t\mapsto p(t)$ and Proposition 3.4
imply that there must be a time $\bar{t}$ such that
$B_{r}(p(t))\subset\Sigma_{t}^{c}$ for every $t\leq\bar{t}$ and a point
$\bar{x}\in\partial B_{r}(p(\bar{t}))\cap\Sigma_{\bar{t}}$.

Let us now consider a sequence $r_{j}\searrow 0$ and define the dilations

$\mathbf{M}^{j}=\mathcal{D}_{r_{j}}(\mathbf{M}-(\bar{x},\bar{t})).$

Since $\mathbf{M}$ has bounded maximal density ratio, the compactness theorems
in [20, Section 10] yield that, up to passing to a subsequence,
$\mathbf{M}^{j}$ converges to a limit Brakke flow $\mathbf{M}^{\infty}$. Since
$\Sigma_{\mathbf{M}}\subset N$, we also have
$\Sigma_{\mathbf{M}^{\infty}}\subset T:=T_{\bar{x}}N_{\bar{t}}$. We claim,
that, actually, $\Sigma_{\mathbf{M}^{\infty}}$ is included in the half-plane
$T^{+}=T_{\bar{x}}(N_{\bar{t}}\setminus B_{r}(p(\bar{t})))\subset T$. Indeed,
if that were not the case, then there would be a point $(z,\tau)\in(T\setminus
T^{+})\times(-\infty,0]$ and $\rho>0$ such that
$M^{\infty}_{\tau}(B_{\rho}(z))>0$. By weak convergence, this yields

$\liminf_{j}\frac{M_{\bar{t}+r_{j}^{2}\tau}(B_{\rho
r_{j}}(\bar{x}+r_{j}z))}{r_{j}^{m}}>0.$

This is, however, impossible, since by continuity of $t\mapsto p(t)$, we have

$B_{\rho r_{j}}(\bar{x}+r_{j}z)\subset
B_{r}(p(\bar{t}))\subset(\Sigma_{\bar{t}+r_{j}^{2}\tau})^{c}$

for $j$ large enough. Therefore $\Sigma_{\mathbf{M}^{\infty}}\subset T^{+}$.
We finally show that this violates the maximum principle. First of all, notice
the fact that $\mathbf{M}$ has bounded maximal density ratio implies that
$(0,0)\in\Sigma_{\mathbf{M}^{\infty}}$. Without loss of generality, say
$T^{+}=\\{x_{m+1}=\dots=x_{d}=0\mbox{ and }x_{m}>0\\}$ and let

$f(x,t)=A|T^{\perp}x|^{2}-|T(x+(1-t)\mathbf{e}_{m})|$

for some $A>0$. Then $f|_{\Sigma_{\mathbf{M}^{\infty}}}$ has a local maximum
at $(0,0)$, but the conclusion of Proposition 3.7 is violated if $A$ is chosen
large enough, depending on $m$. This concludes Step 1.

Step 2. By Step 1, Proposition 3.7 and since $N$ is $C^{1,\beta}$,
$\\{N_{t}\\}_{t\in(-1,0]}$ is a viscosity solution to the mean curvature flow.
In particular, classical Schauder estimates imply that, for every
$t\in(-1,0]$, $N_{t}\setminus\Gamma$ is smooth. Now, by Definition 2.3, for
almost every $t$, there is an integer-valued function $\theta_{t}\in
L^{1}_{loc}(N_{t}\setminus\Gamma)$ so that

$M_{t}=\theta_{t}(\cdot)\mathcal{H}^{m}\llcorner N_{t}.$

By testing (2.1) with vector fields $X\in
C^{1}_{c}(B_{1}\setminus\Gamma;{\mathbb{R}}^{d})$ that are tangent to $N_{t}$,
one deduces that for almost every $t$, $\theta_{t}(\cdot)$ is an integer-
valued $W^{1,1}_{loc}$ function on $N_{t}\setminus\Gamma$. Since $N_{t}$ is
connected, $\theta_{t}(\cdot)$ must be constant for almost every $t$.
Moreover, $\theta_{t}\neq 0$ for almost every $t$, otherwise Step 1 would
fail.

This concludes the proof in the case $\Gamma=\varnothing$. Otherwise, if
$\Gamma\neq\varnothing$, then it holds
$\beta_{M_{t}}=\theta_{t}\mathcal{H}^{m-1}\llcorner\Gamma$, which yields
$\theta_{t}=1$ for almost every $t$. ∎

##  4 Improvement of flatness

This section is the core of the present work. We prove that if a Brakke flow
with boundary is sufficiently flat in $Q_{1}$, then its flatness can be
improved at a smaller universal scale. This is going to allow us to prove the
desired $C^{1,\beta}$ regularity: see Section 7.

We introduce the following notation. We fix a $m$-dimensional subspace of
${\mathbb{R}}^{d}$, which we denote by $S$, and a $(m-1)$-dimensional subspace
of ${\mathbb{R}}^{d}$, which we denote by $\Gamma_{0}$, such that
$\Gamma_{0}\subset S$. Up to changing coordinates in ${\mathbb{R}}^{d}$, we
shall assume for the rest of the present section that
$S=\operatorname{span}\\{\mathbf{e}_{1},\dots,\mathbf{e}_{m}\\}$ and that
$\Gamma_{0}=\operatorname{span}\\{\mathbf{e}_{1},\dots,\mathbf{e}_{m-1}\\}$.
We also let $S^{+}=S\cap\\{x_{m}>0\\}$.

Given a $(m-1)$-dimensional submanifold $\Gamma$ of $B_{1}$, we write
$\Gamma\in\mathcal{F}_{\alpha}(\delta,B_{R})$ if the following hold true:

  * •

$\Gamma$ is a $C^{1,\alpha}$ submanifold of $B_{R}$ and
$[\Gamma]_{C^{1,\alpha}(B_{R})}\leq\delta R^{-\alpha}$.

  * •

$0\in\Gamma$ and $T_{0}\Gamma=\Gamma_{0}$.

In passing, we remark that if $\Gamma\in\mathcal{F}_{\alpha}(\delta,B_{R})$
and $\theta>0$, then $\theta\Gamma\in\mathcal{F}_{\alpha}(\delta,B_{\theta
R})$.

Moreover, if $\Gamma\in\mathcal{F}_{\alpha}(\delta,B_{R})$, and $\delta$ is
smaller than some constant depending only on $\alpha$, then there exists
$\gamma:\Gamma_{0}\cap B_{R}\to\Gamma_{0}^{\perp}$ such that
$|\gamma(0)|=|\nabla\gamma(0)|=0$, $||\gamma||_{C^{1,\alpha}(B_{R})}\leq\delta
R^{-\alpha}$ and

$\Gamma=\\{x+\gamma(x)\colon x\in\Gamma_{0}\cap B_{R}\\}\cap B_{R};$

given $\Gamma\in\mathcal{F}_{\alpha}(\delta,B_{R})$, we will always implicitly
define $\gamma$ as above.

The following is the main result of the present section.

######  Theorem 4.1 (Improvement of flatness).

For every $E_{0}$ and $\alpha$, there exist constants
$\Lambda,\varepsilon_{0},\eta,\beta$ (small) and $C$ (large) with the
following property. Let $\varepsilon\leq\varepsilon_{0}$,
$\Gamma\in\mathcal{F}_{\alpha}(\varepsilon,B_{1})$ and
$\mathbf{M}\in\mathcal{BF}_{m}(B_{1}\times[-\Lambda,0],\Gamma)$ be such that
$(0,0)\in\Sigma_{\mathbf{M}}$,

$\Sigma_{\mathbf{M}}\subset\\{(z,\tau)\colon\operatorname{dist}(z,S^{+})\leq\varepsilon\\},$
$\sup_{t\in[-\Lambda,0]}M_{t}(B_{1})\leq E_{0}$

and

$\int_{B_{1}}\Psi(\cdot,-\Lambda)\,dM_{-\Lambda}\leq\frac{3}{4}.$ (4.1)

Then there exists a half plane $T^{+}$ of the form

$T^{+}=\\{x+w\zeta:x\in\Gamma_{0},w>0\\}$ (4.2)

for some $\zeta\in\Gamma_{0}^{\perp}$ with $|\zeta-\mathbf{e}_{m}|\leq
C\varepsilon$, such that

$\Sigma_{\mathbf{M}}\cap
Q_{\eta}\subset\bigg{\\{}(x,t)\colon\operatorname{dist}(x,T^{+})\leq\eta^{1+\beta}\varepsilon\bigg{\\}}.$
(4.3)

The proof of Theorem 4.1 is based on a contradiction and compactness argument.
If one assumes the conclusion does not hold, then it is possible to find a
sequence of Brakke flows which are flatter and flatter and satisfy the other
assumptions of Theorem 4.1, for which, however, no half-plane of the form
(4.2) can be found so that the flatness improves at any smaller scale.
However, for such flows, one shows that, after an appropriate rescaling, the
space-time tracks must converge in the Hausdorff distance to the graph of a
solution to the heat equation. It is then sufficient to use Schauder estimates
for the heat equation with Dirichlet boundary condition to recover the
conclusion.

The central point of the proof is to obtain the desired compactness. This is
achieved via the two following results. The first one provides a control over
the oscillations near $\Gamma$ of the space-time support of a Brakke flow
satisfying the assumptions of Theorem 4.1.

######  Proposition 4.2 (Boundary behaviour).

For every $E_{0}$ and $\alpha$, there exist small constants $c_{1}$ and
$r_{1}$ with the following property. Let $\mathbf{M}$ and $\Gamma$ satisfy the
assumptions of Theorem 4.1. Then

$\Sigma\cap
Q_{r_{1}}\subset\bigg{\\{}(x-\gamma(x^{\prime\prime}))\cdot\mathbf{e}_{m}\geq-\varepsilon^{2}+c_{1}\frac{|S^{\perp}(x-\gamma(x^{\prime\prime}))|^{2}}{2\varepsilon^{2}}\bigg{\\}}.$

Here, $x^{\prime\prime}$ denotes the point
$(x_{1},\dots,x_{m-1},0,\dots,0)\in\Gamma_{0}$.

With the above results at hand, we may prove that, if the Brakke flow is flat
enough, then assumption (4.1) gives a Holder-type modulus of continuity in
parabolic cylinders whose radii are controlled from below by some power of the
flatness $\varepsilon$.

######  Proposition 4.3 (Decay of oscillations).

For every $E_{0}$ and $\alpha$, there exist small constants $\varsigma,C_{2}$
and $r_{2}$ with the following property. Let $\mathbf{M}$ and $\Gamma$ satisfy
the assumptions of Theorem 4.1 and let $(x,t),(y,s)\in\Sigma\cap Q_{r_{2}}$.
If $\min\\{x_{m},y_{m}\\}\geq 2\varepsilon$ and

$\rho:=\rho((x^{\prime},t),(y^{\prime},s))\geq C_{2}\varepsilon^{\varsigma},$

then

$|S^{\perp}(x-y)|\leq C_{2}\varepsilon\rho^{\varsigma}.$

The two above results are sufficient to prove, via an Arzelà-Ascoli-type
argument, the convergence in the Hausdorff distance which we have described.

Before proceeding, it is worth spending a few words on how the constants in
Theorem 4.1 will be chosen.

  * •

We fix $\Lambda$ once and for all in Proposition 4.4; it will be needed to
prove that $\mathbf{M}$ has bounded maximal density ratio in a smaller
parabolic cylinder, $Q_{r_{3}}$.

  * •

Propositions 4.2, 4.3 and 4.4 hold true provided $\varepsilon_{0}$ is small
enough (depending on $E_{0}$). We will therefore always assume that this is
the case. The final value of $\varepsilon_{0}$ will not be determined
explicitly, as Theorem 4.1 is proved by compactness.

  * •

The constants $r_{1}$ and $r_{2}$ chosen in Propositions 4.2 and 4.3 are
chosen smaller than $r_{3}$ (determined in Proposition 4.4) and they depend on
$E_{0}$ and $\alpha$. These two constants will give upper bounds for $\eta$.
We will then give a further upper bound for $\eta$ coming from the regularity
properties of the heat equation.

  * •

Lastly, the constants $C$ and $\beta$ depend only on $\alpha$ and on
regularity properties for the heat equation.

We now briefly describe the rest of the present section. The proof of Theorem
4.1 is given in Subsection 4.2. In Subsection 4.1, we state and prove some
lemmas which will be useful in the following. The proofs of Proposition 4.2
and Proposition 4.3 are postponed to Section 5 and Section 6, respectively.

###  4.1 Preliminaries to the proof of Theorem 4.1

Some remarks on the assumptions of Theorem 4.1 will be needed for the proofs
of Proposition 4.2, Proposition 4.3 and, ultimately, of Theorem 4.1 itself. We
begin by showing that (4.1) propagates in the interior of the domain.

######  Proposition 4.4 (Propagation of small density).

For every $E_{0}$ and $\alpha$, there is $r_{3}$ small with the following
property. Let $\mathbf{M}$ and $\Gamma$ satisfy the assumptions of Theorem
4.1. Then, for every $(x,t)\in Q_{r_{3}}$ and for every
$\tau\in(-r_{3}^{2},0)$, it holds

$\int_{B_{r_{3}}(x)}\Psi(\cdot-x,\tau)\,dM_{t}\leq\frac{7}{8}+\frac{1}{2}\chi_{\Gamma^{c}}(x).$

Proof. We fix positive constants $r_{3}\leq\frac{1}{8}$, $\varepsilon$,
$\Lambda$ and $\delta$, all of which we will determine later; we always assume
that $r_{3}$ is much smaller than $\Lambda$. For simplicity of notation, in
this proof we set $r=r_{3}$. For $(x,t)\in Q_{r}$ and $\tau\in(-r^{2},0)$, we
let $t_{0}=t-\tau$. Then, by Proposition 3.1, it holds

$\displaystyle\int_{B_{r}(x)}\Psi(\cdot-x,t-t_{0})\,dM_{t}\leq\int\Psi_{1/8}(\cdot-x,t-t_{0})\,dM_{t}$
$\displaystyle\qquad\leq\int\Psi_{1/8}(\cdot-x,-\Lambda-t_{0})\,dM_{-\Lambda}$
(4.4)
$\displaystyle\qquad\quad+\int_{-\Lambda}^{t}\int\nu_{M}\cdot\nabla\Psi_{1/8}(\cdot-x,s-t_{0})\,d\Gamma\,d\tau+CE_{0}(t-\Lambda).$

By Lemma 3.2, if $\varepsilon$ and $\Lambda$ are small enough and $r$ is much
smaller than $\Lambda$, then

$\displaystyle\int_{-\Lambda}^{t}\int\nu_{M}\cdot\nabla\Psi_{1/8}(\cdot-x,s-t_{0})\,d\Gamma\,d\tau$
$\displaystyle\qquad\leq\int_{t_{0}-2\Lambda}^{t_{0}}\int|T_{y}\Gamma^{\perp}\nabla\Psi_{1/8}(y-x,s-t_{0})|\,d\Gamma(y)\,d\tau$
$\displaystyle\qquad\leq\frac{1}{2}\chi_{\Gamma^{c}}(x)+\delta.$

We then take $\Lambda$ even smaller so that $CE_{0}(t-\Lambda)\leq
CE_{0}(\Lambda+r^{2})\leq 2CE_{0}\Lambda\leq\delta$ (recall that $r$ is much
smaller than $\Lambda$).

So far, we have fixed $\varepsilon$ and $\Lambda$ depending only on $E_{0}$
and $\delta$, and we have assumed that $r$ is much smaller than $\Lambda$. The
last step is to choose $r$ even smaller in order to bound (4.4) from above. To
this end, we let $L$ be the Lipschitz constant of $\Psi$ restricted to
${\mathbb{R}}^{d}\times(-\infty,-\Lambda/2]$. Since $r$ is much smaller than
$\Lambda$, then $-\Lambda-t_{0}\leq-\Lambda/2$ and we can estimate, for every
$y\in B_{1/4}(x)$,

$\displaystyle\Psi_{1/8}(y-x,-\Lambda-t_{0})$
$\displaystyle\leq\Psi(y-x,-\Lambda-t_{0})$
$\displaystyle\leq\Psi(y,-\Lambda)+L(|x|+|t_{0}|)$
$\displaystyle\leq\Psi(y,-\Lambda)+2Lr.$

Let now $b=b(\Lambda)>0$ be so small that $\Psi(y,-\Lambda)\geq b$ if $|y|\leq
1/2$. In particular, assuming that $r\leq 1/4$, for every $y\in B_{1/4}(x)$,
it holds

$\Psi_{1/8}(y-x,-\Lambda-
t_{0})\leq\bigg{(}1+\frac{2Lr}{b}\bigg{)}\Psi(y,-\Lambda)$

The same bound holds, trivially, for any $y$ such that $|y-x|\geq 1/4$. We now
choose $r$ even smaller, if needed, so that $\frac{2Lr}{b}\leq\delta$.
Therefore we may bound

$\int\Psi_{1/8}(\cdot-x,t_{0}+\Lambda)\,dM_{-\Lambda}\leq(1+\delta)\int_{B_{1}}\Psi(\cdot,-\Lambda)\,dM_{-\Lambda}\leq\frac{3}{4}(1+\delta),$

which yields the desired conclusion, up to choosing $\delta$ small universal.
∎

######  Corollary 4.5 (Bound on $\operatorname{mdr}(\mathbf{M})$).

Under the assumptions of Proposition 4.4, there exist $E_{1}$ universal such
that, for every $t\in[-r_{3}^{2},0]$ and every $B_{r}(x)\subset B_{r_{3}}$, it
holds

$M_{t}(B_{r}(x))\leq E_{1}r^{m}.$ (4.5)

In particular, for every $(x,t)\in\Sigma_{\mathbf{M}}\cap Q_{r_{3}}$ and for
every $r>0$ small enough, it holds

$M_{t-c_{1}r^{2}}(B_{r}(x))\geq c_{2}r^{m}$ (4.6)

for some $c_{1},c_{2}$ small universal.

Proof. Let $x,t$ and $r$ as in the statement. Then

$M_{t}(B_{r}(x))\leq Cr^{m}\int_{B_{r}(x)}\Psi(\cdot-x,-r^{2})\,dM_{t}\leq
2Cr^{m}.$

(4.6) follows from (4.5) and Proposition 3.4. ∎

###  4.2 Proof of Theorem 4.1

As stated earlier, we are going to argue by contradiction and compactness.
Namely, we fix $E_{0}$ and $\alpha$, we let $\Lambda$ be as specified in
Proposition 4.4 and we assume there exist $\varepsilon_{j}\searrow 0$ and two
sequences $\\{\Gamma^{j}\\}$, $\\{\mathbf{M}^{j}\\}$ such that, for every $j$,
$\mathbf{M}^{j}$ and $\Gamma^{j}$ satisfy the assumptions of Theorem 4.1 with
$\varepsilon_{0}$ replaced by $\varepsilon_{j}$.

In particular, we assume that
$\Gamma^{j}\in\mathcal{F}_{\alpha}(\varepsilon_{j},B_{1})$ and

$\Sigma_{\mathbf{M}^{j}}\subset\\{(z,\tau)\colon\operatorname{dist}(z,S^{+})\leq\varepsilon_{j}\\}.$
(4.7)

We also assume, for the sake of contradiction, that for no $j$ (4.3) is
satisfied for any choice of $T^{+}$, $\eta$ and $\beta$.

In the following, we let $\gamma^{j}:\Gamma_{0}\cap
B_{1}\to\Gamma_{0}^{\perp}$ be such that $\Gamma^{j}\cap
B_{1}\subset\mathop{\rm graph}\gamma^{j}$, as in the definition of
$\mathcal{F}_{\alpha}(\varepsilon_{j},B_{1})$, and we let
$\Sigma^{j}:=\Sigma_{\mathbf{M}^{j}}$. We also fix
$r_{0}=\min\\{r_{1},r_{2},r_{3}\\}$, so that the conclusions of Propositions
4.2, 4.3, 4.4 and of Corollary 4.5 hold true in $Q_{r_{0}}$.

######  Lemma 4.6 (Compactness and convergence to hyperplane).

There exists a subsequence (not relabeled) such that, for almost every
$t\in(-r_{0}^{2},0]$,

$M^{j}_{t}\rightharpoonup\mathcal{H}^{m}\llcorner S^{+}$

as Radon measures in $B_{r_{0}}$.

Proof. By the Arzelà-Ascoli theorem, $\gamma^{j}\to 0$ in $C^{1}$ up to
subsequences. By Corollary 4.5, we may apply the compactness theorems proven
in [20, Theorems 10.1 and 10.2] and find a further subsequence (not relabeled)
and $\mathbf{M}^{\infty}\in\mathcal{BF}_{m}(Q_{r_{0}},\Gamma_{0})$ such that,
for every $t\in(-r_{0}^{2},0]$,

$M^{j}_{t}\rightharpoonup M^{\infty}_{t}.$

In particular, the weak convergence stated above and (4.7) yield

$M^{\infty}_{t}((S^{+})^{c})=0$

for every $t\in(-r_{0}^{2},0]$. Moreover, by (4.6),
$(0,0)\in\Sigma_{\mathbf{M}^{\infty}}$. By Theorem 3.10, the proof is
concluded. ∎

Before stating the next result, we define some objects that we will use in the
rest of the subsection. First of all, let
$F_{\varepsilon}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be the map

$F_{\varepsilon}(x)=\bigg{(}Sx,\frac{1}{\varepsilon}S^{\perp}x\bigg{)};$

with a small abuse of notation, we use the same notation for the map
$F_{\varepsilon}\colon{\mathbb{R}}^{d,1}\to\mathbb{R}^{d,1}$ such that
$F_{\varepsilon}(x,t)=(F_{\varepsilon}(x),t)$. We now define

$\widetilde{\Sigma}^{j}=F_{\varepsilon_{j}}(\Sigma^{j}).$

Notice that, by (4.7),
$\widetilde{\Sigma}^{j}\subset\\{(x,t)\colon|S^{\perp}x|\leq 1\\}$ for every
$j$. For $j\in{\mathbb{N}}$ and $(x^{\prime},t)\in Q_{r_{0}}$, we define

$u^{j}(x^{\prime},t)=\Big{\\{}z\in\overline{B_{1}^{d-m}}\colon((x^{\prime},z),t)\in\widetilde{\Sigma}^{j}\Big{\\}};$

notice that such a set may well be empty or have more than one element. We
also define $\widetilde{\gamma}^{j}=F_{\varepsilon_{j}}\circ\gamma^{j}$; it is
clear that

$\widetilde{\gamma}^{j}\cdot\mathbf{e}_{m}\to 0\qquad\mbox{in }C^{1,\alpha}.$

Furthermore, since $||\widetilde{\gamma}^{j}||_{C^{1,\alpha}(B_{r_{0}})}\leq
1$, by the Arzelà-Ascoli theorem and up to passing to a subsequence (which we
do not relabel) we may find $g\colon B_{r_{0}}^{m-1}\to\overline{B_{1}^{d-m}}$
such that, for every $0<\varsigma<\alpha$,

$S^{\perp}\widetilde{\gamma}^{j}\to g\qquad\mbox{in }C^{1,\varsigma}$

and $\left\|g\right\|_{C^{1,\varsigma}}\leq 1$.

In order to keep the notation light, in the following we denote by
$E=\overline{B^{m}_{r_{0}}}\times\overline{B_{1}^{d-m}}\times[-r_{0}^{2},0]\subset{\mathbb{R}}^{d,1}$
and $E^{\prime}=S(E)=\overline{Q^{m}_{r_{0}}}\subset{\mathbb{R}}^{m,1}$. We
also let $E^{\prime}_{+}=E^{\prime}\cap\\{x_{m}\geq 0\\}$.

######  Lemma 4.7 (Uniform convergence).

There exists a subsequence (not relabeled) and $u\colon
E^{\prime}_{+}\to\overline{B_{1}^{d-m}}$ with the following properties:

  1. (i)

it holds

$d_{H}(\widetilde{\Sigma}_{j}\cap E;\mathop{\rm graph}u)\to 0$ (4.8)

as $j\to\infty$.

  2. (ii)

For every $(x^{\prime\prime},t)\in\overline{Q_{r_{0}}^{m-1}}$ it holds
$u((x^{\prime\prime},0),t)=g(x^{\prime\prime})$.

  3. (iii)

For every $X^{\prime},Y^{\prime}\in E^{\prime}_{+}$,

$|u(X^{\prime})-u(Y^{\prime})|\leq
2C_{2}\rho(X^{\prime},Y^{\prime})^{\varsigma},$

where $C_{2}$ and $\varsigma$ are as in Proposition 4.3.

In (4.8), by $\mathop{\rm graph}u$ we mean the set
$\\{(x^{\prime},u(x^{\prime},t),t)\colon(x^{\prime},t)\in
E^{\prime}_{+}\\}\subset E$.

Proof. Step 1: Hausdorff convergence. By Lemma 4.6,
$\widetilde{\Sigma}^{j}\cap E\neq\varnothing$ eventually. Thus one may extract
a subsequence (not relabeled) so that $\widetilde{\Sigma}^{j}\cap E$ converges
in the Hausdorff distance to some closed set $\widetilde{\Sigma}\subset E$.
Since, by assumption,
$\widetilde{\Sigma}^{j}\subset\\{x_{m}\geq-\varepsilon_{j}\\}$, it must also
be $\widetilde{\Sigma}\subset\\{x_{m}\geq 0\\}$. We define the set-valued
function

$\displaystyle
u(x^{\prime},t)=\big{\\{}y\in\overline{B_{1}^{d-m}}\colon((x^{\prime},y),t)\in\widetilde{\Sigma}\big{\\}}$
(4.9)

for $(x^{\prime},t)\in E^{\prime}_{+}$.

Step 2: $u(x^{\prime},t)\neq\varnothing$ for every $(x^{\prime},t)\in
E^{\prime}_{+}$. Assume, by contradiction, that there exists
$(x^{\prime},t)\in E^{\prime}_{+}\setminus S(\widetilde{\Sigma})$ (recall the
notation $S(x,t)=(Sx,t)=(x^{\prime},t)$). Then, since $S(\widetilde{\Sigma})$
is closed, there exists an open neighborhood $U^{\prime}$ of $(x^{\prime},t)$
such that $U^{\prime}\subset(S(\widetilde{\Sigma}))^{c}$. If we let
$U=S^{-1}(U^{\prime})\subset{\mathbb{R}}^{d,1}$, then by Lemma 4.6 and Fatou’s
lemma

$\displaystyle
0<\mathcal{H}^{m,1}(U\cap(S^{+}\times{\mathbb{R}}))\leq\liminf_{j}M^{j}(U),$

thus $M^{j}(U)>0$ eventually. In particular, by taking smaller and smaller
neighborhoods, one can pick a subsequence $j_{\ell}\to\infty$ and a sequence
$X_{\ell}\in\Sigma^{j_{\ell}}$ so that $S(X_{\ell})\to(x^{\prime},t)$. By
using the maps $F_{\varepsilon_{j}}$ defined above, we rescale in the
directions of $S^{\perp}$ and find that, up to subsequences, there exists
$z\in\overline{B_{1}^{d-m}}$ such that

$\widetilde{\Sigma}^{j_{\ell}}\ni
F_{\varepsilon_{j_{\ell}}}(X_{\ell})\to((x^{\prime},z),t).$

By Step 1, $((x^{\prime},z),t)\in\widetilde{\Sigma}$, which contradicts the
fact that $u(x^{\prime},t)=\varnothing$.

Step 3: $u((x^{\prime\prime},0),t)=\\{g(x^{\prime\prime})\\}$.  Let
$(x^{\prime\prime},t)\in\overline{Q_{r_{0}}^{m-1}}$. If $y\in
u((x^{\prime\prime},0),t)$, then by Step 1 there exists a sequence
$(x_{j},t_{j})\in\widetilde{\Sigma}^{j}$ such that
$x_{j}\to((x^{\prime\prime},0),y)$ and $t_{j}\to t$. In particular, by
Proposition 4.2, it holds

$\displaystyle|S^{\perp}(x_{j}-\widetilde{\gamma}^{j}(x_{j}^{\prime\prime}))|$
$\displaystyle=\frac{1}{\varepsilon_{j}}\big{|}S^{\perp}\big{(}F_{\varepsilon_{j}}^{-1}(x_{j})-\gamma^{j}(x_{j}^{\prime\prime})\big{)}\big{|}$
$\displaystyle\leq
C|x_{j}\cdot\mathbf{e}_{m}+\varepsilon_{j}+\varepsilon_{j}^{2}|^{1/2}$
$\displaystyle\longrightarrow 0$

as $j\to\infty$. Since $S^{\perp}\widetilde{\gamma}^{j}$ converges uniformly
to $g$ and $S^{\perp}x_{j}\to y$, it must be

$u((x^{\prime\prime},0),t)=\\{g(x^{\prime\prime})\\}.$

Step 4: $u(x^{\prime},t)$ is a singleton and Item iii holds true.  For
$i=1,2$, let $X_{i}=(x_{i},t_{i})\in\widetilde{\Sigma}$. Let also
$\rho:=\rho(S(X_{1}),S(X_{2}))$ and, without loss of generality, assume
$(x_{2})_{m}\geq(x_{1})_{m}$.

Case 1: $(x_{1})_{m}=0$. By Step 1 and Proposition 4.2, we have

$|S^{\perp}x_{2}-g(x_{2}^{\prime\prime})|\leq C(x_{2})_{m}^{1/2}\leq
C\rho^{1/2}.$

Moreover,
$|S^{\perp}x_{1}-g(x_{2}^{\prime\prime})|=|g(x_{1}^{\prime\prime})-g(x_{2}^{\prime\prime})|\leq
C\rho$. Thus

$|S^{\perp}x_{2}-S^{\perp}x_{1}|\leq C\rho^{1/2}+C\rho\leq C\rho^{\varsigma}.$

Case 2: $(x_{1})_{m}>0$ and $\rho=0$. In this case, we prove that
$S^{\perp}(x_{1})=S^{\perp}(x_{2})$. Fix $\omega$ much smaller than
$(x_{1})_{m}$. By Steps 1 and 2, we may pick $j$ large enough and three points
$Y_{1},Y_{2},W=(w,\tau)\in\widetilde{\Sigma}^{j}$ such that
$\rho(X_{i},Y_{i})\leq\omega$ and $\rho(S(W),S(X_{i}))\geq 2\omega$. Up to
choosing $j$ larger, we may assume that $\omega\geq
C\varepsilon_{j}^{\varsigma}$ and $(y_{i})_{m}\geq(x_{i})_{m}-\omega\geq
2\varepsilon_{j}$. Therefore, by Proposition 4.3, since
$\rho(S(W),S(Y_{i}))\geq\omega$, we estimate

$\displaystyle|S^{\perp}(x_{1}-x_{2})|$
$\displaystyle\qquad\leq|S^{\perp}(x_{1}-y_{1})|+|S^{\perp}(x_{2}-y_{2})|+|S^{\perp}(y_{1}-w)|+|S^{\perp}(y_{2}-w)|$
$\displaystyle\qquad\leq 2\omega+C\omega^{\varsigma}.$

Since $\omega>0$ is arbitrary, it holds $S^{\perp}(x_{1})=S^{\perp}(x_{2})$.
In particular, $u(x^{\prime},t)$ is a singleton for every $(x^{\prime},t)\in
E^{\prime}_{+}$. With a small abuse of notation, from here onwards, we will
denote by $u(x^{\prime},t)\in{\mathbb{R}}^{d-m}$ the only element of the set
defined in (4.9).

Case 3: $(x_{1})_{m}>0$ and $\rho>0$. By Steps 1 and 2, we may choose $j$
large enough and two points $Y_{1},Y_{2}$ such that the following hold true:

  1. 1.

$Y_{1},Y_{2}\in\widetilde{\Sigma}^{j}$;

  2. 2.

for $i=1,2$, $\rho(X_{i},Y_{i})<\rho/8$;

  3. 3.

$C\varepsilon_{j}^{\varsigma}\leq\rho/2$;

  4. 4.

for $i=1,2$, $(y_{i})_{m}\geq 2\varepsilon_{j}$.

Then, by Proposition 4.3, it holds

$\displaystyle|u(SX_{1})-u(SX_{2})|$
$\displaystyle\leq|u(SX_{1})-S^{\perp}(y_{1})|+|u(SX_{2})-S^{\perp}(y_{2})|+|S^{\perp}(y_{1}-y_{2})|$
$\displaystyle\leq 2\frac{\rho}{8}+C\rho^{\varsigma}\leq 2C\rho^{\varsigma},$

as desired.

∎

The rest of the proof consists in proving that $u$ defined in Lemma 4.7 solves
the heat equation in the interior of $E^{\prime}_{+}$. To this end, we recall
some facts about the heat equation. First, recall that
$E^{\prime}_{+}=\overline{Q_{r_{0}}^{m}}\cap\\{x_{m}\geq 0\\}$ and let us
introduce the sets

$\displaystyle\operatorname{Int}_{p}E^{\prime}_{+}=E^{\prime}_{+}\setminus\partial_{p}E^{\prime}_{+},$
$\displaystyle(E^{\prime}_{+})_{r}=\\{x^{\prime}\in{\mathbb{R}}^{m}\colon|x^{\prime}|\leq
r_{0}-r\mbox{ and }x^{\prime}_{m}\geq r\\}\times[-r_{0}^{2}+r^{2},0].$

Notice that
$\operatorname{Int}_{p}E^{\prime}_{+}=\bigcup_{r>0}(E^{\prime}_{+})_{r}$.

######  Lemma 4.8 (Interior regularity for the heat equation).

Let $g\in C(\partial_{p}E^{\prime}_{+})$. Then there exists $h\in
C^{\infty}(\operatorname{Int}_{p}E^{\prime}_{+})\cap C(E^{\prime}_{+})$ such
that

$\begin{cases}\partial_{t}h-\Delta h=0\qquad&\mbox{in
}\operatorname{Int}_{p}E^{\prime}_{+}\\\ h=g&\mbox{on
}\partial_{p}E^{\prime}_{+}.\end{cases}$

Moreover, for every $r>0$ there exists $C>0$ such that, for every
$(x^{\prime},t)\in(E^{\prime}_{+})_{r}$, it holds

$\max\\{|h(x^{\prime},t)|,|\nabla
h(x^{\prime},t)|,|D^{2}h(x^{\prime},t)|,|\partial_{t}h(x^{\prime},t)|\\}\leq
C\left\|g\right\|_{L^{\infty}(\partial_{p}E^{\prime}_{+})}.$

We now proceed with the proof of Theorem 4.1.

######  Lemma 4.9.

Let $u$ be as in Lemma 4.7. Then $u\in
C^{\infty}(\operatorname{Int}_{p}E^{\prime}_{+};{\mathbb{R}}^{d-m})\cap
C(E^{\prime}_{+};{\mathbb{R}}^{d-m})$ and

$u_{t}-\Delta u=0$

in $\Omega$.

Proof. We take as a model the proof of [11, Lemma 2.4]. We show that $u$ is
equal to the solution $h:E^{\prime}_{+}\to{\mathbb{R}}^{d-m}$ to the boundary
value problem

$\begin{cases}\partial_{t}h-\Delta h=0\qquad&\mbox{in
}\operatorname{Int}_{p}E^{\prime}_{+}\\\ h=u&\mbox{on
}\partial_{p}E^{\prime}_{+}.\end{cases}$

whose existence is guaranteed by Lemma 4.8. If not, there exist $r,\omega$
small and positive so that the function

$E^{\prime}_{+}\ni(x^{\prime},t)\mapsto|u(x^{\prime},t)-h(x^{\prime},t)|+\omega|x^{\prime}|^{2}$

achieves its maximum at $(x_{0}^{\prime},t_{0})\in(E^{\prime}_{+})_{2r}$.
Since $\widetilde{\Sigma}^{j}$ converges in the Hausdorff distance to $u$, for
some large $j$ we may find $X_{1}=(x_{1},t_{1})\in\Sigma^{j}$ such that
$(x^{\prime}_{1},t_{1})\in(E^{\prime}_{+})_{r}$ and the restriction to
$\Sigma^{j}$ of

$H(x,t):=\bigg{|}\frac{S^{\perp}x}{\varepsilon_{j}}-h(Sx,t)\bigg{|}+\omega|Sx|^{2}$

achieves its maximum at $X_{1}$.

We claim that, if $\varepsilon_{j}$ is small enough, depending on $r$ and
$\omega$, then for every $m$-dimensional subspace $T$, it holds
$T:D^{2}H(X_{1})>\partial_{t}H(X_{1})$. This would contradict Proposition 3.7,
thus concluding the proof. To prove the claim, we define
$f(x,t)=\frac{1}{\varepsilon_{j}}S^{\perp}x-h(Sx,t)$ and, with some
straightforward computations, we write

$H(x,t)=G_{1}(x,t)+G_{2}(x,t),$

where

$\displaystyle
G_{1}(x,t)=|f(X_{1})|^{2}+2f(X_{1})\cdot(f(x,t)-f(X_{1}))+\omega|Sx|^{2},$
$\displaystyle G_{2}(x,t)=|f(x,t)-f(X_{1})|^{2}.$

Notice that, by Lemma 4.8, there exists $C$ depending on $r$ such that

$|D^{2}G_{1}(X_{1})|\leq C(\omega+|f(X_{1})||D^{2}h(SX_{1})|)\leq C.$

Then, just as in [11], it is easy to show that, if $|T-S|\leq c\omega$, then

$T:D^{2}G_{1}(X_{1})>\partial_{t}H(X_{1})$

and $D^{2}G_{2}(X_{1})\geq 0$, thus in this case
$T:D^{2}H(X_{1})>\partial_{t}H(X_{1})$. On the other hand, if $|T-S|\geq
c\omega$, then there exists a unit-vector $\nu\in T$ such that
$S^{\perp}\nu\geq c\omega$. In particular, since $D^{2}G_{2}=2\nabla
f(X_{1})\nabla f(X_{1})^{T}$, it holds

$T:D^{2}G_{2}(X_{1})\geq-|S\nu|^{2}|\nabla
h(SX_{1})|^{2}+\frac{1}{\varepsilon^{2}}|S^{\perp}\nu|^{2}\geq\frac{c\omega^{2}}{\varepsilon_{j}^{2}}.$

We now conclude by remarking that
$\partial_{t}H(X_{1})=2f(X_{1})\cdot\partial_{t}h(SX_{1})\leq C$ and
$T:D^{2}G_{1}(X_{1})\geq-|D^{2}G_{1}(X_{1})|\geq-C$, thus

$T:D^{2}H(X_{1})\geq-C+\frac{c\omega^{2}}{\varepsilon_{j}^{2}}>\partial_{t}H(X_{1}),$

provided $\varepsilon_{j}$ is chosen small enough depending on $\omega$ and
$C$ (which, in turns, is a large constant depending on $r$). ∎

Once proven that $u$ is a solution to the heat equation, it is sufficient to
apply the following classical estimate:

######  Lemma 4.10 (Boundary regularity for the heat equation).

For every $\alpha\in(0,1)$, there exist positive constants $C$ and $\beta$
with the following property. Let $u\in
C^{2}(\operatorname{Int}_{p}E^{\prime}_{+})\cap C(E^{\prime}_{+})$ be such
that

$\partial_{t}u-\Delta u=0\qquad\mbox{in
}\operatorname{Int}_{p}E^{\prime}_{+}.$

Assume, moreover, that for all $t$, $u(\cdot,t)|_{\\{x_{m}=0\\}}=g\in
C^{1,\alpha}(B_{r_{0}}\cap\\{x_{m}=0\\})$, that $|g(0)|=|Dg(0)|=0$ and that
$|u|\leq 1$ everywhere. Then there exists a linear operator
$L:{\mathbb{R}}^{m}\to{\mathbb{R}}^{d-m}$ with $|L|\leq C$ such that, for
every $\eta\in(0,1/4)$,

$|u(x^{\prime},t)-L(x^{\prime})|\leq C\eta^{1+\beta}$

in $(B_{\eta}^{m}\cap\\{x_{m}\geq 0\\})\times(-\eta^{2},0]$.

Proof. See [16, Theorem 2.1]. ∎

######  Remark 4.11.

From the fact that $g\in C^{1,\alpha}$ and that $Dg(0)=0$, it follows that
$L(x^{\prime})=0$ if $x_{m}=0$.

Conclusion of the proof of Theorem 4.1.  By Lemma 4.9 and Lemma 4.10, there
exists $L:{\mathbb{R}}^{m}\to{\mathbb{R}}^{d-m}$ linear such that
$L(x^{\prime})=0$ if $x^{\prime}_{m}=0$, $|L|\leq C$ and, for every $\eta$
small, it holds

$\widetilde{\Sigma}\cap(B^{m}_{2\eta}\times
B_{1}^{d-m}\times(-4\eta^{2},0])\subset\\{(x,t):|S^{\perp}x-L(Sx)|\leq
C\eta^{1+\beta}\\}.$

We fix $\eta$ small, to be specified later and we choose $j$ sufficiently
large so that the Hausdorff distance between $\widetilde{\Sigma}$ and
$\widetilde{\Sigma}^{j}$ is smaller than $\eta^{1+\beta}$. We now let
$T=\\{x\in{\mathbb{R}}^{d}\colon S^{\perp}x=\varepsilon_{j}L(Sx)\\}$. Then it
holds

$\Sigma^{j}\cap Q_{\eta}\subset\\{|T^{\perp}x|\leq
C^{\prime}\varepsilon_{j}\eta^{1+\beta}\\}.$

Moreover, by Proposition 4.2 and the fact that
$|\gamma_{m}(x^{\prime\prime})|\leq\varepsilon|x^{\prime\prime}|^{1+\alpha}$,
it holds

$\Sigma^{j}\cap
Q_{\eta}\subset\\{x_{m}\geq-\varepsilon_{j}\eta^{1+\alpha}-\varepsilon_{j}^{2}\\},$

provided $\eta\leq r_{2}$. We choose $j$ large enough so that
$\varepsilon_{j}^{2}\leq\eta^{1+\beta}$. Since $\beta$ can be chosen smaller
than $\alpha$, we have

$\Sigma^{j}\cap
Q_{\eta}\subset\\{x_{m}\geq-2\varepsilon_{j}\eta^{1+\beta}\\}\cap\\{|T^{\perp}x|\leq
C\varepsilon_{j}\eta^{1+\beta}\\}.$

Up to choosing $j$ larger, the above inclusion yields

$\Sigma^{j}\cap Q_{\eta}\subset\\{\operatorname{dist}(\cdot,T^{+})\leq
2C\eta^{1+\beta}\\}.$

We conclude the proof by choosing $\beta^{\prime}>\beta$ and $\eta$ so small
that $2C\eta^{1+\beta}\leq\eta^{1+\beta^{\prime}}$ and we recover (4.3) (with
$\beta^{\prime}$ instead of $\beta$). This contradicts the assumption made at
the beginning of the present subsection, thus concluding the proof. ∎

##  5 Boundary behavior

We now prove Proposition 4.2. The setting is the following. Let $E_{0}$ and
$\alpha$ be given and let $r_{3}$ be the constant given in Proposition 4.4.
Assume $\mathbf{M}$ and $\Gamma$ satisfy the assumptions of Theorem 4.1. Then
$\operatorname{mdr}(\mathbf{M},Q_{r_{3}})<\infty$, therefore Proposition 4.2
follows from the following, more general, statement:

######  Proposition 5.1 (Boundary behavior at scale $R$).

There exist $c$ and $\varepsilon_{1}$ depending only on $\alpha$ with the
following property. Let $0<\delta<\varepsilon\leq\varepsilon_{1}$,
$\Gamma\in\mathcal{F}_{\alpha}(\delta,B_{R})$ and
$\mathbf{M}\in\mathcal{BF}_{m}(Q_{R},\Gamma)$ be such that

$\Sigma\cap
Q_{R}\subset\\{(x,t)\colon\operatorname{dist}(x,S^{+})\leq\varepsilon R\\}$

and

$\operatorname{mdr}(\mathbf{M},Q_{R})<\infty.$ (5.1)

Then

$\Sigma\cap Q_{R/2}\subset\Bigg{\\{}(x,t)\colon
x_{m}\geq\gamma_{m}(x^{\prime\prime})-R\delta^{2}+cR\frac{|S^{\perp}(x-\gamma(x^{\prime\prime}))|^{2}}{2(\varepsilon
R)^{2}}\Bigg{\\}}.$

######  Remark 5.2.

The role of (5.1) is to guarantee that the maximum principle (Proposition 3.7)
holds true.

Proof. By a simple rescaling argument, it is sufficient to prove the result in
the case $R=1$. We fix $c$ small and $\varepsilon_{1}\leq c$, to be specified
later. By contradiction, assume there exist
$0<\delta\leq\varepsilon\leq\varepsilon_{1}$, $\Gamma$ and $\mathbf{M}$ as
above, and a point $(\bar{x},\bar{t})\in\Sigma\cap Q_{1/2}$ such that

$0<\omega:=\frac{c}{2\varepsilon^{2}}|S^{\perp}(\bar{x}-\gamma(\bar{x}^{\prime\prime}))|^{2}-\delta^{2}+\gamma_{m}(\bar{x}^{\prime\prime})-\bar{x}_{m}.$

We show that, if this is the case, then we may build a family of surfaces
sliding in the direction of $\mathbf{e}_{m}$ that touch $\Sigma$ at some point
where the conclusion of Proposition 3.7 fails.

In order to do so, we first define the functions
$g:{\mathbb{R}}^{d-m}\to{\mathbb{R}}$ and $h:{\mathbb{R}}^{m}\to{\mathbb{R}}$
as

$\displaystyle
g(z)=c\frac{|z-S^{\perp}\gamma(\bar{x}^{\prime\prime})|^{2}}{2\varepsilon^{2}}$
$\displaystyle
h(y)=P(y^{\prime\prime})-|y^{\prime\prime}-\bar{x}^{\prime\prime}|^{2}-y_{m},$

where

$P(y^{\prime\prime})=\gamma_{m}(\bar{x}^{\prime\prime})+\nabla\gamma_{m}(\bar{x}^{\prime\prime})\cdot(x^{\prime\prime}-\bar{x}^{\prime\prime})-\delta^{2}-C|y^{\prime\prime}-\bar{x}^{\prime\prime}|^{2},$

and $C$ is a constant depending only on $\alpha$ chosen so that

$P(x^{\prime\prime})\leq\gamma_{m}(x^{\prime\prime})$ (5.2)

(to show that such $C$ depending only on $\alpha$ exists, use the fact that
$\gamma\in C^{1,\alpha}(B_{R})$ and Young’s inequality). Then, choose a smooth
function $f\colon{\mathbb{R}}\to{\mathbb{R}}$ such that $f(-1)=-4c$,
$f|_{t\geq-1/4}\geq-\frac{\omega}{2}$, $f<0$ everywhere and $f^{\prime}(t)\leq
8c$ everywhere. We now set

$H(x,t)=g(S^{\perp}x)+h(Sx)+f(t).$

This way, the zero-level set of $H$ is a surface sliding in the
$\mathbf{e}_{m}$-direction. Notice that

$H(\bar{x},\bar{t})=\omega+f(\bar{t})>0.$ (5.3)

We now show that, if
$(x,t)\in\Sigma\cap((\Gamma\times{\mathbb{R}})\cup\partial_{p}Q_{1})$, then
$H(x,t)\leq 0$.

  1. 1.

If $x\in\Sigma_{-1}$, then

     * •

$g(S^{\perp}x)\leq c(\varepsilon+\delta)^{2}/(2\varepsilon^{2})=2c$, since
$|S^{\perp}x|\leq\varepsilon$ and
$|\gamma(x^{\prime\prime})|\leq\delta\leq\varepsilon$;

     * •

by (5.2), $h(Sx)\leq\gamma_{m}(x^{\prime\prime})-x_{m}\leq 2\varepsilon$.

The two above facts, along with the assumption $f(-1)=-4c$, yield

$H(x,-1)\leq 2c+2\varepsilon-4c\leq 0$

provided $\varepsilon\leq c$.

  2. 2.

If $x\in\partial{B_{1}}\cap\Sigma_{t}$, then $|S^{\perp}x|\leq\varepsilon$ and
$x_{m}\geq-\varepsilon$, thus

$|x^{\prime\prime}|\geq\sqrt{1-\varepsilon^{2}-x_{m}^{2}}\geq\frac{3}{4}-x_{m},$

provided $\varepsilon$ is small enough. In particular,
$|x^{\prime\prime}-\bar{x}^{\prime\prime}|\geq\frac{1}{4}-x_{m}$. Hence:

     * •

since $\left\|\gamma\right\|_{C^{1,\alpha}(B_{1})}\leq\delta$, we have

$h(Sx)\leq
2\delta-(C+1)|x^{\prime\prime}-\bar{x}^{\prime\prime}|^{2}-x_{m}\leq
2\delta-(C+1)\bigg{(}\frac{1}{4}-x_{m}\bigg{)}^{2}-x_{m};$

     * •

as in Item 1, $g(S^{\perp}x)\leq 2c$;

     * •

$f(t)\leq 0$;

Therefore

$\displaystyle H(x,t)$ $\displaystyle\leq
2c+2\delta-(C+1)\bigg{(}\frac{1}{4}-x_{m}\bigg{)}^{2}-x_{m}\leq
2c+2\delta-\frac{C}{4(1+C)}\leq 0$

provided $C\geq 1$ and $c,\delta$ are small enough.

  3. 3.

Lastly, for every $x\in\Gamma$ and $t\in(-1,0)$, under the assumptions
$\delta\leq\varepsilon$ and $c\leq 1$, it holds

$\displaystyle g(S^{\perp}x)$
$\displaystyle=\frac{c}{2\varepsilon^{2}}|S^{\perp}(\gamma(x^{\prime\prime})-\gamma(\bar{x}^{\prime\prime}))|^{2}$
$\displaystyle\leq\frac{c}{2\varepsilon^{2}}\left\|\nabla\gamma\right\|_{\infty}^{2}|x^{\prime\prime}-\bar{x}^{\prime\prime}|^{2}$
$\displaystyle\leq|x^{\prime\prime}-\bar{x}^{\prime\prime}|^{2}.$

Since $f\leq 0$ and
$h(Sx)\leq\gamma_{m}(x^{\prime\prime})-|x^{\prime\prime}-\bar{x}^{\prime\prime}|^{2}-x_{m}$,
we have

$H(x,t)\leq\gamma_{m}(x^{\prime\prime})-x_{m}=0.$

Points 1-3 above and (5.3) show that there must exist $Y=(y,s)\in
Q_{1}\cap\Sigma$ with $y\notin\Gamma$ such that $H|_{\\{t\leq s\\}}$ has a
local maximum at $(y,s)$.

We now show that one can choose $c$ even smaller, if needed, so that the
existence of such a point would contradict the maximum principle. Indeed,
since $|S^{\perp}y|\leq\varepsilon$, if $c$ is small enough then $|\nabla
h(Sy)|^{2}/|\nabla g(S^{\perp}y)|^{2}\geq\varepsilon$, thus

$|S^{\perp}\nabla H(Y)|=|\nabla g(S^{\perp}y)|\leq(1-\varepsilon)|\nabla
H(Y)|.$

Therefore, if $T$ is a $m$-dimensional subspace of ${\mathbb{R}}^{d}$ such
that $T\perp\nabla H(Y)$, then

$T\colon S^{\perp}\geq\varepsilon$

and

$T\colon D^{2}H(Y)=T\colon\begin{pmatrix}D^{2}h&0\\\
0&D^{2}g\end{pmatrix}\geq-|D^{2}h(Sy)|+\varepsilon|D^{2}g(S^{\perp}y)|.$

Now, simple computations show that, up to multiplications by constants
depending only on $m$ and $\alpha$, $|D^{2}h(Sy)|\leq 1$ and
$|D^{2}g(S^{\perp}y)|\geq\frac{c}{\varepsilon^{2}}$ where $c$ is the constant
we fixed at the beginning of the proof. Therefore, if $\varepsilon$ is much
smaller than $c$, then $T\colon D^{2}H(y)\geq\frac{c}{2\varepsilon}$. However,
by Proposition 3.7, it holds

$\inf_{T\perp\nabla H(Y)}T\colon
D^{2}H(Y)\leq\partial_{t}H(Y)=f^{\prime}(s)\leq 8c,$

which is a contradiction. ∎

##  6 Decay of oscillations: proof of Proposition 4.3

In the present section, we prove Proposition 4.3.

We begin by giving the following definition:

######  Definition 6.1.

Let $u:{\mathbb{R}}^{m,1}\to[-\infty,1]$ be an upper-semicontinuous function.
Assume that, whenever a smooth function
$\varphi:{\mathbb{R}}^{m,1}\to{\mathbb{R}}$ touches $u$ from above at some
$(x_{0}^{\prime},t_{0})\in U\times I$ (according to the terminology set in
Subsection 3.2) and
$|\nabla\varphi(x_{0}^{\prime},t_{0})|,|D^{2}\varphi(x_{0}^{\prime},t_{0})|$
are smaller than some fixed universal constant $\delta_{0}$, then

$\partial_{t}\varphi-\mathcal{M}^{+}(D^{2}\varphi)\leq 0$ (6.1)

at $(x^{\prime}_{0},t_{0})$ (see Subsection 3.2 for the definition of
$\mathcal{M}^{+}$). Then $u$ is said to be a viscosity subsolution to (6.1) in
$U\times I$.

The reader should notice that the classical definition of viscosity solution
is slightly different than ours, in that the test function $\varphi$ usually
has no restrictions on the magnitude of $|\nabla\varphi|$ and $|D^{2}\varphi|$
at the touching point.

The proof of Proposition 4.3 is achieved in three steps:

  1. 1.

First of all, one sees that the support of a $\mathbf{M}$ behaves, in some
sense, like the graph of a viscosity subsolution to (6.1), as in the
definition above; this was proved in Corollary 3.8.

  2. 2.

By exploiting the results in [17], one shows that, if a $\Sigma$ has a point
far enough from $S$, then the mass of $\mathbf{M}$ near that point cannot be
too small.

  3. 3.

If $\Sigma$ does not have the decay of oscillations stated in Proposition 4.3,
then by the previous step the mass of $\mathbf{M}$ in some parabolic cylinder
must be large; this contradicts the small density assumption (4.1).

Before proceeding, we introduce some notations that we are going to use in the
present subsection. Given $\theta\in(0,1)$, we define the set

$\mathcal{P}_{1}^{\theta}=\bigg{\\{}(x^{\prime},t)\in{\mathbb{R}}^{m,1}\colon|x^{\prime}|^{2}<\frac{-t}{\theta^{2}}<1\bigg{\\}}.$

One should compare these sets with those which, in [17], are called “parabolic
balls”. Our definition slightly differs from theirs; notice that with our
choice $\mathcal{P}_{1}^{\theta}\subset B^{m}_{1}\times(-\theta^{2},0)$.

######  Lemma 6.2 (Measure estimate, [17]).

For every $\theta>0$ and $\mu\in(0,1)$, there exist small constants
$\eta^{\prime},r$ with the following property. Let
$u\colon{\mathbb{R}}^{m,1}\to[-\infty,1]$ be a viscosity solution to (6.1) in
$B_{1}^{m}\times(-\theta^{2},0)$ and assume that

$u(Y_{0})\geq 1-\eta^{\prime}$

for some $Y_{0}\in B^{m}_{r}\times(-\theta^{2}r^{2},0)$. Then

$\mathcal{L}^{m,1}(\\{u\geq
1-\mu\\}\cap\mathcal{P}_{1}^{\theta})\geq(1-\mu)\mathcal{L}^{m,1}(\mathcal{P}_{1}^{\theta}).$
(6.2)

Proof. This result corresponds, essentially, to [17, Lemma 4.3]. Apart from
some trivial adjustment of constants, there are two caveats:

  * •

The results in [17] are stated with the classical definition of viscosity
solutions, where no bound on the test function at the touching point is
required. However, it is easy to see that the results are valid for our
definition of viscosity solution, as well.

  * •

In our setting, we allow $u$ to be merely upper-semicontinuous and, possibly,
take infinite values, while in [17] $u$ is required to be continuous. This
minor point can be easily overcome by looking at the sup-convolution of $u$:

$u_{\delta}(x,t)=\sup\bigg{\\{}u(y,s)-\frac{1}{\delta}(|x-y|^{2}+(t-s)^{2})\bigg{\\}},$

which conserves the property of being a viscosity subsolution to (6.1) and for
which (6.2) holds true, by [17, Lemma 4.3]. Letting $\delta\searrow 0$ gives
the desired conclusion.

∎

Before stating the next result, we fix some further notations. For any closed
set $\Sigma\subset{\mathbb{R}}^{d,1}$ and any
$\Omega\subset{\mathbb{R}}^{d,1}$, we let

$\operatorname{osc}(\Sigma,\Omega)=\inf\bigg{\\{}h>0\colon\mbox{there is
$y\in{\mathbb{R}}^{d}$ such that
}\Sigma\cap\Omega\subset\\{x\colon|S^{\perp}(x-y)|\leq h\\}\bigg{\\}}.$

We also let

$C_{r}=\big{\\{}x\in{\mathbb{R}}^{d}\colon|Sx|<r\big{\\}}.$

######  Lemma 6.3 (Harnack inequality).

For every $\delta\in(0,1)$, there exist small constants
$\varepsilon_{2},\theta,r,\eta$ with the following property. Let
$\varepsilon\leq\varepsilon_{2}$ and
$\mathbf{M}\in\mathcal{BF}_{m}(C_{1}\times(-\theta^{2},0])$ be such that:

$\displaystyle\Sigma\subset\\{|S^{\perp}x|\leq\varepsilon\\},$ (6.3)
$\displaystyle\int_{C_{1}}\Psi(\cdot,t)\,dM_{t}\leq 2-\delta\qquad\mbox{for
all }t\in(-\theta^{2},0),$ (6.4)

and

$\operatorname{mdr}(\mathbf{M},C_{1}\times(-\theta^{2},0])<\infty.$ (6.5)

Then

$\operatorname{osc}(\Sigma,C_{r}\times(-\theta^{2}r^{2},0])\leq(1-\eta)\varepsilon.$
(6.6)

The proof of the above result involves some technical estimates. It is
therefore convenient to give an overview of the strategy. If (6.6) does not
hold, then one finds two points $Y_{1}$ and $Y_{2}$ in $\Sigma$ that are far
enough in $S^{\perp}$. By applying Lemma 6.2 twice, we find that in
$C_{1}\times(-\theta^{2},0)$ the mass of $\mathbf{M}$ must be almost that of
two $m$-dimensional disks. This contradicts (6.4), which encodes the fact that
the mass of $\mathbf{M}$ must not exceed by too much that of a single disk.

Proof of Lemma 6.3.  Let $\delta\in(0,1)$ be given. Fix $\theta$ and $\mu$,
which we will specify later, and let $r$ and $\eta^{\prime}$ be chosen
accordingly as in Lemma 6.2. Moreover, fix $\varepsilon$ much smaller than
$\mu$ and $\eta\leq\eta^{\prime}$, to be specified later. Assume, by
contradiction, that there exist
$\mathbf{M}\in\mathcal{BF}_{m}(C_{1}\times(-\theta^{2},0])$ that satisfies the
assumptions of the present result with the choices made above, and two points
$Y_{1}=(y_{1},s_{1}),Y_{2}=(y_{2},s_{2})\in\Sigma\cap(C_{r}\times(-\theta^{2}r^{2},0])$
with $|S^{\perp}y_{1}-S^{\perp}y_{2}|\geq 2(1-\eta)\varepsilon$. For every
$(x^{\prime},t)\in B_{1}^{m}\times(-\theta^{2},0]$ and for $i=1,2$, let

$u_{i}(x^{\prime},t)=\frac{1}{2\varepsilon}\sup\Big{\\{}|z-S^{\perp}y_{i}|\colon
z\in S^{\perp}\mbox{ and }(x^{\prime},z)\in\Sigma_{t}\Big{\\}}.$

Notice that $u_{1}$ and $u_{2}$ are upper-semicontinuous and, for every
$(x^{\prime},t)$, either $u_{1}(x^{\prime},t),u_{2}(x^{\prime},t)\in[0,1]$ or
$u_{1}=u_{2}=-\infty$. By Corollary 3.8 and (6.5), both $u_{1}$ and $u_{2}$
are viscosity subsolutions to (6.1). Moreover,

$u_{1}(Sy_{2},s_{2})\geq\frac{1}{2\varepsilon}\big{|}S^{\perp}y_{2}-S^{\perp}y_{1}\big{|}\geq
1-\eta\geq 1-\eta^{\prime},$

hence, by Lemma 6.2,

$\mathcal{L}^{m,1}(\\{u_{1}\geq
1-\mu\\}\cap\mathcal{P}_{1}^{\theta})\geq(1-\mu)\mathcal{L}^{m,1}(\mathcal{P}_{1}^{\theta}).$

With the same argument, one also obtains

$\mathcal{L}^{m,1}(\\{u_{2}\geq
1-\mu\\}\cap\mathcal{P}_{1}^{\theta})\geq(1-\mu)\mathcal{L}^{m,1}(\mathcal{P}_{1}^{\theta}).$
(6.7)

We now want to estimate

$\int_{C_{1}\times(-\theta^{2},0)}\Psi\,dM.$

We first define, for $i=1,2$, the sets

$A_{i}=\\{(x,t)\in{\mathbb{R}}^{d,1}\colon(Sx,t)\in\mathcal{P}_{1}^{\theta},|S^{\perp}(x-y_{i})|\leq\varepsilon/2\mbox{
and }t\leq-2\varepsilon^{2}/\delta\\}.$

Notice that $A_{1}\cap A_{2}=\varnothing$ and, by (6.3), for $M$-a.e.
$(x,t)\in A_{i}$, it holds

$\displaystyle\Psi(x,t)$
$\displaystyle=\exp\bigg{(}\frac{|S^{\perp}x|^{2}}{4t}\bigg{)}\Psi^{\prime}(Sx,t)$
$\displaystyle\geq\exp\bigg{(}-\frac{\varepsilon^{2}}{8\varepsilon^{2}/\delta}\bigg{)}\Psi^{\prime}(Sx,t)$
$\displaystyle\geq\bigg{(}1-\frac{\delta}{8}\bigg{)}\Psi^{\prime}(Sx,t),$

where $\Psi^{\prime}(x^{\prime},t)\colon=\Psi((x^{\prime},0),t)$.

Therefore we have

$\displaystyle\int_{C_{1}\times(-\theta^{2},0)}\Psi\,dM$
$\displaystyle\qquad\geq\int_{A_{1}}\Psi\,dM+\int_{A_{2}}\Psi\,dM$
$\displaystyle\qquad\geq\bigg{(}1-\frac{\delta}{8}\bigg{)}\Bigg{(}\int_{A_{1}}\Psi^{\prime}(Sx,t)\,dM(x,t)+\int_{A_{2}}\Psi^{\prime}(Sx,t)\,dM(x,t)\Bigg{)}.$
(6.8)

Moreover, by Lemma 3.6 and by the coarea formula,

$\displaystyle\int_{A_{i}}\Psi^{\prime}(Sx,t)\,dM(x,t)\geq\int_{S(A_{i}\cap\Sigma)}\Psi^{\prime}(x^{\prime},t)\,d\mathcal{L}^{m,1}(x^{\prime},t).$
(6.9)

We may assume that $\mu$ and $\eta$ are smaller that some universal constant
so that, if $z\in{\mathbb{R}}^{d-m}$ is such that
$\frac{1}{2\varepsilon}|z-S^{\perp}y_{2}|\geq 1-\mu$, then

$|z-S^{\perp}y_{1}|\leq\frac{\varepsilon}{2}.$

In particular, we have

$\displaystyle S(A_{1}\cap\Sigma)\supset\\{u_{2}\geq
1-\mu\\}\cap\mathcal{P}_{1}^{\theta}\cap\\{t\leq-2\varepsilon^{2}/\delta\\}$

which, together with (6.7), yields that $S(A_{1}\cap\Sigma)$ covers a large
portion of $\mathcal{P}_{1}^{\theta}$: namely

$\displaystyle\mathcal{L}^{m,1}\bigg{(}S(A_{1}\cap\Sigma)\bigg{)}$
$\displaystyle\geq\mathcal{L}^{m,1}\bigg{(}\mathcal{P}_{1}^{\theta}\cap\\{u_{2}\geq
1-\mu\\}\cap\\{t\leq-2\varepsilon^{2}/\delta\\}\bigg{)}$
$\displaystyle\geq\mathcal{L}^{m,1}\bigg{(}\mathcal{P}_{1}^{\theta}\cap\\{u_{2}\geq
1-\mu\\}\bigg{)}-\frac{2\varepsilon^{2}}{\delta}$
$\displaystyle\geq(1-2\mu)\mathcal{L}^{m,1}\big{(}\mathcal{P}_{1}^{\theta}\big{)},$

provided $\varepsilon^{2}\leq c\delta\mu$ for some $c$ small universal.

We are now ready to choose $\mu$, depending on $\delta$, so that the above
inequality and the fact that $\Psi\in
L^{1}(\mathcal{L}^{m,1}\llcorner\mathcal{P}_{1}^{\theta})$ yield

$\displaystyle\int_{S(A_{1}\cap\Sigma)}\Psi^{\prime}\,d\mathcal{L}^{m,1}$
$\displaystyle\qquad\geq\int_{\mathcal{P}_{1}^{\theta}}\Psi^{\prime}\,d\mathcal{L}^{m,1}-\frac{\delta\theta^{2}}{8}$
$\displaystyle\qquad=\theta^{2}\int_{B_{1}^{m}}\Psi^{\prime}(\cdot,-\theta^{2})\,d\mathcal{L}^{m}-\frac{\delta\theta^{2}}{8}.$
(6.10)

Finally, we also choose $\theta$ small such that

$\int_{B_{1}^{m}}\Psi^{\prime}(\cdot,-\theta^{2})\,d\mathcal{L}^{m}\geq\int_{{\mathbb{R}}^{m}}\Psi^{\prime}(\cdot,-\theta^{2})\,d\mathcal{L}^{m}-\frac{\delta}{8}=1-\frac{\delta}{8}.$
(6.11)

By (6.10) and (6.11), it holds

$\int_{S(A_{1}\cap\Sigma)}\Psi^{\prime}\,d\mathcal{L}^{m,1}\geq\theta^{2}\bigg{(}1-\frac{\delta}{4}\bigg{)}.$
(6.12)

The same argument can be repeated for $A_{2}$, thus giving

$\int_{S(A_{2}\cap\Sigma)}\Psi^{\prime}\,d\mathcal{L}^{m,1}\geq\theta^{2}\bigg{(}1-\frac{\delta}{4}\bigg{)}.$
(6.13)

We conclude the proof by combining (6.8), (6.9), and (6.12), (6.13), obtaining

$\displaystyle\int_{C_{1}\times(-\theta^{2},0)}\Psi\,dM$
$\displaystyle\qquad\geq
2\theta^{2}\bigg{(}1-\frac{\delta}{8}\bigg{)}\bigg{(}1-\frac{\delta}{4}\bigg{)}$
$\displaystyle\qquad\geq\theta^{2}\bigg{(}2-\frac{3}{4}\delta\bigg{)}$

which contradicts (6.4). This concludes the proof. ∎

A simple rescaling argument allows one to iterate Lemma 6.3 and obtain the
following

######  Proposition 6.4.

For every $\delta\in(0,1)$ there exist $C$ (large) and $\varsigma$ (small)
with the following property. Let
$\mathbf{M}\in\mathcal{BF}_{m}(C_{R}\times(-R^{2},R^{2}))$ be such that

$\operatorname{mdr}(\mathbf{M},C_{R}\times(-R^{2},R^{2}))<\infty$ (6.14)

and assume that, for every $(x,t)\in C_{R/2}\times(-R^{2}/4,R^{2}/4))$ and
every $s\in(t-R^{2}/4,t)$, it holds

$\int_{C_{R/2}(x)}\Psi(\cdot-x,s-t)\,dM_{s}\leq 2-\delta.$ (6.15)

If $\varepsilon=\operatorname{osc}(\Sigma,C_{R}\times(-R,R))$, then for any
couple $(x,t),(y,s)\in C_{R/2}\times(-R^{2}/4,R^{2}/4)\cap\Sigma$ such that
$\rho=\rho(X^{\prime},Y^{\prime})\geq
CR^{1-\varsigma}\varepsilon^{\varsigma}$, it holds

$|S^{\perp}(x-y)|\leq C\varepsilon\bigg{(}\frac{\rho}{R}\bigg{)}^{\varsigma}.$

Proof. We prove the result for $R=1$, as the general case follows by replacing
$\mathbf{M}$ with $\mathcal{D}_{R}\mathbf{M}$.

Let $\varepsilon_{2},\theta,r,\eta$ be the constants given in Lemma 6.3 in
correspondence to $\delta$. Consider the rescaled flows
$\mathbf{M}^{k}=\mathcal{D}_{r^{k}}(\mathbf{M}-X)$. By induction, the
assumptions of Lemma 6.3 are in place for every integer $k$ such that

$\bigg{(}\frac{1-\eta}{r}\bigg{)}^{k}\varepsilon\leq\varepsilon_{2}.$ (6.16)

Therefore, scaling back to the original flow, we see that for those $k$:

$\operatorname{osc}\big{(}\Sigma_{\mathbf{M}},C_{r^{k}}(x)\times(t-\theta^{2}r^{2k},t]\big{)}\leq(1-\eta)^{k}\varepsilon.$

Let now $X=(x,t)$ and $Y=(y,s)$ be two points in
$C_{1/2}\times(-1/4,1/4)\cap\Sigma$ and let
$\rho=\rho((x^{\prime},t),(y^{\prime},s))$. Without loss of generality, we may
assume that $t\geq s$. Furthermore, by taking $C\geq 2/\theta$, we may clearly
reduce ourselves to the case $\rho\leq\theta/2$. By choosing $\varsigma$ small
enough and $C$ larger than the choice made above, if necessary, we infer from
$\rho\geq C\varepsilon^{\varsigma}$ that there exists $k\in{\mathbb{N}}$
satisfying (6.16) such that $r^{k+1}\leq 2\rho/\theta\leq r^{k}$. Thus

$Y\in C_{2\rho}(x)\times(t-4\rho^{2},t]\subset
C_{r^{k}}(x)\times(t-\theta^{2}r^{2k},t]$

hence it must be $|S^{\perp}(x-y)|\leq 2(1-\eta)^{k}\varepsilon$. We conclude
the proof by taking $C$ larger and $\varsigma$ smaller, if needed, so that
$2(1-\eta)^{k}\leq C\rho^{\varsigma}$. ∎

We finally prove Proposition 4.3.

Proof of Proposition 4.3.  Let $r_{2}=\frac{1}{2}\min\\{r_{1},r_{3}\\}$, where
$r_{1}$ and $r_{3}$ are given in Propositions 4.2 and 4.4, respectively. Let
also $X=(x,t),Y=(y,s)$ be two points in $\Sigma\cap Q_{r_{2}}$. Without loss
of generality, we assume that $R:=x_{m}\geq y_{m}\geq 2\varepsilon$. Let
$\rho=\rho((x^{\prime},t),(y^{\prime},s))$; finally, let $\varsigma$ be the
constant determined in Proposition 6.4 corresponding to $\delta=\frac{1}{2}$.
We shall distinguish two cases.

If $\rho\leq\frac{R}{8}$, then we may find $t^{\prime}\in(-r_{2}^{2},0]$ such
that $X,Y\in C_{R/4}(x)\times(t^{\prime}-R^{2}/16,t^{\prime}+R^{2}/16)$ and
$C_{R/4}(x)\subset\Gamma^{c}$. Since $R\leq 1$, the assumption $\rho\geq
C\varepsilon^{\varsigma}$ yields $\rho\geq
CR^{1-\varsigma}\varepsilon^{\varsigma}$. By Proposition 4.4 and Corollary
4.5, (6.14) and (6.15) hold true, thus Proposition 6.4 applies and we obtain

$|S^{\perp}(x-y)|\leq
C\bigg{(}\frac{\rho}{R}\bigg{)}^{\varsigma}\operatorname{osc}(\Sigma,\mathcal{U}(X)),$

where
$\mathcal{U}(X)\colon=C_{R/4}(x)\times(t^{\prime}-R^{2}/16,t^{\prime}+R^{2}/16)$.
By Proposition 4.2, we may estimate

$\displaystyle\operatorname{osc}(\Sigma,\mathcal{U}(X))\leq
2C\varepsilon\sqrt{2R+\varepsilon+\varepsilon^{2}}+CR\left\|\nabla\gamma\right\|_{\infty}\leq
C\varepsilon R^{1/2},$

since $\varepsilon\leq R/2$ and $\left\|\nabla\gamma\right\|_{\infty}\leq
C\varepsilon$. Thus

$|S^{\perp}(x-y)|\leq C\varepsilon R^{1/2-\varsigma}\rho^{\varsigma}\leq
C\varepsilon\rho^{\varsigma},$

since $\varsigma$ can be chosen smaller than $1/2$.

On the other hand, if $\rho\geq\frac{R}{8}$, then it is sufficient to use
Proposition 4.2 twice and the fact that
$\left\|\nabla\gamma\right\|_{\infty}\leq C\varepsilon$ to estimate:

$\displaystyle|S^{\perp}(x-\gamma(x^{\prime\prime}))|\leq
C\varepsilon(R+\varepsilon+\varepsilon^{2})^{1/2}\leq C\varepsilon\rho^{1/2},$
$\displaystyle|S^{\perp}(y-\gamma(y^{\prime\prime}))|\leq
C\varepsilon(2R+\varepsilon+\varepsilon^{2})^{1/2}\leq
C\varepsilon\rho^{1/2},$
$\displaystyle|S^{\perp}(\gamma(y^{\prime\prime})-\gamma(x^{\prime\prime}))|\leq
C\varepsilon\rho.$

We therefore conclude

$|S^{\perp}(x-y)|\leq C\varepsilon\rho^{1/2}\leq
C\varepsilon\rho^{\varsigma},$

which is the desired result. ∎

##  7 $C^{1,\beta}$ regularity

In the present section, we prove the following $\varepsilon$-regularity
theorem:

######  Theorem 7.1 ($C^{1,\beta}$ regularity).

For every $E_{0}$ and $\alpha$, there are small constants
$\varepsilon_{0},\Lambda,\eta$ and $\beta$ with the following property. Let
$\Gamma\in\mathcal{F}_{\alpha}(\varepsilon_{0},B_{1})$,
$\mathbf{M}\in\mathcal{BF}_{m}(B_{1}\times[-\Lambda,0],\Gamma)$ be such that
$(0,0)\in\Sigma_{\mathbf{M}}$,

$\displaystyle\Sigma_{\mathbf{M}}\subset\\{(z,\tau)\colon\operatorname{dist}(z,S^{+})\leq\varepsilon_{0}\\},$
$\displaystyle\sup_{t\in[-\Lambda,0]}M_{t}(B_{1})\leq E_{0}$

and

$\int_{B_{1}}\Psi(\cdot,-\Lambda)\,dM_{-\Lambda}\leq\frac{3}{4}.$

Then $\Sigma_{\mathbf{M}}\cap Q_{\eta}$ is a $C^{1,\beta}$-regular single
sheet flow.

Before proving the above result, we record the following consequence of
Theorem 4.1

######  Proposition 7.2 (Iteration of the improvement of flatness).

Under the assumptions of Theorem 4.1, for every $X=(x,t)\in\Sigma\cap
Q_{\eta}$:

  * •

If $x\in\Gamma$, then there exists a $m$-dimensional half plane $T^{+}_{X}$
such that $\partial T^{+}_{X}=T_{x}\Gamma$ and

$\Sigma\cap
Q_{\eta^{k}}(X)\subset\\{\operatorname{dist}(\cdot,x+T^{+}_{X})\leq
2\varepsilon\eta^{k(1+\beta)}\\}$

for every $k\in\mathbb{N}$;

  * •

If $x\notin\Gamma$, then there exists a $m$-dimensional plane $T_{X}$ such
that

$\Sigma\cap Q_{\eta^{k}}(X)\subset\\{\operatorname{dist}(\cdot,x+T_{X})\leq
2\varepsilon\eta^{k(1+\beta)}\\}$

for every $k\in\mathbb{N}$.

Proof. This result is a straightforward consequence of an iteration of Theorem
4.1. Namely, given $X\in\Sigma\cap Q_{\eta}$ with $x\in\Gamma$, we may find a
sequence of half-planes $T^{+}_{k}$ such that

$\Sigma\cap
Q_{\eta^{k}}\subset\\{\operatorname{dist}(\cdot,T_{k}^{+})\leq\eta^{k(1+\beta)}\varepsilon\\}.$

Moreover, $|T_{k}^{+}-T^{+}_{k-1}|\leq
C\varepsilon\frac{\eta^{k(1+\beta)}}{\eta^{k}}$ for some $C$ depending only on
$E_{0}$ and $\alpha$. Therefore, $\\{T_{k}^{+}\\}$ converges to some half
plane $T_{X}^{+}$ for which the conclusion of the proposition holds true.

For the case $x\notin\Gamma$, one may see [15] or replicate the techniques of
the previous sections. ∎

######  Remark 7.3.

Given $x\in\Gamma$ and $T_{(x,t)}^{+}$ as in Proposition 7.2, throughout the
rest of the present section, we let $T_{(x,t)}$ be the $m$-dimensional plane
obtained by reflecting $T_{(x,t)}^{+}$ across $T_{x}\Gamma$. We remark the
following conclusion of Theorem 1.1: there is $C$ depending only on $E_{0}$
and $\alpha$ such that, for every $X\in\Sigma\cap Q_{\eta}$, it holds

$|T_{X}-S|\leq C\varepsilon.$

We are now ready to prove Theorem 7.1.

Proof of Theorem 7.1.  Up to a rotation, we may assume, without loss of
generality, that $T_{(0,0)}$ defined in Proposition 7.2 coincides with the
plane $S$ that satisfies the assumptions of the present result.

Step 1: Hölder decay of the tangent planes.  We prove that, for any two
$X=(x,t),Y=(y,s)\in\Sigma\cap Q_{\eta/4}$, it holds

$|T_{X}-T_{Y}|\leq C_{0}\varepsilon\rho(X,Y)^{\beta}$

for some $C_{0}$ universal large enough.

Let $\rho=\rho(X,Y)$ and, without loss of generality, say $t\geq s$. By
Proposition 7.2, since $Q_{\rho}(Y)\subset Q_{2\rho}(X)$,

$\Sigma\cap
Q_{\rho}(Y)\subset\bigg{\\{}(z,\tau):\operatorname{dist}(z-x,T_{X})\leq\varepsilon(2\rho)^{1+\beta}\bigg{\\}}.$

We may apply Proposition 7.2 to the flow $\mathcal{D}_{\rho}(\mathbf{M}-Y)$.
Then, by Remark 7.3, we have

$|T_{Y}-T_{X}|\leq C\rho^{\beta},$

as desired.

Step 2: $\Sigma$ is included in the graph of a $C^{1,\beta}$ function over
$S$. Let $X\in\Sigma\cap Q_{\eta}^{d}$ and, for simplicity of notation, let
$T=T_{X}$ as defined in Remark 7.3; recall that $|T_{X}-S|\leq C\varepsilon$.
For any other point $Y\in\Sigma\cap Q_{\eta}^{d}$, we may write

$\displaystyle|S^{\perp}(x-y)|$
$\displaystyle\leq|T^{\perp}(x-y)|+|S^{\perp}-T^{\perp}||x-y|$
$\displaystyle\leq C\varepsilon\rho(X,Y)^{1+\beta}+C\varepsilon\rho(X,Y)$
$\displaystyle\leq 2C\varepsilon\rho(X,Y).$

If $\varepsilon$ is smaller than some universal constant, we conclude

$|S^{\perp}(x-y)|\leq 3C\varepsilon\rho(SX,SY).$

In passing, we record that the above inequality, together with Proposition
7.2, yields

$|T_{X}^{\perp}(x-y)|\leq C\varepsilon\rho(SX,SY)^{1+\beta}.$ (7.1)

Now, it may be checked by direct computations that

$(I-S^{\perp}T)S^{\perp}(x-y)=S^{\perp}TS(x-y)+S^{\perp}T^{\perp}(x-y).$

By Remark 7.3, $|S-T|\leq C\varepsilon$, thus $(I-S^{\perp}T)$ is invertible
and $|(I-S^{\perp}T)^{-1}|\leq 2$ provided $\varepsilon$ is small enough. In
particular, by letting $L=(I-S^{\perp}T)^{-1}S^{\perp}$, we have $|L|\leq 2$
and

$\big{|}S^{\perp}x-S^{\perp}y-LT(Sx-Sy)\big{|}\leq|LT^{\perp}(x-y)|\leq
2|T^{\perp}(x-y)|\leq C\varepsilon\rho(SX,SY)^{1+\beta}.$ (7.2)

We may repeat the discussion above for any other couple of points $X,Y$. In
particular, by (7.1), $\Sigma\cap Q_{\eta}$ is included in the graph of some
function $u:S(\Sigma\cap Q_{\eta}^{d})\to{\mathbb{R}}^{d-m}$. Moreover, by
(7.2), $u$ is differentiable, and for every $X\in Q_{\eta}^{d}\cap\Sigma$, we
have

$\nabla
u(Sx)v^{\prime}=(I-S^{\perp}T_{X})^{-1}S^{\perp}T_{X}S^{-1}(v^{\prime})$

for every $v^{\prime}\in{\mathbb{R}}^{m}$. In particular, provided
$\varepsilon$ is small enough, it holds

$|\nabla u(SX)-\nabla u(SY)|\leq C|T_{X}-T_{Y}|\leq
C\varepsilon\rho(SX,SY)^{\beta}$

for every $X,Y\in\Sigma\cap Q_{\eta}^{d}$. It is straightforward to see that
$u$ can be extended to $Q_{\eta}^{m}$ and

$\left\|u\right\|_{C^{1,\alpha}(Q_{\eta})}\leq C\varepsilon.$

Step 3: conclusion. We let

$E_{+}:=\\{(x,t)\in Q_{\eta}^{d}\colon
x_{m}>\gamma_{m}(x^{\prime\prime})\\},\qquad E_{-}:=\\{(x,t)\in
Q_{\eta}^{d}\colon x_{m}<\gamma_{m}(x^{\prime\prime})\\}.$

By Step 2 and Theorem 3.10, since $\Gamma\cap E_{+}=\varnothing$, either
$(\Sigma\setminus\Gamma)\cap E_{+}=\mathop{\rm graph}u\cap E_{+}$ or
$(\Sigma\setminus\Gamma)\cap E_{+}=\varnothing$. The same holds true for the
component of $\Sigma\setminus\Gamma$ inside $E_{-}$; however, by Proposition
5.1 applied with $R=\eta$, it must be $(\Sigma\setminus\Gamma)\cap
E_{-}=\varnothing$. On the other hand, since $(0,0)\in\Sigma$, by Proposition
3.4 it cannot be $(\Sigma\setminus\Gamma)\cap E_{+}=\varnothing$ as well.
Therefore

$\Sigma\cap Q_{\eta}=\overline{E_{+}}\cap\mathop{\rm graph}u$

and

$M=\mathcal{H}^{m,1}\llcorner(\mathop{\rm graph}u\cap E_{+}).$

We conclude the proof by showing that $(x^{\prime},u(x^{\prime},t))\in\Gamma$
whenever $x^{\prime}\in S(\Gamma)$. Indeed, if that was not the case for some
$(x^{\prime},t)\in S(\Gamma)\times(-\eta^{2},0]$, then, by continuity of $u$,
$\mathop{\rm graph}u$ would detach from $\Gamma$ in a set of positive
$\mathcal{H}^{m-1,1}$ measure. This however would violate the fact that
$M_{t}\in\mathcal{V}_{m}(B_{\eta},\Gamma)$ for almost every $t$, thus reaching
a contradiction. ∎

##  Appendix A Proof of Lemma 3.2

Up to rescaling and translating, it is sufficient to prove that there exist
$A$ and $\Lambda$ small and positive, depending only on $\delta$ and $\alpha$,
such that, if $\Gamma$ is a $(m-1)$-dimensional properly embedded submanifold
of $B_{2}$ with $[\Gamma]_{C^{1,\alpha}(B_{2})}\leq A$, then

$\int_{-\Lambda}^{0}\int|T_{y}\Gamma^{\perp}\nabla\Psi_{1}(y,t)|\,d\Gamma(y)\,dt\leq\frac{1}{2}\chi_{\Gamma^{c}}(0)+\delta.$
(A.1)

For brevity, we denote by $\Gamma_{y}$ the space $T_{y}\Gamma$. Throughout the
proof, $C$ will denote constants (possibly changing from one expression to
another) depending only on $m,d,\alpha$.

Case 1: $0\in\Gamma$.  We start by remarking that

$\displaystyle\int_{-\Lambda}^{0}\int|\Gamma_{y}^{\perp}\nabla\Psi_{1}(\cdot,t)|\,d\Gamma\,dt$
$\displaystyle\qquad\leq\int_{-\Lambda}^{0}\int_{B_{2}}|\Gamma_{y}^{\perp}\nabla\Psi(\cdot,t)|\,d\Gamma(y)\,dt+C\Gamma(B_{2})\int_{-\Lambda}^{0}\frac{e^{1/(4t)}}{(-t)^{m/2}}\,dt.$

If $A$ is smaller than some universal constant, then $\Gamma(B_{2})\leq C$,
thus we may take $\Lambda$ small depending on $\delta$ so that the last term
in the above inequality is smaller than $\delta/2$. Therefore we reduce
ourselves to proving that, if $A$ is small, then

$\displaystyle
I_{1}:=\int_{-\Lambda}^{0}\int_{B_{2}}|\Gamma_{y}^{\perp}\nabla\Psi(\cdot,t)|\,d\Gamma\,dt\leq\frac{\delta}{2}.$

Since $[\Gamma]_{C^{1,\alpha}(B_{2})}\leq A$ is small, for every
$(y,t)\in\Gamma\times(-\infty,0)$:

$|\Gamma_{y}^{\perp}\nabla\Psi(y,t)|\leq
C\frac{e^{|y|^{2}/t}}{(-t)^{1+m/2}}|\Gamma_{y}^{\perp}y|\leq
CA\frac{e^{|y|^{2}/t}}{(-t)^{1+m/2}}|y|^{1+\alpha}.$

We then use the fact that, if $A$ is smaller than some universal constant,
then $\Gamma\cap B_{2}$ is the graph over $\Gamma_{0}$ of some function
$\gamma:{\mathbb{R}}^{m-1}\to{\mathbb{R}}^{d-m+1}$ such that
$\left\|\gamma\right\|_{C^{1,\alpha}}\leq CA$. In particular, by using the
area formula and the fact that the $||\nabla\gamma||_{L^{\infty}(B_{2})}\leq
1$ for $A$ small enough, we obtain

$\displaystyle\int_{B_{2}}|\Gamma_{y}^{\perp}\nabla\Psi(y,t)|\,d\Gamma(y)\leq
CA\frac{1}{(-t)^{1+m/2}}\int_{{\mathbb{R}}^{m-1}}|y|^{1+\alpha}e^{|y|^{2}/t}\,d\mathcal{L}^{m-1}(y)=CA(-t)^{\frac{\alpha}{2}-1}$

for some $C$ depending only on $m$ and $\alpha$. Therefore, assuming
$\Lambda\leq 1$,

$I_{1}\leq CA\int_{-1}^{0}(-t)^{\frac{\alpha}{2}-1}\,dt\leq CA.$

We conclude the proof in the case $0\in\Gamma$ by choosing
$A\leq\frac{\delta}{2C}$.

Case 2: $0\notin\Gamma$. Let $E_{\Gamma}$ be the $m$-dimensional Hausdorff
measure restricted to the exterior cone

$C_{\gamma}:=\\{\lambda y\colon\lambda\geq 1\mbox{ and }y\in\Gamma\\}$

with multiplicity, as defined in [20, Section 7]. With similar computations to
those in the proof of [20, Theorem 7.1], we may show that

$\displaystyle\int_{-\Lambda}^{0}\int|\Gamma_{y}^{\perp}\nabla\Psi_{1}(y,t)|\,d\Gamma(y)\,dt$
$\displaystyle\qquad\leq-\lim_{\tau\nearrow
0}\int\Psi_{1}(\cdot,\tau)\,dE_{\Gamma}+\int\Psi_{1}(\cdot,-\Lambda)\,dE_{\Gamma}+C\Lambda
E_{\Gamma}(B_{2})$
$\displaystyle\qquad=\int\Psi_{1}(\cdot,-\Lambda)\,dE_{\Gamma}+C\Lambda
E_{\Gamma}(B_{2}),$ (A.2)

Where the last equality comes from the fact that $0\notin\Gamma$.

In order to prove (A.1), we argue by contradiction: assume there is a sequence
$\\{\Gamma^{j}\\}$ with $0\notin\Gamma^{j}$ such that
$\left\|\Gamma^{j}\right\|_{C^{1,\alpha}(B_{2})}\leq\frac{1}{j}$ for which the
left-hand side of (A.2) is greater than $\frac{1}{2}+\delta$. One may show
that, up to extracting a subsequence, $E_{\Gamma^{j}}$ converges weakly to
$\mathcal{H}^{m}\llcorner S^{+}$, where $S^{+}$ is some $m$-dimensional half
plane such that $0\notin\operatorname{Int}(S^{+})$. Therefore

$\limsup_{j\to\infty}\Bigg{\\{}\int\Psi_{1}(\cdot,-\Lambda)E_{\Gamma^{j}}+C\Lambda
E_{\Gamma^{j}}(B_{2})\Bigg{\\}}\leq\int_{S^{+}}\Psi_{1}(\cdot,-\Lambda)\,d\mathcal{H}^{m}+C\Lambda\mathcal{H}^{m}(\overline{B_{2}}\cap
S^{+}).$

Since $0\notin\operatorname{Int}(S^{+})$, the integral in the right-hand side
of the above inequality is smaller than $\frac{1}{2}$ for every choice of
$\Lambda$. On the other hand, $\Lambda$ may be chosen so small that

$C\Lambda\mathcal{H}^{m}(\overline{B_{2}}\cap S^{+})\leq\frac{\delta}{2},$

which contradicts the assumption made above, thus concluding the proof.

##  Appendix B Proof of Lemma 3.6

We refer the reader to [9, Lemma 9.4] for a detailed proof of Lemma 3.6. Since
some minor modifications are needed, in this section we sketch the outline of
the proof.

Let $U\subset{\mathbb{R}}^{d}$ be open, $I\subset{\mathbb{R}}$ be a non-empty
interval, $\Gamma$ be a $(m-1)$-dimensional $C^{1,\alpha}$ submanifold of $U$
and let $\mathbf{M}\in\mathcal{BF}_{m}(U\times I,\Gamma)$. We assume that
$\mathbf{M}$ satisfies a bound of the form

$\operatorname{mdr}(\mathbf{M},U\times I)\leq E_{1}<\infty$ (B.1)

and we let $\Sigma=\Sigma_{\mathbf{M}}$ be its space-time support.

Before proceeding, by virtue of Proposition 3.4, we fix small constants
$c_{1}$, $c_{2}$ and $R_{0}$, depending on $E_{1}$ and $\Gamma$, such that for
every $(x,t)\in\Sigma$ and every $R\leq R_{0}$ such that
$B_{R}(x)\times(t-c_{1}R^{2},t)\subset\subset U\times I$, it holds

$M_{t-c_{1}R^{2}}(B_{R/2}(x))\geq c_{2}R^{m}.$

By Definition 2.3, for almost every $t\in I$ there exist a $m$-dimensional
rectifiable set $E\subset U$ and a positive, integer valued function
$\theta_{t}:E_{t}\to\mathbb{N}$ such that
$M_{t}=\theta_{t}(\cdot)\mathcal{H}^{m}\llcorner E_{t}$ We choose a time $t$
as above, with the additional condition that $s\mapsto M_{s}(\varphi)$ is
continuous at $t$ for every $\varphi\in C_{c}(U)$. By [15, Proposition 3.3],
almost every $t\in I$ satisfies the latter condition.

We claim that, for every such $t$ and for every $B_{3r}(x_{0})\subset\subset
U$, it holds

$\mathcal{H}^{m}((\Sigma_{t}\setminus E_{t})\cap B_{r}(x_{0}))=0;$ (B.2)

this clearly implies (3.6).

In order to prove (B.2), we argue by contradiction. Assume that there is
$(x_{0},t_{0})\in U\times I$ and $r>0$ such that $B_{3r}\subset\subset U$ and
$\mathcal{H}^{m}(A\cap B_{r}(x_{0}))>0$. Without loss of generality, we may
take $x_{0}=0$, $t=0$ and set $A:=\Sigma_{0}\setminus E_{0}$.

Let

$A_{k}:=\big{\\{}x\in A\cap B_{r}\colon M_{0}(B_{R}(x))\leq
c_{2}{R^{m}}/{2}\mbox{ for all }R\in(0,{r}/{k})\big{\\}}.$

Since, for $\mathcal{H}^{m}$-a.e. $x\in E_{0}^{c}$, it holds

$\lim_{R\searrow 0}\frac{M_{0}(B_{R}(x))}{R^{m}}=0,$

we have

$0<\mathcal{H}^{m}(A\cap
B_{r})=\mathcal{H}^{m}(\bigcup_{k\in\mathbb{N}}A_{k}).$

Therefore we may find $k\in\mathbb{N}$ such that
$b_{0}:=\mathcal{H}^{m}(A_{k})>0$.

By standard measure-theoretic arguments, it is not hard to show that there
exists $c$ small universal such that, for every $R$ small enough, we may find
$N\in\mathbb{N}$ and a finite collection of points
$\\{x_{j}\\}_{j=1}^{N}\subset A_{k}$ such that $\\{B_{R}(x_{j})\\}$ are
mutually disjoint and

$NR^{m}\geq cb_{0}.$ (B.3)

By definition of $A$, since $x_{j}\in A_{k}$, we have

$M_{0}\bigg{(}\bigcup_{j=1}^{N}B_{R}(x_{j})\bigg{)}\leq
Nc_{2}\frac{R^{m}}{2}.$ (B.4)

On the other hand, by Proposition 3.4 and the fact that $x_{j}\in\Sigma_{0}$,
we have

$M_{-c_{1}R^{2}}\bigg{(}\bigcup_{j=1}^{N}B_{R/2}(x_{j})\bigg{)}\geq
Nc_{2}R^{m}.$ (B.5)

We now fix a cut-off function $\varphi\in C^{\infty}_{c}(B_{1})$ such that
$\varphi\in[0,1]$ everywhere, $\varphi|_{B_{1/2}}\equiv 1$ and
$|\nabla\varphi|\leq 4$. Then, given $R$ small, we let
$\varphi_{0}(x)=\varphi(x/(2r))$, $\varphi_{j}(x)=\varphi((x-x_{j})/R)$ and

$\widetilde{\varphi}=\varphi_{0}-\sum_{j=1}^{N}\varphi_{j}.$

Then clearly $\widetilde{\varphi}\in[0,1]$ everywhere and
$|\nabla\widetilde{\varphi}|\leq C/R$. Notice, moreover, that

$\sum_{j=1}^{N}\chi_{B_{R/2}(x_{j})}\leq\sum_{j=1}^{N}\varphi_{j}\leq\sum_{j=1}^{N}\chi_{B_{R}(x_{j})}$

For brevity, set $s=-c_{1}R^{2}$. By (B.4) and (B.5), we have

$\displaystyle M_{0}(\varphi_{0})-M_{s}(\varphi_{0})$
$\displaystyle=\big{(}M_{0}(\widetilde{\varphi})-M_{s}(\widetilde{\varphi})\big{)}+\bigg{(}M_{0}\big{(}\sum\varphi_{j}\big{)}-M_{s}\big{(}\sum\varphi_{j}\big{)}\bigg{)}$
$\displaystyle\leq\big{(}M_{0}(\widetilde{\varphi})-M_{s}(\widetilde{\varphi})\big{)}+(Nc_{2}R^{m}/2-Nc_{2}R^{m})$
$\displaystyle\leq\big{(}M_{0}(\widetilde{\varphi})-M_{s}(\widetilde{\varphi})\big{)}-c_{3}b_{0}$
(B.6)

for some $c_{3}$ small, where (B.3) was used in the last inequality.

We now estimate, by using Definition 2.3,

$\displaystyle M_{0}(\widetilde{\varphi})-M_{s}(\widetilde{\varphi})$
$\displaystyle\leq\int_{s}^{0}\int
H\cdot\nabla\widetilde{\varphi}\,dM_{t}\,dt$
$\displaystyle\leq\bigg{(}\int_{s}^{0}\int_{B_{2r}}|H|^{2}\bigg{)}^{1/2}\bigg{(}\int_{s}^{0}\int|\nabla\widetilde{\varphi}|^{2}\bigg{)}^{1/2}.$
(B.7)

By (B.1) and the fact that $s=-c_{1}R^{m}$, we have, for some $C$ large,

$\int_{s}^{0}\int|\nabla\widetilde{\varphi}|^{2}\,dM_{t}\,dt\leq(-s)||\nabla\widetilde{\varphi}||_{\infty}M_{t}(B_{2r})\leq
CE_{1}r^{m},$

therefore (B.6) and (B.7) yield

$\int_{-c_{2}R^{2}}^{0}\int_{B_{2r}}|H|^{2}\,dM_{t}\,dt\geq\frac{1}{CE_{1}r^{m}}\bigg{(}M_{0}(\varphi_{0})-M_{-c_{1}R^{2}}(\varphi_{0})+c_{3}b_{0}\bigg{)}.$

By assumption, $t\mapsto M_{t}(\varphi_{0})$ is continuous at $0$. Thus we may
choose $R$ so small that the right-hand side of the above inequality is larger
than $c\frac{b_{0}}{E_{1}r^{m}}$ for some $c$ small enough.

Finally, we consider the function $\widehat{\varphi}=\varphi(x/(3r))$. By
Definition 2.3 and the Cauchy-Schwarz inequality, we have

$\displaystyle M_{0}(\widehat{\varphi})-M_{-c_{1}R^{2}}(\widehat{\varphi})$
$\displaystyle\leq\int_{-c_{1}R^{2}}^{0}\int_{B_{3r}}\big{(}-\widehat{\varphi}|H|^{2}+\nabla\widehat{\varphi}\cdot
H\big{)}\,dM_{t}\,dt$
$\displaystyle\leq-\int_{-c_{1}R^{2}}^{0}\int_{B_{3r}}\frac{1}{2}\widehat{\varphi}|H|^{2}\,dM_{t}\,dt+\int_{-c_{1}R^{2}}^{0}\int_{B_{3r}}\frac{|\nabla\widehat{\varphi}|^{2}}{2\widehat{\varphi}}\,dM_{t}\,dt$
$\displaystyle\leq-\frac{1}{2}\int_{-c_{1}R^{2}}^{0}\int_{B_{2r}}|H|^{2}\,dM_{t}\,dt+CE_{1}r^{m}R^{2}$
$\displaystyle\leq-c\frac{b_{0}}{E_{1}r^{m}}$

provided $R$ is chosen small enough. This contradicts the continuity of
$t\mapsto M_{t}(\widehat{\varphi})$ at $0$, thus concluding the proof.

## References

  * [1] William K. Allard  “On the first variation of a varifold”  In _Ann. of Math. (2)_ 95, 1972, pp. 417–491
  * [2] William K. Allard  “On the first variation of a varifold: boundary behavior”  In _Ann. of Math. (2)_ 101, 1975, pp. 418–446
  * [3] Luigi Ambrosio and H. Soner  “A measure-theoretic approach to higher codimension mean curvature flows”  In _Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)_ 25.1-2, 1997, pp. 27–49
  * [4] Theodora Bourni  “Allard-type boundary regularity for $C^{1,\alpha}$ boundaries”  In _Adv. Calc. Var._ 9.2, 2016, pp. 143–161
  * [5] Kenneth A. Brakke  “The motion of a surface by its mean curvature” 20, Mathematical Notes  Princeton University Press, Princeton, N.J., 1978
  * [6] Luis A. Caffarelli and Xavier Cabré  “Fully nonlinear elliptic equations” 43, American Mathematical Society Colloquium Publications  American Mathematical Society, Providence, RI, 1995
  * [7] Nick Edelen  “The free-boundary Brakke flow”  In _J. Reine Angew. Math._ 758, 2020, pp. 95–137
  * [8] Or Hershkovits and Brian White  “Avoidance for Set-Theoretic Solutions of Mean Curvature-Type Flows”  arXiv, 2018  URL: https://arxiv.org/abs/1809.03026
  * [9] Kota Kasai and Yoshihiro Tonegawa  “A general regularity theory for weak mean curvature flow”  In _Calc. Var. Partial Differential Equations_ 50.1-2, 2014, pp. 1–68
  * [10] N.. Krylov  “Lectures on elliptic and parabolic equations in Hölder spaces” 12, Graduate Studies in Mathematics  American Mathematical Society, Providence, RI, 1996
  * [11] O. Savin  “Viscosity solutions and the minimal surface system”  In _Nonlinear analysis in geometry and applied mathematics. Part 2_ 2, Harv. Univ. Cent. Math. Sci. Appl. Ser. Math.  Int. Press, Somerville, MA, 2018, pp. 135–145
  * [12] Ovidiu Savin  “Small perturbation solutions for elliptic equations”  In _Comm. Partial Differential Equations_ 32.4-6, 2007, pp. 557–578
  * [13] Leon Simon  “Lectures on geometric measure theory” 3, Proceedings of the Centre for Mathematical Analysis, Australian National University  Australian National University, Centre for Mathematical Analysis, Canberra, 1983
  * [14] Salvatore Stuvard and Yoshihiro Tonegawa  “An existence theorem for Brakke flow with fixed boundary conditions”  In _Calc. Var. Partial Differential Equations_ 60.1, 2021, pp. Paper No. 4353
  * [15] Yoshihiro Tonegawa  “Brakke’s mean curvature flow” An introduction, SpringerBriefs in Mathematics  Springer, Singapore, 2019
  * [16] Lihe Wang  “On the regularity theory of fully nonlinear parabolic equations. II”  In _Comm. Pure Appl. Math._ 45.2, 1992, pp. 141–178
  * [17] Yu Wang  “Small perturbation solutions for parabolic equations”  In _Indiana Univ. Math. J._ 62.2, 2013, pp. 671–697
  * [18] Brian White  “A local regularity theorem for mean curvature flow”  In _Ann. of Math. (2)_ 161.3, 2005, pp. 1487–1519
  * [19] Brian White  “Controlling area blow-up in minimal or bounded mean curvature varieties”  In _J. Differential Geom._ 102.3, 2016, pp. 501–535
  * [20] Brian White  “Mean curvature flow with boundary” Id/No 4  In _Ars Inven. Anal._ 2021, 2021, pp. 43
  * [21] Brian White  “The topology of hypersurfaces moving by mean curvature”  In _Comm. Anal. Geom._ 3.1-2, 1995, pp. 317–333