# Rotational symmetry of solutions of mean curvature flow coming out of a
double cone II

Letian Chen  Department of Mathematics, Johns Hopkins University, 3400 N.
Charles Street, Baltimore, MD 21218<EMAIL_ADDRESS>
(Date: January 25, 2022)

###### Abstract.

Given a double cone $\mathcal{C}$ with entropy at most two that is symmetric
across some hyperplane, we show that any integral Brakke flow coming out of
the cone must inherit the reflection symmetry for all time, provided the flow
is smooth for a short time. As a corollary we prove that any such flow coming
out of a rotationally symmetric double cone must stay rotationally symmetric
for all time. We also show the existence of a non-self-similar flow coming out
of a double cone with entropy at most two, and give an example of such a flow
with a finite time singularity. Additionally, we show the existence of self-
expanders with triple junctions, which are exceptions to our main theorem.

##  1\. Introduction

A family of properly embedded hypersurface $\\{\Sigma_{t}\\}_{t\in I}$
satisfies the mean curvature flow (MCF) equation if:

$\displaystyle\left(\frac{\partial x}{\partial
t}\right)^{\perp}=H_{\Sigma_{t}}(x),\;\;x\in\Sigma_{t}.$

Here $H_{\Sigma_{t}}$ is the mean curvature vector of $\Sigma_{t}$, $x$ is the
position vector and $\perp$ denotes the projection onto the normal bundle. In
this paper we study mean curvature flows (MCF) that come out of (smooth)
cones. Due to the singularity at the origin, there could be multiple distinct
mean curvature flows coming out of a given cone $\mathcal{C}$. We are
interested in how symmetries of the cone influence these solutions.

The simplest solutions coming out of cones are self-expanders. We say a
properly embedded hypersurface $\Sigma^{n}\subset\mathbb{R}^{n+1}$ is a self-
expander if

$\displaystyle H_{\Sigma}(x)=\frac{x^{\perp}}{2}.$ (1.1)

Equivalently, $\Sigma$ is an self-expander if and only if
$\\{\sqrt{t}\Sigma\\}_{t\in(0,\infty)}$ is a solution to the MCF. Self-
expanders are of particular importance in the study of singularities of MCF as
they arise naturally as models of how MCFs flow through conical singularities.

There exist, however, other solutions coming out of the cone that are not
self-similar. A class of such solutions was first sketched by Bernstein–Wang
[10] as Morse flow lines between unstable and stable self-expanders asymptotic
to the same cone. Note that the existence of such solution requires the
existence of an unstable self-expander asymptotic to $\mathcal{C}$, which is
not guaranteed.

In the present article, we study flows coming out of a double cone with
reflection symmetry. Our main result roughly says that any MCF coming out of a
suitable cone with a reflection symmetry across some hyperplane must inherit
the symmetry for all future times, provided the flow is initially smooth for a
short time (see Section 2 for terminologies).

######  Theorem 1.1.

Let $\Pi\subset\mathbb{R}^{n+1}$ be a hyperplane passing through the origin
and $\mathbb{H}$ an open half-space with $\partial\mathbb{H}=\Pi$. Let
$\mathcal{C}\subset\mathbb{R}^{n+1}$ be a smooth double cone with
$\lambda[\mathcal{C}]<2$ such that $\mathcal{C}\cap\mathbb{H}$ is a Lipschitz
graph over $\Pi$. Let $\mathcal{M}=\\{\mu_{t}\\}_{t\in[0,\infty)}$ be an
integral, unit-regular and cyclic Brakke flow coming out of $\mathcal{C}$;
that is,

$\displaystyle\lim_{t\to 0}\mu_{t}=\mathcal{H}^{n}\llcorner\mathcal{C}$

as Radon measures. Suppose also $\mathcal{M}$ is smooth on $(0,T)$ for some
$T>0$. If $\mathcal{C}$ is symmetric across $\Pi$, then so is $\mathcal{M}$
for $t\in[0,\infty)$. Moreover, $\mathcal{M}$ is smooth away from $\Pi$.

Applying the above to rotationally symmetric double cones we obtain:

######  Corollary 1.2.

Let $\mathcal{C}\subset\mathbb{R}^{n+1}$ be a smooth rotationally symmetric
double cone with $\lambda[\mathcal{C}]<2$ (see Section 2 for the precise
definitions). Let $\mathcal{M}=\\{\mu_{t}\\}_{t\in[0,\infty)}$ be an integral,
unit-regular and cyclic Brakke flow coming out of $\mathcal{C}$ that is smooth
on $(0,T)$. Then $\mathcal{M}$ is rotationally symmetric with the same axis of
symmetry as $\mathcal{C}$. Moreover, $\mathcal{M}$ is smooth away from its
axis of symmetry.

######  Remark.

In our previous work [16], we only showed that the flow is rotationally
symmetric up to the first singular time $T$, so the point of the current work
is to show that the symmetry still holds after any singularity, which, as part
of the conclusion, must lie on the axis of symmetry.

######  Remark.

The entropy condition $\lambda[\mathcal{C}]<2$ is likely redundant. See
Section 7.

In fact, if the cone $\mathcal{C}\subset\mathbb{R}^{n+1}$ is of the form

$\displaystyle
x_{1}^{2}=m^{2}\left(x_{2}^{2}+\cdots+x_{n+1}^{2}\right),\;\;m>0,$ (1.2)

where $m$ is the parameter determined by the cone angle, a much stronger
conclusion holds:

######  Corollary 1.3.

Suppose $\mathcal{C}$ is of the form Equation 1.2 and has entropy
$\lambda[\mathcal{C}]<2$. Suppose $\mathcal{M}$ is an integral, unit-regular
and cyclic Brakke flow coming out of $\mathcal{C}$. If $\mathcal{M}$ is smooth
on $(0,T)$, then $\mathcal{M}$ is rotationally symmetric across the
$x_{1}$-axis. The only possible singularity model of $\mathcal{M}$ is the
round cylinder $\mathbb{R}\times\mathbb{S}^{n-1}$. Moreover, there can be at
most one of such singularity, which, if it exists, must occur at the origin.

We can also apply 1.1 to cones with $O(p+1)\times O(n-p+1)$ symmetry to prove:

######  Corollary 1.4.

Suppose $n\geq 2$ and $1\leq p\leq n-1$. Let
$\mathcal{C}\subset\mathbb{R}^{n+2}$ be a cone invariant under $O(p+1)\times
O(n-p+1)$ with $\lambda[\mathcal{C}]<2$. Let
$\mathcal{M}=\\{\mu_{t}\\}_{t\in[0,\infty)}$ be an integral, unit-regular and
cyclic Brakke flow coming out of $\mathcal{C}$. If there is $T>0$ such that
$\mathcal{M}$ is smooth on $(0,T)$, then $\mathcal{M}$ inherits the
$O(p+1)\times O(n-p+1)$ symmetry (with the same axes of symmetry).

There are many other related works on the rotational symmetry of self-
expanders. Observe that if there is a unique self-expander asymptotic to a
rotationally symmetric cone, then it must inherit the rotational symmetry. The
first nontrivial result for double cones is obtained by Fong–McGrath [25].
They proved that a mean-convex self-expander (i.e. $H_{\Sigma}>0$) asymptotic
to a rotationally symmetric double cone is rotationally symmetric. This is
later generalized by Bernstein–Wang (Lemma 8.3 in [8]) to weakly stable self-
expanders. In our previous work [16], we showed in full generality that any
smooth self-expander asymptotic to a rotationally symmetric double cone is
rotationally symmetric. Rotational symmetry of self-expanding solitons of
other geometric flows has been studied by Chodosh [19] and Chodosh–Fong [14].

Next we briefly comment on some of the assumptions made in 1.1. First of all,
the only extra assumption over the smooth case is the entropy bound
$\lambda[\mathcal{C}]<2$. This is due to the complicated nature of
singularities of higher multiplicities arising from Brakke flows. In
particular, the entropy condition is essential to the maximum principle B.1
(in which the entropy controls the Gaussian density) and the proof of 3.2.

Secondly, the cyclicity of $\mathcal{M}$ is needed to ensure singularities
modeled on triple junctions do not appear along the flow. Standard Schauder
estimates and pseudolocality arguments can only guarantee smoothness outside
of a large ball (see for example A.2) but do not rule out formations of triple
junctions. This condition is explicitly used in 3.2. In fact, we will show
that there exists a self-expander with triple junctions in Section 6. Our
moving plane method will not work for such self-expanders. However, if we
assume that $\mathcal{C}$ is also symmetric across the hyperplane
perpendicular to its axis of symmetry, then there is still some hope that all
self-expanders with triple junctions are rotationally symmetric (See Section
7).

Finally, it is not immediately clear that our theorem contains more flows than
[16], which includes smooth self-expanders and low entropy flow lines of
Bernstein–Wang [10]. For this reason we will establish an existence result of
a non-self-similar flow coming out of a cone $\mathcal{C}$ with
$\lambda[\mathcal{C}]<2$. We warn the readers that, while the flow constructed
below agrees with the construction from [10] on the smooth part, they are not
necessarily the same flow. This is due to the lack of understanding how
expander mean convexity is preserved after the singularity.

######  Theorem 1.5.

Let $\mathcal{C}\subset\mathbb{R}^{n+1}$ be a smooth cone with
$\lambda[\mathcal{C}]<2$. Let $\Sigma$ be a self-expander
$C^{2,\alpha}$-asymptotic to $\mathcal{C}$. If $\Sigma$ is unstable, there
exists a non-self-similar immortal integral Brakke flow
$\mathcal{M}=\\{\mu_{t}\\}_{t\in(0,\infty)}$ such that

$\displaystyle\lim_{t\to 0}\mu_{t}=\mathcal{H}^{n}\llcorner\mathcal{C}.$

Moreover, there is $T>0$ such that $\mathcal{M}$ is smooth on $(0,T)$.

######  Remark.

This existence theorem is weaker than the one obtained in [2] in which no
entropy assumption is made, but is enough for our purposes. Thanks to the
entropy bound, it is much simpler to analyze our situation as the flows we
produce are automatically matching. A similar construction for self-shrinkers
is carried out by Chodosh–Choi–Mantoulidis–Schulze [13].

######  Remark.

Without the more restrictive entropy bound, it is not enough to produce the
flow line as a smooth MCF — Some unstable connected self-expanders under the
flow will necessarily disconnect in order to reach the stable double-disk
solution. See 1.3 and 5.7.

Let us now briefly discuss the proof of 1.1. As in [16], the proof is based on
the moving plane method, first used by Alexandrov to prove that embedded,
compact CMC hypersurfaces are round spheres. The method was then employed by
Gidas–Ni–Nirenberg [26] who proved radial symmetry of positive solutions to
the Yamabe problem, and by Schoen [38] who proved the uniqueness of catenoids.
Recently, there are a number of other results on MCF that utilize the moving
plane method, including [37] and [17]. Unfortunately since we are dealing with
potentially singular flows, these methods for smooth flows are no longer
sufficient. Instead we will use a novel variant without smoothness, recently
developed by [18] (for parabolic equations) and [28] (for elliptic equations).
The key ingredient in the proof is the Hopf lemma without smoothness B.2,
which allows us to upgrade regularity of the flow concurrently with the
symmetry.

In order to apply the (geometric) moving plane method to noncompact objects,
it is mandatory to have a well-controlled asymptote at infinity. In the
asymptotically cylindrical case [18] (see also the preceding [17]), a fine
neck analysis is carried out in order to determine the asymptotic expansion
near infinity. However, to our advantage, in the asymptotically conical case,
the asymptote near infinity is entirely determined by the given cone
$\mathcal{C}$, and it is an immediate consequence of the pseudolocality
theorem that our flows are nice $C^{2,\alpha}$ normal graphs over
$\mathcal{C}$ outside of a large ball with appropriate decay rates (see
Appendix A or the works of Bernstein–Wang [9] or [7]). Knowing this, we can
then carry out the moving plane method with the usual maximum principle and
Hopf lemma replaced by B.1 and B.2 respectively (barring some technicality,
e.g. the tameness assumption 3.2).

We will prove 1.1 in Section 3. In Section 4 we apply 1.1 to prove the various
corollaries mentioned above. In Section 5 we give a construction for 1.5. In
Section 6 we will prove an ODE existence result of self-expanders with triple
junctions, which shows that the cyclic assumption in 1.1 is indeed necessary.
In Appendix A we review the construction of smooth Morse flow line following
[10]. Finally in Appendix B we recall the key maximum principle and Hopf lemma
from Section 3 of [18].

### Acknowledgment

I would like to thank my advisor, Jacob Bernstein, for many helpful comments
on the paper and continuous encouragement. I would also like to thank Junfu
Yao for helpful conversations, Kyeongsu Choi and Or Hershkovits for explaining
some arguments in [18], and the referee for many constructive comments on the
first draft of this article.

##  2\. Preliminaries

###  2.1. Notations

Throughout the paper lower case letters such as $x$ denote points in
$\mathbb{R}^{n+1}$, while upper case letters such as $X$ denote points in the
spacetime $\mathbb{R}^{n+1}\times[0,\infty)$. $B_{r}(x)$ denotes the Euclidean
ball of radius $r$ centered at $x$, and $P(X,r)$ denotes the parabolic ball
centered at $X=(x,t)$ of radius $r$, i.e.

$\displaystyle P(X,r)=B_{r}(x)\times(t-r^{2},t].$

$\mathcal{T}_{r}(A)$ denotes the tubular neighborhood of
$A\subset\mathbb{R}^{n+1}$ of radius $r$. Finally, for
$x=(x^{\prime},x_{n+1})$, let

$\displaystyle
B_{r}^{n}(x)=\\{(y^{\prime},y_{n+1})\in\mathbb{R}^{n+1}\mid\left|y^{\prime}-x^{\prime}\right|<r,y_{n+1}=x_{n+1}\\},$

and $C_{r}(x)$ be the open cylinder of height $r$ over $B_{r}^{n}(x)$, i.e.

$\displaystyle
C_{r}(x)=\\{(y^{\prime},y_{n+1})\in\mathbb{R}^{n+1}\mid\left|x^{\prime}-y^{\prime}\right|<r,\left|x_{n+1}-y_{n+1}\right|<r\\}.$

By a (hyper)cone we mean a set $\mathcal{C}\subset\mathbb{R}^{n+1}$ that is
invariant under dilation, i.e. $\rho\mathcal{C}=\mathcal{C}$ for all $\rho>0$.
The link of $\mathcal{C}$ is
$\mathcal{L}(\mathcal{C})=\mathcal{C}\cap\mathbb{S}^{n}$. We say $\mathcal{C}$
is smooth if $\mathcal{L}(\mathcal{C})$ is a smooth hypersurface of
$\mathbb{S}^{n}$. A double cone is a cone $\mathcal{C}$ whose link
$\mathcal{L}(\mathcal{C})$ has two connected components lying in opposite
hemispheres of $\mathbb{S}^{n}$. A hypersurface $\Sigma$ is
$C^{k,\alpha}$-asymptotically conical to $\mathcal{C}$ if

$\displaystyle\lim_{\rho\to 0^{+}}\rho\Sigma=\mathcal{C}\text{ in
}C^{k,\alpha}_{loc}(\mathbb{R}^{n+1}\setminus\\{0\\}).$

We will simply say $\Sigma$ is asymptotically conical to $\mathcal{C}$ if it
is smoothly asymptotically conical. In our applications, the cone
$\mathcal{C}$ is almost always assumed to be smooth and the asymptotically
conical hypersurfaces will come from solutions to MCF, which are in fact
smoothly asymptotic to $\mathcal{C}$ by standard Schauder theory.

Given a hypersurface $\Sigma\subset\mathbb{R}^{n+1}$, the Gaussian surface
area of $\Sigma$ is

$\displaystyle
F[\Sigma]=\int_{\Sigma}e^{-\frac{\left|x\right|^{2}}{4}}d\mathcal{H}^{n}.$

Following Colding–Minicozzi [20], the entropy of $\Sigma$ is

$\displaystyle\lambda[\Sigma]=\sup_{\rho\in\mathbb{R}^{+},x_{0}\in\mathbb{R}^{n+1}}F[\rho\Sigma+x_{0}]=\sup_{\rho,x_{0}}\frac{1}{(4\pi\rho)^{n/2}}\int_{\Sigma}e^{-\frac{\left|x-x_{0}\right|^{2}}{4\rho}}d\mathcal{H}^{n}.$

By Huisken’s monotonicity formula, the entropy is nonincreasing along a MCF.

###  2.2. Integral Brakke flows

For the rest of the article we will be using notations from geometric measure
theory. We refer to [39] and [30] for the relevant definitions.

Since we are dealing with potentially singular MCF, we need to generalize the
classes of MCF to make sense of the flow and hence the symmetries past
singularities. The measure-theoretic generalization of MCF is the Brakke flow
[4] (see also the more recent book of Tonegawa [41]), which is a flow of
varifolds. Given an integral $n$-rectifiable Radon measure $\mu$, let $V(\mu)$
denote the associated integral varifold, and $H$ its generalized mean
curvature vector given by the formula:

$\displaystyle\int\operatorname{div}_{V(\mu)}Xd\mu=-\int H\cdot Xd\mu.$

where $X$ is a compactly supported $C^{1}$ vector field.

Given an open set $U\subset\mathbb{R}^{n+1}$, by an integral $n$-Brakke flow
in $U$ we mean a family of integral $n$-rectifiable Radon measures
$\mathcal{M}=\\{\mu_{t}\\}_{t\in I}$ such that:

  1. (a)

For a.e. $t\in I$, $V(\mu)$ has locally bounded first variation and its
generalized mean curvature vector $H$ is orthogonal to the approximating
tangent space of $V(\mu)$ $x$-a.e.

  2. (b)

For any bounded interval $[a,b]\subset I$ and compact set $K\subset U$,

$\displaystyle\int_{a}^{b}\int_{K}(1+\left|H\right|^{2})d\mu_{t}dt<\infty.$

  3. (c)

For $[a,b]\subset I$ and every $\phi\in
C_{c}^{1}(U\times[a,b],\mathbb{R}^{+})$,

$\displaystyle\int\phi d\mu_{b}-\int\phi
d\mu_{a}\leq\int_{a}^{b}\int\left(-\phi\left|H\right|^{2}+\left|H\right|\cdot\nabla\phi+\frac{\partial\phi}{\partial
t}\right)d\mu_{t}dt.$

Since we are working in codimension one we will drop the dependence on $n$ in
the definition above and simply refer to it as a Brakke flow.

Brakke flow has two main drawbacks. First, due to the inequality in condition
(c), a Brakke flow can vanish abruptly (in fact some evolution must involve
such vanishing). In order to avoid this technical difficulty in Section 5 we
will employ the notion of a matching flow which, in some sense, prevents
certain sudden loss of mass. Secondly, and as part of our motivation, Brakke
flow does not have to be unique. In fact, there could be multiple (smooth)
self-expanders coming out of a fixed cone, which is singular at the origin.

A Brakke flow $\mathcal{M}$ is unit-regular if, for a spacetime point
$X=(x,t)$, $\mathcal{M}$ is smooth and has no sudden mass loss if a tangent
flow at $X$ is a multiplicity one hyperplane. We say $\mathcal{M}$ is cyclic
if the associated mod-2 flat chain $[V(\mu_{t})]$ (see eg. [44]) has no
boundary. By works of Ilmanen [30], Brakke flows produced by elliptic
regularization are unit-regular and cyclic.

Throughout our presentation, given a Brakke flow
$\mathcal{M}=\\{\mu_{t}\\}_{t\in I}$, we will write
$\mathcal{M}_{t}=\operatorname{supp}\mu_{t}$ for $t\in I$.

##  3\. Reflection symmetry

In this section we prove 1.1. Fix a smooth double cone $\mathcal{C}$ with
$\lambda[\mathcal{C}]<2$ and an integral, unit-regular Brakke flow
$\mathcal{M}=\\{\mu_{t}\\}$ satisfying the assumptions of 1.1; that is,

$\displaystyle\lim_{t\to 0}\mu_{t}=\mathcal{H}^{n}\llcorner\mathcal{C},$

and $\mathcal{M}$ is smooth on $(0,T)$ for some $T>0$. Given a hyperplane
$\Pi$, if $\mathcal{C}\cap\mathbb{H}$ is graphical over $\Pi$ and symmetric
across $\Pi$, the main theorem in our previous work [16] implies that
$\mathcal{M}$ is symmetric across $\Pi$ until its first singular time, past
which the usual moving plane argument stops to work. For this reason, we will
prove 1.1 using a version of the moving plane method without assuming
smoothness, recently developed by [18] (see also [28]).

A technical ingredient we need for the moving plane method is the notion of
tameness, which we now define.

######  Definition 3.1 (Definition 3.1 in [18]).

For an integral Brakke flow $\mathcal{M}$ in $\mathbb{R}^{n+1}$, we say
$X\in\mathcal{M}$ is a tame point of the flow if the $-1$ time slice of every
tangent flow at $X$ is smooth with multiplicity one away from a singular set
$\mathcal{S}$ with $\mathcal{H}^{n-1}(\mathcal{S})=0$. We say $\mathcal{M}$ is
a tame flow if every point $X\in\mathcal{M}$ is a tame point.

For instance, a tame flow should not have a singularity modeled on a triple
junction (that is, three hyperplanes meeting at equal angles) or a
multiplicity two plane. Tameness is a key assumption to apply the Hopf lemma
without smoothness B.2, which, in turn, is crucial to the moving plane method.
The next proposition establishes tameness of $\mathcal{M}$.

######  Proposition 3.2.

Let $\mathcal{M}$ be as above. Then $\mathcal{M}$ is a tame flow.

###### Proof.

It suffices to check the definition. Let
$\mathcal{X}=\\{\nu_{t}\\}_{t\in(-\infty,0]}$ be a tangent flow at
$(x_{0},t_{0})$. Since $\lambda[\mathcal{C}]<2$, $\mathcal{X}$ has
multiplicity 1 (i.e. the Gaussian density is 1 $\mathcal{H}^{n-1}$-a.e. on
$\mathcal{X}$, $t$-a.e.).

If $\mathcal{X}$ is static or quasi-static, then
$\nu_{-1}=\mathcal{H}^{n}\llcorner\Gamma$ for some stationary cone $\Gamma$.
If $\Gamma$ splits off $(n-2)$-lines, then
$\nu_{-1}=\mu_{\mathbb{R}^{n-2}}\times\nu^{\prime}$ where $\nu^{\prime}$ is a
one-dimensional stationary cone in $\mathbb{R}^{2}$. Hence $\nu^{\prime}$ is a
union of half-rays. Since $\lambda[\nu^{\prime}]=\lambda[\nu_{-1}]<2$, there
are at most 3 rays. Since $\mathcal{M}$ is cyclic, $\nu^{\prime}$ cannot be 3
rays meeting at the origin. This is because the triple junction is not cyclic,
and a cyclic Brakke flow cannot have a singularity modeled on a non-cyclic
singularity by [44]. Therefore $\nu^{\prime}$ consists of 2 lines and in fact
$\nu_{-1}$ is smooth. So any singular cone $\nu_{-1}$ can split off at most
$(n-3)$-lines, and consequently the singular part of $\nu_{-1}$ has
codimension at least 3.

If $\mathcal{X}$ is a non-flat self-shrinker, then any tangent cone
$\nu^{\prime}$ to $\nu$ is a stationary cone with entropy at most 2 (here we
used the fact that self-shrinkers are minimal surfaces with respect to the
metric $g_{ij}=e^{-\frac{\left|x\right|^{2}}{2n}}\delta_{ij}$ which is
conformal to the Euclidean metric). It follows from the above discussion that
$\nu^{\prime}$ can split off at most $(n-3)$-lines (if $\nu^{\prime}$ splits
of $(n-2)$-lines then it is a multiplicity 1 hyperplane, which, by Allard
regularity theorem, means that $\mathcal{S}(\mathcal{X})\subset\\{0\\}$ and
consequently $\mathcal{X}$ is a multiplicity 1 hyperplane), and so $\nu_{-1}$
is smooth away from a set of Hausdorff dimension at most $(n-2)$. Thus
$\mathcal{M}$ is tame. ∎

######  Remark.

With a more restrictive entropy bound it is possible to refine the codimension
of the singular set even more. See [6] or Section 4 of [18].

Next we establish some properties of $\mathcal{M}$. The next proposition says
that the flow $\mathcal{M}$ stays asymptotically conical for all future time.

######  Proposition 3.3.

Let $\mathcal{M}$ be as above. Then $\mathcal{M}$ is asymptotically conical to
$\mathcal{C}$ for all $t\in(0,\infty)$. Consequently, for every $t>0$ there is
$R=R(t)$ such that $\mathcal{M}_{t}\setminus B_{R}(0)$ is a smooth MCF.

###### Proof.

This follows from pseudolocality for Brakke flows and parabolic Schauder
estimates as in A.2 (see also Proposition 4.4 of [7]). ∎

Fix an open half space $\mathbb{H}$ and let $\Pi=\partial\mathbb{H}$. We now
use the pseudolocality theorem to prove that if $\mathcal{C}$ is graphical
over $\Pi$, then so is $\mathcal{M}$ outside of a large compact set. This will
serve as an asymptotic expansion at infinity and be upgraded into interior
graphicality via the moving plane method. For the next two lemmas we write
$\mathcal{M}_{t}^{+}=\mathcal{M}_{t}\cap\mathbb{H}$.

######  Lemma 3.4.

Suppose $\mathcal{C}\cap\mathbb{H}$ is a Lipschitz graph over $\Pi$. let
$\mathcal{M}$ be as above, then for every $t>0$ there is $R=R(t)$ and a smooth
function $u$ on $\mathcal{C}$ such that

$\displaystyle\mathcal{M}_{t}^{+}\setminus
B_{R}(0)\subset\\{p+u(p)\nu_{\mathcal{C}}(p)\mid p\in\mathcal{C}\\}.$

and $\left|u(p)\right|\leq C\left|p\right|^{-1}$ for $p\in\mathcal{C}$ for
some constant $C=C(t)$.

###### Proof.

Fix a time $t_{0}$. Let us first show that $\mathcal{M}_{t}^{+}\setminus
B_{R}(0)$ can be written as a smooth normal graph over $\mathcal{C}$. By 3.3,
there exists $R=R(t)>0$ such that $\mathcal{M}_{t}\setminus B_{R}(0)$ is
asymptotically conical to $\mathcal{C}$. Since $\mathcal{M}_{0}=\mathcal{C}$,
by pseudolocality theorem for Brakke flows (Theorem 1.5 of [31]), given
$\eta>0$ there exists $t_{1}$ such that for $0<t<t_{1}$ and
$x\in\mathcal{C}\setminus B_{1}(0)$, $\mathcal{M}_{t}\cap C_{\sqrt{t_{1}}}(x)$
can be written as a normal graph over $B_{\sqrt{t_{1}}}^{n}(x)\cap
T_{x}\mathcal{C}$ with Lipschitz constant bounded by $\eta$. By parabolic
rescaling, we see that, for $0<t<2t_{0}$ and $x\in\mathcal{C}\setminus
B_{\sqrt{2t_{0}t_{1}^{-1}}}(0)$, $\mathcal{M}_{t}\cap C_{\sqrt{2t_{0}}}(x)$
can be written as a normal graph over $B_{\sqrt{2t_{0}}}^{n}(x)$ with
Lipschitz constant bounded by $\eta$. In particular putting $t=t_{0}$ gives
the desired graphicality. The regularity of $u$ follows from 3.3.

To see that the function $u$ decays near infinity, by a similar argument as in
A.1, there exists $N$ such that for all $R>1$ we have

$\displaystyle\mathcal{M}_{t_{0}}\setminus
B_{NR\sqrt{t_{0}+1}}(0)\subset\mathcal{T}_{R^{-1}\sqrt{t_{0}+1}}(\mathcal{C}).$

Equivalently, for $R>N\sqrt{t_{0}+1}$,

$\displaystyle\mathcal{M}_{t_{0}}\setminus
B_{R^{\prime}}(0)\subset\mathcal{T}_{N(t_{0}+1)(R^{\prime})^{-1}}(\mathcal{C}).$

Enlarge $R$ if needed so that $\mathcal{M}_{t_{0}}^{+}\setminus B_{R}(0)$ is a
normal graph over $\mathcal{C}$. We see that $u$ satisfies
$\left|u(p)\right|\leq C\left|p\right|^{-1}$ for $p\in\mathcal{C}$ and
$u(p)\in\mathcal{M}_{t}^{+}\setminus B_{2R}(0)$. ∎

######  Lemma 3.5.

Suppose $\mathcal{C}\cap\mathbb{H}$ is a Lipschitz graph over $\Pi$. Let
$\mathcal{M}$ be as above. For every $t>0$ there is $R=R(t)$ such that
$\mathcal{M}_{t}^{+}\setminus B_{R}(0)$ can be written as a graph over $\Pi$;
that is, the projection $\pi:\mathcal{M}_{t}^{+}\setminus B_{R}(0)\to\Pi$ is
injective.

###### Proof.

By 3.4, for every $\eta>0$, there is $R=R(t)$ such that
$\mathcal{M}_{t}^{+}\setminus B_{R}(0)$ is a normal graph over $\mathcal{C}$
with Lipschitz constant bounded by $\eta$. Since $\mathcal{C}\cap\mathbb{H}$
is a Lipschitz graph over $\Pi$, the unit normal vector $\nu_{\Pi}$ is not
contained in any tangent space to
$x^{\prime}\in(\mathcal{C}\cap\mathbb{H})\setminus\\{0\\}$. Therefore by
taking $\eta$ sufficiently small we may make sure that $\nu_{\Pi}$ is also not
contained in any tangent space to $x\in\mathcal{M}_{t}^{+}\setminus B_{R}(0)$
(here $R=R(t,\eta)$, but of course $\eta$ in turn depends on $t$). This proves
that $\mathcal{M}_{t}^{+}\setminus B_{R}(0)$ is graphical over $\Pi$ as well.
∎

3.2 and 3.5 allow us to use the moving plane method without smoothness, which
we now carry out. Let

$\displaystyle\Pi^{s}=\\{(x,x_{n+1})\in\mathbb{R}^{n+1}\mid
x_{n+1}=s\\}\times[0,\infty)\subset\mathbb{R}^{n+1}\times[0,\infty)$

be the hyperplane at level $s$ in spacetime. Given a set
$A\subset\mathbb{R}^{n+1}\times[0,\infty)$ and $s\in[0,\infty)$ we let

$\displaystyle A^{s+}=\\{(x,x_{n+1},t)\in A\mid x_{n+1}>s\\}\text{ and
}A^{s-}=\\{(x,x_{n+1},t)\in A\mid x_{n+1}<s\\}$

be the parts of $A$ lying above $\Pi_{s}$ and below $\Pi_{s}$ respectively.
Finally, the set

$\displaystyle A^{s*}=\\{(x,x_{n+1},t)\mid(x,2s-x_{n+1},t)\in A\\}$

is the reflection of $A$ across $\Pi_{s}$. We say $A>B$ for
$A,B\subset\mathbb{R}^{n+1}\times[0,\infty)$ provided for any $(x,s,t)\in A$
and $(x,s^{\prime},t)\in B$ we have $s>s^{\prime}$. In contrast, a subscript
$t$ will continue to denote the time $t$ slice of a spacetime set.

To set up the proof of 1.1, WLOG we may assume the hyperplane is
$\\{x_{n+1}=0\\}$. Fix a time $T_{0}>0$. We consider $\mathcal{M}$ on
$[0,T_{0})$ as its spacetime track; namely,

$\displaystyle\mathcal{M}=\bigcup_{t=0}^{T_{0}}\mathcal{M}_{t}\times\\{t\\}\subset\mathbb{R}^{n+1}\times[0,T_{0}].$

Finally, let

$\displaystyle
S=\\{s\in(0,\infty)\mid(\mathcal{M}^{s+})^{*}>\mathcal{M}^{s-},\text{ and
}(\mathcal{M}^{s+})_{t}\text{ is graphical over }(\Pi^{s})_{t}\text{ for
}t\in[0,T_{0}]\\}.$

Here graphicality means that the projection
$\pi_{s}:(\mathcal{M}^{s+})_{t}\to(\Pi^{s})_{t}$ is injective for $t\in[0,T]$.
Since each $(\mathcal{M}^{s+})_{t}$ is countably $n$-rectifiable, graphicality
is equivalent to that the unit normal $e_{n+1}=(0,\ldots,0,1)$ of
$(\Pi^{s})_{t}$ is not contained in the approximate tangent space of
$(\mathcal{M}^{s+})_{t}$ for $t\in[0,T_{0}]$. Observe that
$(\mathcal{M}^{s+})^{*}$ is asymptotically conical to the translated cone
$(\mathcal{C}+2se_{n+1})\times[0,\infty)$ (in the sense of 3.4 — this ensures
that a hypersurface cannot be simultaneously asymptotic to two distinct
cones).

First we need a lemma about smoothness of the top part of the flow similar to
Proposition 7.4 of [18].

######  Lemma 3.6.

Suppose $s>0$ and $s\in S$. Then $\mathcal{M}^{s+}$ is a smooth MCF asymptotic
to $\mathcal{C}$. Moreover, every point on $\mathcal{M}\cap\\{x_{n+1}=s\\}$ is
a regular point of the flow.

###### Proof.

Let

$\displaystyle I_{s}=\\{s^{\prime}\geq s\mid\mathcal{M}^{s^{\prime}+}\text{ is
smooth}\\}.$

By 3.3, there is $R=R(t)$ such that $\mathcal{M}_{t}\setminus B_{R}(0)$ is a
smooth MCF asymptotic to $\mathcal{C}$. So for sufficiently large $s$
depending on $T_{0}$ we see that $\mathcal{M}^{s+}$ is asymptotic to
$\mathcal{C}\times[0,T_{0}]$. This shows $I_{s}$ is not empty.

Let $s_{0}=\inf I_{s}$. We first argue that
$\mathcal{M}\cap\\{x_{n+1}=s_{0}\\}$ consists of regular points. Let
$(x_{0},t_{0})\in\mathcal{M}\cap\\{x_{n+1}=s_{0}\\}$ and let $\mathcal{M}^{*}$
be the flow reflected across $\Pi_{s}$. We wish to apply the Hopf lemma B.2 to
$\mathcal{M}$, $\mathcal{M}^{*}$ and $\mathbb{H}=\\{x_{n}<s\\}$ to conclude
that $(x_{0},t_{0})$ is a regular point. To this end we must check that the
conditions are satisfied. Tameness follows from 3.2. We may also assume that
$\partial\mathbb{H}$ is not a tangent flow to either $\mathcal{M}$ or
$\mathcal{M}^{*}$ at $(x_{0},t_{0})$, because otherwise the entropy bound
together with Brakke regularity theorem implies $(x_{0},t_{0})$ is a regular
point. Finally, we claim $\operatorname{reg}\mathcal{M}_{t}\cap\mathbb{H}$ and
$\operatorname{reg}\mathcal{M}^{*}_{t}\cap\mathbb{H}$ are disjoint for $t$
sufficiently close to $t_{0}$. Suppose not, then there must be a first time of
contact:

$\displaystyle
t_{1}=\inf\\{t\mid(\mathcal{M}^{s_{0}-}_{t})\cap(\mathcal{M}^{s_{0}+})^{*}_{t}\cap\mathbb{H}\neq\emptyset\\}$
(3.1)

in $\mathbb{H}$. Given any point

$\displaystyle(x_{1},t_{1})\in(\mathcal{M}^{s_{0}-}_{t})\cap(\mathcal{M}^{s_{0}+})^{*}_{t}\subset\mathbb{H}.$

By definition of $s_{0}$ we know $(\mathcal{M}^{s_{0}+})^{*}_{t_{1}}$ is in
fact a smooth MCF, so maximum principle B.1 implies that
$\mathcal{M}^{s_{0}-}$ agrees with $(\mathcal{M}^{s_{0}+})^{*}$ in some
parabolic cylinder around $(x_{1},t_{1})$. The same reasoning applied to any
other point in $\mathcal{M}^{s_{0}-}\cap(\mathcal{M}^{s_{0}+})^{*}$ shows that
a connected component of $\mathcal{M}^{s_{0}-}$ agrees with a connected
component of $(\mathcal{M}^{s_{0}+})^{*}$. This implies that
$(\mathcal{M}^{s_{0}+})^{*}$ is simultaneously asymptotic to
$\mathcal{C}\times[0,\infty)$ and
$(\mathcal{C}+2s_{0}e_{n+1})\times[0,\infty)$, a contradiction. Hence the last
condition in order to apply B.2 is satisfied and we conclude that
$\mathcal{M}\cap\\{x_{n+1}=s_{0}\\}$ is regular.

Lastly we show that $s_{0}=s$. This is a consequence of the fact that
$\mathcal{M}\cap\Pi^{s_{0}}$ is compact. Using small balls as barriers similar
to A.1, one sees that there exists some constant $N_{1}$ such that

$\displaystyle\mathcal{M}_{t}\setminus
B_{N_{1}R\sqrt{t+1}}(0)\subset\mathcal{T}_{R^{-1}\sqrt{t+1}}(\mathcal{C})$

for $R>1$. On the other hand, for a fixed $t$ there is a constant $N_{2}$ such
that

$\displaystyle\mathcal{M}_{t}\cap
B_{N_{1}\sqrt{t+1}}(0)\subset\mathcal{T}_{N_{2}}(\mathcal{C}).$

as the first set is clearly compact. These two facts together imply the
existence of a constant $N_{3}$ such that

$\displaystyle\mathcal{M}_{t}\cap\\{x_{n+1}=s_{0}\\}\subset\mathcal{T}_{N_{3}}(\mathcal{C})\cap\\{x_{n+1}=s_{0}\\}.$

This shows that $\mathcal{M}_{t}\cap\\{x_{n+1}=s_{0}\\}$ is compact (as
$\mathcal{C}\cap\\{x_{n+1}=s_{0}\\}$ is compact), and since $t\in[0,T_{0}]$,
$\mathcal{M}\cap\Pi^{s_{0}}$ is compact as well. To finish the proof, note
that by the previous paragraph $\mathcal{M}\cap\\{x_{n+1}=s_{0}\\}$ consist of
regular points only. At each regular point $(x_{0},t_{0})$ there is some
$r=r(x_{0},t_{0})$ such that $\mathcal{M}$ is smooth in $P((x_{0},t_{0}),r)$.
Since $\mathcal{M}\cap\\{x_{n+1}=s_{0}\\}$ is compact, $r$ is uniformly
bounded below away from 0, and this is a contradiction unless $s_{0}=s$. ∎

###### Proof of 1.1.

To finish the proof we must show $S$ is nonempty, $S$ is open, and $S$ is
closed.

Again by 3.3, for sufficiently large $s$ we can make sure that
$\mathcal{M}^{s+}$ is a smooth MCF,

$\displaystyle(\mathcal{M}^{s+})^{*}\cap((\mathcal{M}^{s-})\cap\\{x_{n+1}\geq
0\\})=\emptyset,$

and that for any $(x,s_{1},t)\in(\mathcal{M}^{s+})^{*}$ and
$(x,s_{2},t)\in\mathcal{M}^{0-}$ it holds that $s_{1}-s_{2}\geq 2s-1$. These
two facts imply that for sufficiently large $s$ the inequality
$(\mathcal{M}^{s+})^{*}>\mathcal{M}^{s-}$ is valid. On the other hand, by 3.5,
there is $R=R(T_{0})$ such that $(\mathcal{M}^{0+})_{T_{0}}\setminus B_{R}(0)$
is graphical over $(\Pi^{0})_{T_{0}}$. So for $s>R$ we have
$(\mathcal{M}^{s+})_{t}$ is graphical over $(\Pi^{s})_{t}$ for all
$t\in[0,T_{0})$. This shows $S$ is not empty.

It is clear that $(\mathcal{M}^{s+})^{*}>\mathcal{M}^{s-}$ is an open
condition. To see that the graphicality condition is also an open condition,
let $\theta_{t}(x)$ be the angle between the unit normal to the approximate
tangent space at a point $x\in\mathcal{M}_{t}$ and $e_{n+1}$. Suppose that
$s\in S$, then graphicality is equivalent to $\theta_{t}(x)<\frac{\pi}{2}$ for
all $t\in[0,T_{0})$ and $x\in(\mathcal{M}^{s+})_{t}$. Since the flow
$\mathcal{M}$ is $C^{2,\alpha}$-asymptotically conical, for given $t$ there
exists $\varepsilon>0$ such that $\theta_{t}(x)<\pi/2$ for all
$x\in(\mathcal{M}^{s^{\prime}+})_{t}$ where
$\left|s^{\prime}-s\right|<\varepsilon$. Since the time interval is compact,
there is a universal $\varepsilon$ such that the above holds for all
$t\in[0,T_{0})$. This shows openness of $S$.

Finally we show $S$ is closed. Obviously if $s\in S$ then $[s,\infty)\subset
S$. So we assume $(s,\infty)\subset S$ and suppose for a contradiction that
$s\not\in S$. At level $s$, either
$(\mathcal{M}^{s+})^{*}\cap\mathcal{M}^{s-}\neq\emptyset$ or there is some
$t_{0}\in[0,T_{0})$ such that $(\mathcal{M}^{s+})_{t_{0}}$ fails to be
graphical over $(\Pi^{s})_{t_{0}}$.

In the first case, $s$ is necessarily the first level of contact. By choosing
$r$ small enough we can ensure $(\mathcal{M}^{s+})^{*}$ and $\mathcal{M}^{s-}$
are graphical in $P(X,r)$ where
$X\in(\mathcal{M}^{s+})^{*}\cap\mathcal{M}^{s-}$. Moreover, by 3.6, the
reflected part $(\mathcal{M}^{s+})^{*}$ is a smooth MCF, so all the conditions
of the maximum principle B.1 are satisfied (note that the Gaussian density
bound is automatic from the entropy bound). Applying B.1, we see
$(\mathcal{M}^{s+})^{*}$ and $\mathcal{M}^{s-}$ agree in a neighborhood of
$X$. Now an identical argument as in the proof of 3.6 shows that
$(\mathcal{M}^{s_{0}+})^{*}$ is simultaneously asymptotic to
$\mathcal{C}\times[0,\infty)$ and
$(\mathcal{C}+2s_{0}e_{n+1})\times[0,\infty)$, which is again a contradiction.

In the second case, WLOG we may assume $t_{0}$ is the first time the
graphicality condition fails. Then there necessarily exists a point
$X=(x,s,t_{0})\in\mathcal{M}_{t_{0}}\cap\\{x_{n+1}=s\\}$ whose tangent space
contains the vector $e_{n+1}$. We again check the condition to apply Hopf
lemma B.2 to $\mathcal{M}^{1}=(\mathcal{M}^{s+})^{*}$,
$\mathcal{M}^{2}=\mathcal{M}^{s-}$ and $\mathbb{H}=\\{x_{n+1}<s\\}$ as in the
proof of 3.6. Tameness follows from 3.2. Since $e_{n+1}$ is normal to the
hyperplane $\\{x_{n+1}=s\\}$ and $X$ is a regular point of $\mathcal{M}$ by
3.6, we see that $\partial\mathbb{H}$ is not the tangent flow to either
$\mathcal{M}^{1}$ or $\mathcal{M}^{2}$ (here we used the fact that the tangent
flow at a regular point agrees with the static flow of the tangent plane). The
disjointness of the regular parts of $\mathcal{M}^{1}$ and $\mathcal{M}^{2}$
in $\mathbb{H}$ follows identically as in the proof of 3.6. Hence, we may
apply B.2 to conclude that $\mathcal{M}^{1}$ and $\mathcal{M}^{2}$ have
distinct tangents, which is a contradiction since the tangent spaces agree at
$X$. This concludes the proof that $S$ is closed.

This shows that $S=(0,\infty)$. At $s=0$, one sees that the graphicality
condition is preserved (alternatively one can run the moving plane method from
the other side, i.e. $s<0$), but the strict inequality
$(\mathcal{M}^{0+})^{*}>\mathcal{M}^{0-}$ does not hold anymore, which implies
that $(\mathcal{M}^{0+})^{*}\cap\mathcal{M}^{0-}\neq\emptyset$. Applying the
maximum principle B.1 once again we conclude
$(\mathcal{M}^{0+})^{*}=\mathcal{M}^{0-}$, and this is the required reflection
symmetry across $\Pi^{0}=\\{x_{n+1}=0\\}$. ∎

##  4\. Rotational symmetry

In this section we prove 1.2 and 1.4. First let $\mathcal{C}$ be a smooth,
rotationally symmetric double cone. A typical example of such a $\mathcal{C}$
is given by Equation 1.2 which has the $x_{1}$ axis as its axis of symmetry.
Our theorem allows more general cones of the form

$\displaystyle
x_{1}^{2}=\begin{cases}m_{1}^{2}(x_{2}^{2}+x_{3}^{2}+\cdots+x_{n}^{2}),&x_{1}\geq
0\\\ m_{2}^{2}(x_{2}^{2}+x_{3}^{2}+\cdots+x_{n}^{2}),&x_{1}<0\end{cases},$

where $m_{1},m_{2}>0$ (i.e. the top and bottom parts of the cone can have
different cone angles). A cone $\mathcal{C}$ of the above form should also
satisfy $\lambda[\mathcal{C}]<2$, but this is not explicitly known. See
Section 7 for more on the entropy of cones.

###### Proof of 1.2.

WLOG we may assume the axis of symmetry is the $x_{1}$-axis. Observe that
rotational symmetry is equivalent to reflection symmetry across every
hyperplane containing the $x_{1}$-axis. Up to a rotation it suffices to show
that $\mathcal{M}$ is symmetric across the hyperplane $\\{x_{n+1}=0\\}$. Since
$\mathcal{C}$ is a smooth graph over $\\{x_{n+1}=0\\}$, the desired conclusion
follows from 1.1. ∎

###### Proof of 1.3.

The rotational symmetry is 1.2. Since $\mathcal{C}$ is symmetric across the
hyperplane $\\{x_{1}=0\\}$ as well, we can apply 1.1 to
$\mathbb{H}=\\{x_{1}>0\\}$ to conclude that $\mathcal{M}$ is smooth away from
$\\{x_{1}=0\\}$. Together with 1.2 we infer that the only possible singularity
of $\mathcal{M}$ is at the origin. Moreover, any tangent flow $\mathcal{X}$ at
the first singular time must be rotationally symmetric. By the classification
of rotationally symmetric self-shrinkers of Kleene and Møller [33],
$\mathcal{X}$ has to be one of the following: a round sphere, a round cylinder
$\mathbb{R}\times\mathbb{S}^{n-1}$ or a smooth embedded
$\mathbb{S}^{1}\times\mathbb{S}^{n-1}$. Since $\mathcal{M}$ is not closed, we
conclude that $\mathcal{X}$ has to be a round cylinder. The uniqueness of the
cylinder follows from the work of Colding–Minicozzi [21]. ∎

######  Remark.

The above corollary does not guarantee the existence of a cylindrical
singularity, as it is entirely possible that the flow remains smooth for all
times. See 5.7 for sufficient conditions for the flow to have a singularity.

Next we apply the same method to cones with more general symmetry groups. For
this part we will work, for convenience, in $\mathbb{R}^{n+2}$ instead. Let
$O(p)$ denote the symmetry group of $\mathbb{S}^{p-1}\subset\mathbb{R}^{p}$.
Fix an integer $1\leq p\leq n-1$ and suppose $\mathcal{C}$ is a smooth double
cone with $\lambda[\mathcal{C}]<2$ that has symmetry group $O(p+1)\times
O(n-p+1)$. Typical examples are cones $\mathcal{C}_{n,p}$ over the families of
minimal hypersurfaces in $\mathbb{S}^{n}$ given by

$\displaystyle\mathcal{S}_{n,p}=\sqrt{\frac{p}{n}}\mathbb{S}^{p}\times\sqrt{\frac{n-p}{n}}\mathbb{S}^{n-p}\subset\mathbb{S}^{n}.$

The cones $\mathcal{C}_{n,p}$ are known as Simons-type cones. An immediate
consequence of 1.4 is:

######  Corollary 4.1.

Any smooth self-expander coming out of a Simons-type cone $\mathcal{C}_{n,p}$
inherits the $O(p+1)\times O(n-p+1)$-symmetry of $\mathcal{C}_{n,p}$.

######  Remark.

Similar results for minimal surfaces have been obtained by Mazet [36], using
the elliptic moving plane method.

###### Proof of 1.4.

Write $(x_{1},\ldots,x_{p+1},y_{1},\ldots,y_{n-p+1})$ the standard coordinates
on $\mathbb{R}^{n+2}$. WLOG we may assume
$\mathcal{L}(\mathcal{C)}=\sigma_{1}\times\sigma_{2}$ where $\sigma_{1}$ is
rotationally symmetric across the $x_{1}$ axis and $\sigma_{2}$ rotationally
symmetric across the the $y_{1}$-axis. Evidently showing the rotational
symmetry in $x$-coordinates is enough, as the identical argument works for the
$y$-coordinates. It suffices to show that the reflection symmetry is preserved
through all hyperplanes of the form

$\displaystyle\sum_{i=2}^{p+1}c_{i}x_{i}=0,$

which, up to an ambient rotation in the $x$-coordinates, we may assume to be
$\\{x_{p+1}=0\\}$. Note that the a cone with $O(p+1)\times O(n-p+1)$ symmetry
takes the form (up to relabeling)

$\displaystyle\sum_{i=1}^{p+1}a_{i}x_{i}^{2}=\sum_{j=1}^{n-p+1}b_{j}y_{j}^{2}$

for suitable choices of coefficients $a_{i}$ and $b_{j}$. It is not hard to
see that $\mathcal{C}\cap\\{x_{p+1}>0\\}$ is a graph over $\\{x_{p+1}=0\\}$
via

$\displaystyle
x_{p+1}=a_{p+1}^{-1/2}\left(\sum_{j=1}^{n-p+1}b_{j}y_{j}^{2}-\sum_{i=1}^{p}a_{i}x_{i}^{2}\right)^{1/2}.$

Hence 1.1 applies and the desired reflection symmetry follows (of course, we
have to put

$\displaystyle\Pi^{s}=\\{(x,x_{p+1},y)\in\mathbb{R}^{n+2}\mid
x_{p+1}=s\\}\times[0,\infty)\subset\mathbb{R}^{n+2}\times[0,\infty).$

and change the dimensions in the proofs accordingly). ∎

##  5\. Construction of the matching motion

In this section we use the ideas of Bernstein–Wang in [10] and [7] to produce
an immortal Brakke flow starting from a cone $\mathcal{C}$ with
$\lambda[\mathcal{C}]<2$ that is not self-expanding. This demonstrates that
1.2 is not void.

###  5.1. Self-expanders

Let us briefly summarize some basic facts about self-expanders. It is often
helpful to consider the variational characterization of self-expanders.
Formally, self-expanders are critical points of the functional:

$\displaystyle
E[\Sigma]=\int_{\Sigma}e^{\frac{\left|x\right|^{2}}{4}}d\mathcal{H}^{n}$

and equation Equation 1.1 corresponds to the Euler-Lagrange equation of
$E[\Sigma]$. We record the second variation of $E$. For a proof see for
example Proposition 4.2 in [12].

######  Theorem 5.1.

If $\\{\phi_{t}\\}_{t\in(-\varepsilon,\varepsilon)}$ is a compactly supported
normal variation of $\Sigma$ with
$\left.\frac{d\phi_{t}}{dt}\right|_{t=0}=f\nu_{\Sigma}$, where $\nu_{\Sigma}$
is the outwards unit normal. Then

$\displaystyle\left.\frac{d^{2}}{dt^{2}}\right|_{t=0}E[\phi_{t}(\Sigma)]=-\int_{\Sigma}fL_{\Sigma}fd\mathcal{H}^{n}$

where $L_{\Sigma}$ is the stability operator of $\Sigma$ given by

$\displaystyle
L_{\Sigma}=\Delta_{\Sigma}+\frac{1}{2}x\cdot\nabla_{\Sigma}+\left|A_{\Sigma}\right|^{2}-\frac{1}{2}.$

A real number $\mu\in\mathbb{R}$ is an eigenvalue for $-L_{\Sigma}$ if there
is a function $u\in W^{1}_{\frac{1}{4}}(\Sigma)\setminus\\{0\\}$ such that
$-L_{\Sigma}u=\mu u$, where

$\displaystyle
W^{1}_{\frac{1}{4}}(\Sigma)=\\{f:\Sigma\to\mathbb{R}\mid\int_{\Sigma}(\left|f\right|^{2}+\left|\nabla
f\right|^{2})e^{\frac{\left|x\right|^{2}}{4}}d\mathcal{H}^{n}<\infty\\}.$

The index of a self-expander is the number of negative eigenvalues of
$-L_{\Sigma}$, which is equal to

$\displaystyle\sup\\{\dim V\mid V\text{ linear subspace }\subset
C_{0}^{2}(\Sigma),-\int_{\Sigma}fL_{\Sigma}f\leq 0\;\forall f\in
V\setminus\\{0\\}\\}.$

We say a self-expander is stable if it has index zero. By Lemma 4.1 in [8],
the operator $L_{\Sigma}$ is formally adjoint in
$W^{0}_{\frac{1}{4}}(\Sigma)$, and $L_{\Sigma}$ has a discrete spectrum; that
is, the eigenvalues of $-L_{\Sigma}$ can be ordered as
$\mu_{1}<\mu_{2}<\cdots<\mu_{n}<\cdots$. Moreover, the space of eigenfunctions
associated to the lowest eigenvalue $\mu_{1}$ is 1-dimensional, and any
eigenfunction $f$ of $\mu_{1}$ has a sign.

We need the following basic distance estimates of asymptotically conical self-
expanders.

######  Proposition 5.2.

Let $\mathcal{C}$ be a smooth cone. Suppose $\Sigma$ is a self-expander
$C^{2,\alpha}$-asymptotic to $\mathcal{C}$, then there is $N>0$ such that
$\Sigma\setminus B_{NR}(0)\subset\mathcal{T}_{R^{-1}}(\mathcal{C})$ for $R>1$.

###### Proof.

Since $\Sigma$ is smooth and $\rho\Sigma\to\mathcal{C}$ in
$C^{2,\alpha}_{loc}(\mathbb{R}^{n}\setminus\\{0\\})$, for sufficiently small
$\rho$ we have on $\rho\Sigma\cap(B_{1}(0)\setminus\\{0\\})$

$\displaystyle\left|A_{\rho\Sigma}\right|\leq C.$

Hence for $x\in\Sigma$ with $\left|x\right|$ sufficiently large depending on
the above,

$\displaystyle\left|A(x)\right|=\rho\left|A_{\rho\Sigma}(\rho x)\right|\leq
C\left|x\right|^{-1}$

if we pick $\rho=\frac{1}{2}\left|x\right|^{-1}$ so that $\rho x\in B_{1}(0)$.
This proves that $\left|A(x)\right|\leq C\left|x\right|^{-1}$ for all
$x\in\Sigma$. Together with the self-expander equation, these imply that there
is $C>0$ with

$\displaystyle\operatorname{dist}(x,\mathcal{C})<C\left|x\right|^{-1}$

for $x\in\mathcal{C}\setminus B_{1}(0)$. Finally by scaling it follows that
there is $N$ such that for $R\geq 1$.

$\displaystyle\Sigma\setminus
B_{NR}(0)\subset\mathcal{T}_{R^{-1}}(\mathcal{C}).$

######  Remark.

Note that similar to the above we can also estimate the derivatives of $A$:

$\displaystyle\left|\nabla^{m}A\right|\leq C\left|x\right|^{-m-1},$

provided the cone is sufficiently regular.

###  5.2. Mean curvature flow with boundary

Since we are dealing with noncompact initial hypersurfaces, many existence
theorems (in particular the unit density theorem 11.4 in [30]) do not apply
directly in our case. To account for this we will utilize White’s recent work
on MCF with boundary [45], which is a generalization of the Brakke flow in
Section 2.2. For simplicity we will only work in the ambient manifold $\mkern
1.5mu\overline{\mkern-1.5muB_{R}(0)\mkern-1.5mu}\mkern 1.5mu$.

Given a hypersurface $\Sigma$ with boundary $\Gamma\subset\partial B_{R}(0)$
in an open set $U\subset\mkern
1.5mu\overline{\mkern-1.5muB_{R}(0)\mkern-1.5mu}\mkern 1.5mu$ and an integral
$n$-rectifiable Radon measure $\mu$, the first variation formula becomes:

$\displaystyle\int\operatorname{div}_{V(\mu)}Xd\mu=-\int H\cdot
Xd\mu+\int\nu_{\mu}\cdot Xd(\mathcal{H}^{n-1}\llcorner\Gamma)$

for any compactly supported $C^{1}$ vector field $X$, where $H$ is the
generalized mean curvature vector and $\nu$ the approximating normal vector to
$\Gamma$. By an integral $n$-Brakke flow with boundary $\Gamma$ in $U$ we mean
a family of integral $n$-rectifiable Radon measures
$\mathcal{M}=\\{\mu_{t}\\}_{t\in I}$ satisfying the items (a), (b) and (c) in
Section 2.2 with the extra condition:

  1. (d)

For a.e. $t\in I$ the normal vector satisfies $\left|\nu_{\mu_{t}}\right|\leq
1$ $\mathcal{H}^{n-1}$-a.e on $\Gamma$.

For simplicity we will refer to the above as Brakke flow with boundary
$\Gamma$. This is unambiguous since, by item (d), the boundary $\Gamma$ stays
unchanged under the Brakke flow.

As before, a Brakke flow with boundary $\Gamma$ is unit-regular if, for a
spacetime point $X=(x,t)$, $\mathcal{M}$ is smooth and has no sudden mass loss
if a tangent flow at $X$ is a multiplicity one plane. $\mathcal{M}$ is cyclic
if the associated mod-2 flat chain $[V(\mu_{t})]$ has boundary equal to
$\Gamma$. By works of White [45], Brakke flows with boundary $\Gamma$ produced
by elliptic regularization are unit-regular and cyclic.

######  Theorem 5.3 (Theorem 1.1, Theorem 14.1 of [45]).

Let $\Sigma\subset\mkern
1.5mu\overline{\mkern-1.5muB_{R}(0)\mkern-1.5mu}\mkern 1.5mu$ be a
hypersurface with boundary $\Gamma\subset\partial B_{R}(0)$. There exists a
unit-regular and cyclic Brakke flow with boundary $\Gamma$,
$\mathcal{M}=\\{\mu_{t}\\}_{t\in[0,\infty)}$ with
$\mu_{0}=\mathcal{H}^{n}\llcorner\Sigma$.

Similarly Brakke flow with boundary needs not be unique, but White’s theorem
says that a unit-regular and cyclic one always exists. White proved in
addition a strong boundary regularity theorem (Theorem 17.1 of [45]) in the
codimension one case, ruling out a scenario where interior singularities could
accumulate to a boundary singularity. Hence the boundary $\Gamma$ truly
remains unchanged in the classical sense.

###  5.3. Level set flows and matching motions

The final ingredient we need is the set-theoretic generalization of MCF,
initially developed by [15] and [24] as viscosity solutions to certain PDEs.
Given a closed set $\Gamma_{0}\subset\mathbb{R}^{n+1}$, we choose any
uniformly continuous function $u_{0}$ such that
$\Gamma_{0}=\\{x\in\mathbb{R}^{n+1}\mid u_{0}(x)=0\\}$. There exists a unique
$u\in C(\mathbb{R}^{n+1}\times[0,\infty))$ which is a viscosity solution to
the problem

$\displaystyle\begin{cases}u_{t}=\sum_{i,j=1}^{n+1}\left(\delta_{ij}-\frac{u_{x_{i}}u_{x_{j}}}{\left|\nabla
u\right|^{2}}\right)u_{x_{i}x_{j}}&\text{ on
}\mathbb{R}^{n+1}\times[0,\infty)\\\ u(x,0)=u_{0}(x)&\text{ on
}\mathbb{R}^{n+1}\times\\{0\\}.\end{cases}$

Let $\Gamma_{t}=\\{x\in\mathbb{R}^{n+1}\mid u(x,t)=0\\}$. We call
$\mathcal{K}=\bigcup_{t\in[0,\infty)}\Gamma_{t}\times\\{t\\}$ the level set
flow of $\Gamma_{0}$.

Since the viscosity solution is unique, level set flow is also unique with
given initial data. However, level set flows might fatten, i.e. $\mathcal{K}$
might develop a non-empty interior (for example the figure eight fattens
immediately). Formally, the level set flow of $\Gamma_{0}$ fattens if
$\mathcal{H}^{n+1}(\Gamma_{t})>0$ for some $t>0$. A theorem of Ilmanen (11.3
in [30]) shows that fattening phenomenon is not generic and can therefore be
perturbed away.

Alternatively, it has been observed that the level set flow can be
characterized as the ”biggest flow” of a closed set satisfying the avoidance
principle. There is a rich literature on this more geometrically intuitive way
of handling set flows and we refer to [43], [29] and [2] for more information
on this approach.

Ilmanen [30] combined ideas from Brakke flows and level set flows and
introduced the notion of a matching motion, which turns out to be the suitable
notion for our purposes. Let $I_{n}(U)$ be the set of integral $n$-current in
$U$.

######  Definition 5.4 (8.1, 9.1 of Ilmanen [30]).

Let $\mathcal{K}\in I_{n+1}(\mathbb{R}^{n+1}\times\mathbb{R}^{+})$,
$\mathcal{M}=\\{\mu_{t}\\}_{t\in[0,\infty)}$ be a Brakke flow and
$\Gamma_{0}\in I_{n}(\mathbb{R}^{n+1})$ with finite mass and empty boundary. A
pair $(\mathcal{K},\mathcal{M})$ is an enhanced motion with initial data
$\Gamma_{0}$ if

  1. (a)

$\partial\mathcal{K}=\Gamma_{0}$ and $\mathcal{K}_{t}\in
I_{n}(\mathbb{R}^{n+1})$ for a.e. $t\geq 0$;

  2. (b)

$\partial\mathcal{K}_{t}=0$ and $t\to\mathcal{K}_{t}$ is continuous in the
flat topology for $t\geq 0$;

  3. (c)

$\mu_{0}=\mu_{\Gamma_{0}}$, $\mathbb{M}[\mu_{t}]\leq\mathbb{M}[\mu_{0}]$ and
$\mu_{\mathcal{K}_{t}}\leq\mu_{t}$ for a.e. $t\geq 0$.

If the pair $(\mathcal{K},\mathcal{M})$ further satisfies

  1. (d)

$\mu_{t}=\mu_{\mathcal{K}_{t}}=\mu_{V(\mu_{t})}$ for $t\geq 0$,

then we call it a matching motion.

Note that in the above definition we have already abused some notation.
Indeed, in our applications $\mathcal{K}$ is going to be the level set flow
from $\Gamma_{0}$. A fundamental result of Ilmanen (Section 12 of [30]) shows
that a nonfattening level set flow is a matching motion, which justifies our
abuse of notation here.

We will use the following result of S. Wang [42] which asserts that limit of
low entropy matching motions is a matching motion. Recall that a sequence of
Brakke flow $\mathcal{M}_{i}=\\{\mu_{t}^{i}\\}_{t\in[0,\infty)}$ converges to
$\mathcal{M}=\\{\mu_{t}\\}_{t\in[0,\infty)}$ if $\mu_{t}^{i}\to\mu_{t}$ as
Radon measures and, after possibly passing to a subsequence,
$V(\mu_{t}^{i})\to V(\mu_{t})$ as varifolds for a.e $t\in[0,\infty)$.

######  Proposition 5.5 (Theorem 3.5 of [42]).

Let $(\mathcal{K}_{i},\mathcal{M}_{i})$ be a sequence of matching motions
converging to an enhanced motion $(\mathcal{K},\mathcal{M})$ with
$\lambda[\mathcal{M}]<2$, then $(\mathcal{K},\mathcal{M})$ is a matching
motion.

######  Remark.

The theorem fails without the entropy assumption as the set-theoretic limit of
a sequence of grim reapers (which has entropy 2) is two lines but the limit in
the sense of currents is empty (as the two lines cancel each other).

###  5.4. Construction of the smooth flow

Before we construct the weak flow, we briefly summarize the construction of
Morse flow line starting from an unstable self-expander [10]. For reader’s
convenience we have also included a summary of the theorems with some proofs
in Appendix A.

Recall that for an unstable self-expander $\Sigma$, the stability operator
$L_{\Sigma}$ has discrete spectrum and eigenfunctions to the lowest eigenvalue
$\mu_{1}<0$ have a sign. Let $f$ be the unique positive eigenfunction of
$\mu_{1}$ with $\left\lVert f\right\rVert_{W^{0}_{\frac{1}{4}}(\Sigma)}=1$.
For $\varepsilon>0$ we form the perturbations of $\Sigma$ by $f$ given by

$\displaystyle\Sigma^{\varepsilon}=\Psi^{\varepsilon}(\Sigma)\text{ where
}\Psi^{\varepsilon}(x)=x+\varepsilon f(x)\nu_{\Sigma}.$ (5.1)

By A.4, there is $N>0$ depending on $\varepsilon$ such that
$\Sigma^{\varepsilon}\setminus
B_{NR}(0)\subset\mathcal{T}_{R^{-1}}(\mathcal{C})$ for all $R>1$. By A.5,
$\Sigma^{\varepsilon}$ is expander mean-convex (see Appendix A for the
definition) for $\varepsilon$ sufficiently small. By the existence theorem A.3
there is a unique MCF starting from an asymptotically conical, expander mean-
convex hypersurface, and the expander-mean-convexity is preserved along the
flow until the first singular time. Applying A.3 to $\Sigma^{\varepsilon}$, we
get for each $\varepsilon$ a unique MCF
$\mathcal{M}^{\varepsilon}=\\{\Sigma^{\varepsilon}_{t}\\}_{t\in[1,T^{\varepsilon})}$
with $\Sigma^{\varepsilon}_{1}=\Sigma^{\varepsilon}$. Moreover, by A.4 and
5.2, $\Sigma^{\varepsilon}$ has uniformly bounded curvature, so the interior
estimates of Ecker–Huisken [23] implies that the interval of existence of is
independent of $\varepsilon$. Moreover, at first singular time
$T^{\varepsilon}$,

$\displaystyle\lim_{t\to T^{\varepsilon}}\sup_{\Sigma^{\varepsilon}_{t}\cap
B_{N^{\prime}\sqrt{t}}}\left|A_{\Sigma_{t}^{\varepsilon}}\right|=\infty$

for some constant $N^{\prime}>0$.

###  5.5. Construction of the Brakke flow

We now turn to our construction of the weak flow. One of the method to
construct a weak flow is via capping off the hypersurfaces
$\Sigma^{\varepsilon}\cap\mkern
1.5mu\overline{\mkern-1.5muB_{R}(0)\mkern-1.5mu}\mkern 1.5mu$ smoothly and
taking a sequence of Brakke flows (or weak set flows) starting from these
capped-off hypersurfaces. However, one has to be careful with the above method
when working with entropy bounds, since the cap might increase the entropy. On
the other hand, the capping method has the advantage that, by a suitable
choice of the caps, the expander mean-convexity is preserved through the cap
and hence in the limits (here with weak set flows one has to interpret the
mean-convexity in the weak sense as well). Such a construction for self-
shrinkers can be found in Section 7 of [13] and, for self-expanders, in [2].

Here we use an alternative approach of Brakke flow with boundary. This
construction is less technical and respects the entropy well. However, it is
also less clear how expander mean-convexity is preserved through the flows and
hence in the limit. This is not needed in our case because we are only
concerned with existence. It is an interesting question to determine whether
the flows constructed from the two methods above agree (they are, of course,
the same notion when smooth, so the point is to determine how each of them
flows past singularities). We believe this should be the case with the entropy
bound $\lambda[\mathcal{C}]<2$, but the general picture might be less clear.

The following proposition is the weak flow analogy of the smooth flow produced
in A.3.

######  Proposition 5.6.

Let $\mathcal{C},\Sigma$ be as in 1.5, and $\Sigma^{\varepsilon}$ be as in
Equation 5.1. There exists $\varepsilon_{0}>0$ such that, for
$\left|\varepsilon\right|<\varepsilon_{0}$, there exists an immortal matching
motion $(\mathcal{K}^{\varepsilon},\mathcal{M}^{\varepsilon})$ where
$\mathcal{K}^{\varepsilon}=\\{\Gamma^{\varepsilon}_{t}\\}_{t\in[1,\infty)}$
and $\mathcal{M}^{\varepsilon}=\\{\mu^{\varepsilon}_{t}\\}_{t\in[1,\infty)}$
such that $\Gamma^{\varepsilon}_{1}=\Sigma^{\varepsilon}$ and
$\mu^{\varepsilon}_{1}=\mathcal{H}^{n}\llcorner\Sigma^{\varepsilon}$. Moreover
the flow $(\mathcal{K}^{\varepsilon},\mathcal{M}^{\varepsilon})$ agrees with
the smooth flow starting from $\Sigma^{\varepsilon}$ for
$t\in[1,T^{\varepsilon})$.

###### Proof.

Suppose $\varepsilon>0$, as the argument for $\varepsilon<0$ is identical. Let
$\Sigma^{\varepsilon,R}=\Sigma^{\varepsilon}\cap\mkern
1.5mu\overline{\mkern-1.5muB_{R}(0)\mkern-1.5mu}\mkern 1.5mu$ be the
hypersurface in $\mkern 1.5mu\overline{\mkern-1.5muB_{R}(0)\mkern-1.5mu}\mkern
1.5mu$ with boundary $\Sigma^{\varepsilon}\cap\partial B_{R}(0)$. By 5.3,
there exists an unit-regular and cyclic Brakke flow with boundary
$\mathcal{M}^{\varepsilon,R}=\\{\mu^{\varepsilon,R}_{t}\\}_{t\in[0,\infty)}$
starting from $\Sigma^{\varepsilon,R}$. The flow
$\mathcal{M}^{\varepsilon,R}\llcorner B_{R/2}(0)$ is therefore a (usual)
Brakke flow inside $B_{R/2}(0)$. Since nonfattening is generic, we may choose
a sequence $R_{i}\to\infty$ such that the associated level set flow of
$\mathcal{M}^{\varepsilon,R_{i}}\llcorner B_{R_{i}/2}(0)$ is nonfattening.
This produces a sequence of matching motions

$\displaystyle(\mathcal{K}^{\varepsilon,R_{i}},\mathcal{M}^{\varepsilon,R_{i}}\llcorner
B_{R_{i}/2}(0)).$

By compactness of Brakke flow we may now pass to a subsequence
$R_{i}\to\infty$ to obtain a limiting enhanced motion
$(\mathcal{K}^{\varepsilon},\mathcal{M}^{\varepsilon})$ in $\mathbb{R}^{n+1}$
starting from $\Sigma^{\varepsilon}$.

By Lemma 3.5 of [9], $\lambda[\Sigma]=\lambda[\mathcal{C}]<2$, and by Lemma
6.2 of [10], for every $\delta>0$ there exists $\varepsilon_{0}$ such that
$\left|\lambda[\Sigma^{\varepsilon}]-\lambda[\Sigma]\right|<\delta$ for
$\varepsilon<\varepsilon_{0}$. Choosing $\delta$ small enough so that
$\lambda[\mathcal{C}]+\delta<2$ and $\varepsilon_{0}$ small according to
$\delta$ ensures that
$\lambda[\mathcal{M}^{\varepsilon}]=\lambda[\Sigma^{\varepsilon}]<2$, and so,
in view of 5.5, $(\mathcal{K}^{\varepsilon},\mathcal{M}^{\varepsilon})$ is
matching.

Finally, using the argument in A.2 with pseudolocality of MCF replaced by that
of Brakke flow there exist $\delta>0$ and $N^{\prime}>0$ such that
$\operatorname{supp}\mu^{\varepsilon}_{t}\setminus
B_{N^{\prime}\sqrt{t}}(0)=\Sigma^{\varepsilon}_{t}\setminus
B_{N^{\prime}\sqrt{t}}(0)$ for $t\in[1,1+\delta]$. Since
$(\mathcal{K}^{\varepsilon},\mathcal{M}^{\varepsilon})$ is matching,
$\Gamma_{t}^{\varepsilon}\setminus
B_{N^{\prime}\sqrt{t}}(0)=\Sigma^{\varepsilon}_{t}\setminus
B_{N^{\prime}\sqrt{t}}(0)$ as well. It follows from uniqueness of level set
flow that $\Gamma_{t}^{\varepsilon}$ agrees with $\Sigma_{t}^{\varepsilon}$.
Using the matching property again we infer that
$\operatorname{supp}\mu_{t}^{\varepsilon}=\Sigma_{t}^{\varepsilon}$. It is
easy to see that these flow agree up to the first singular time of
$\Sigma^{\varepsilon}_{t}$ (i.e. $T^{\varepsilon}$). ∎

Let $(\mathcal{K}^{\varepsilon},\mathcal{M}^{\varepsilon})$ be the matching
motions constructed as above. We can once again take a limit as
$\varepsilon\to 0^{+}$ to obtain a limiting enhanced motion
$(\mathcal{K},\mathcal{M})$ where
$\mathcal{K}=\\{\Gamma_{t}\\}_{t\in[1,\infty)}$ and
$\mathcal{M}=\\{\mu_{t}\\}_{t\in[1,\infty)}$ such that $\Gamma_{0}=\Sigma$ and
$\mu_{0}=\mathcal{H}^{n}\llcorner\Sigma$. However, this limit is not enough to
prove 1.5 as we will only recover the flow of the self-expander $\Sigma$ (as
in the case of the smooth flow). Moreover, the limit does not attain the cone
as its initial data. We need to translate the flow properly so that the
starting time can be extended back to 0, and argue that we do not get the
original flow of the self-expander back in the process.

###### Proof of 1.5.

Again WLOG suppose $\varepsilon>0$. Let
$(\mathcal{K}^{\varepsilon},\mathcal{M}^{\varepsilon})$ be the matching
motions from 5.6. It is convenient to work with the following rescaled MCF:

$\displaystyle\tilde{\mu}^{\varepsilon}_{s}=\mu^{\varepsilon}_{t}\circ
t^{-\frac{1}{2}}\text{ and
}\tilde{\Gamma}^{\varepsilon}_{s}=\Gamma^{\varepsilon}_{t},\;\;s=\log t.$

and let
$(\tilde{\mathcal{K}}^{\varepsilon},\tilde{\mathcal{M}}^{\varepsilon})$ denote
the rescaled flow. Under the rescaling, a smooth MCF will satisfy the rescaled
MCF equation

$\displaystyle\left(\frac{\partial x}{\partial
s}\right)^{\perp}=H_{\Sigma_{s}}-\frac{x^{\perp}}{2},$

which has self-expanders as stable solutions. The flow
$\tilde{\mathcal{M}}^{\varepsilon}$ is defined on the time interval
$[0,\infty)$ with
$\operatorname{supp}\tilde{\mu}^{\varepsilon}_{0}=\Sigma^{\varepsilon}$. Since
$\Sigma$ is unstable, the lowest eigenvalue $\lambda_{1}$ of $-L_{\Sigma}$
satisfies $\lambda_{1}<\frac{1}{2}$, and consequently the
$\mathcal{M}^{\varepsilon}$ ”flows faster” than the parabolic rescaling
$\sqrt{t}$. To be precise, for the the flow of the self-expander $\Sigma$ we
have

$\displaystyle\lim_{\lambda\to
0^{+}}\operatorname{dist}(\Sigma,\lambda\Sigma_{\lambda^{-2}})=\lim_{\lambda\to
0^{+}}\operatorname{dist}(\Sigma,\lambda\sqrt{\lambda^{-2}}\Sigma)=0$

but since we are perturbing $\Sigma$ by its eigenfunction whose eigenvalue is
below $\frac{1}{2}$, we must have

$\displaystyle\lim_{\lambda\to
0^{+}}\operatorname{dist}(\Sigma^{\varepsilon},\lambda\operatorname{supp}\mu^{\varepsilon}_{\lambda^{-2}})=\infty.$

In the rescaled setting, the above means that $\tilde{M}^{\varepsilon}$ moves
out exponentially (the exact rate of which depends on $\lambda_{1}$). As such,
we can find a sequence of time $s_{\varepsilon}$ with

$\displaystyle
d(\operatorname{supp}\tilde{\mu}^{\varepsilon}_{s_{\varepsilon}},x_{0})=\gamma$

for some fixed point $x_{0}\in\Sigma$ and positive constant $\gamma$. On the
other hand $\varepsilon\to 0^{+}$, the rescaled flows
$\tilde{\mathcal{M}}^{\varepsilon}$ converge to the static rescaled flow of
$\Sigma$, so in fact $s_{\varepsilon}\to\infty$ as $\varepsilon\to 0^{+}$,
i.e. one has to go further in time to reach a distance $\gamma$ away from
$x_{0}\in\Sigma$. This fact allows us to time translate $\tilde{\mathcal{M}}$
by $-s_{\varepsilon}$ to obtain a sequence of rescaled MCFs
$\tilde{\mathcal{M}}^{\varepsilon,s_{\varepsilon}}=\\{\tilde{\mu}^{\varepsilon,s_{\varepsilon}}_{s}\\}_{s\in[-s_{\varepsilon},\infty)}$
such that

$\displaystyle\operatorname{supp}\tilde{\mu}^{\varepsilon,s_{\varepsilon}}_{-s_{\varepsilon}}=\Sigma^{\varepsilon}\text{
and }d(\operatorname{supp}\tilde{\mu}^{s,s_{\varepsilon}}_{0},x_{0})=\gamma.$

By compactness of Brakke flows we can take $\varepsilon\to 0^{+}$ to obtain a
limiting flow $\tilde{\mathcal{M}}$ defined on $(-\infty,\infty)$. It is easy
to see that $\tilde{\mathcal{M}}$ is not the flow of $\Sigma$ as
$d(\operatorname{supp}\tilde{\mu}_{0},\Sigma)\geq\gamma$. Moreover, since
$\Sigma$ is asymptotically conical and $\Sigma^{\varepsilon}\to\Sigma$
smoothly as $\varepsilon\to 0$, it follows that

$\displaystyle\lim_{s\to-\infty}\operatorname{supp}\tilde{\mu}_{s}=\lim_{\varepsilon\to
0}\operatorname{supp}\tilde{\mu}_{-s_{\varepsilon}}^{\varepsilon,s_{\varepsilon}}=\lim_{\varepsilon\to
0}\Sigma^{\varepsilon}=\Sigma.$

In view of the rescaling, this proves that the flow achieves $\mathcal{C}$ as
the initial data.

Since $\tilde{\mu}_{s}$ is integral, it follows from strong maximum principle
for varifolds [40] that
$\lim_{s\to-\infty}\tilde{\mu}_{s}=k\mathcal{H}^{n}\llcorner\Sigma$. Since
$\lambda[\mathcal{C}]<2$, $k=1$, so the desired regularity follows from Brakke
regularity theorem. ∎

###  5.6. An example of singularity formation

To conclude the section, we give a sufficient condition when there exists a
flow coming out of $\mathcal{C}$ that has a singularity. The proof makes heavy
use of the structure theory of self-expanders developed by Bernstein and Wang
in a series of papers starting from [12]. It would be interesting to see if a
simpler proof exists.

######  Proposition 5.7.

Suppose $\mathcal{C}\subset\mathbb{R}^{n+1}$, $2\leq n\leq 6$, is a smooth
double cone with $\lambda[\mathcal{C}]<2$ and that the two connected
components of $\mathcal{L}(\mathcal{C})$ are graphs over some (fixed)
hyperplane. Suppose that there is a connected self-expander asymptotic to
$\mathcal{C}$. Then there exists an integral Brakke flow coming out of
$\mathcal{C}$ that has a singularity in finite time.

######  Remark.

We note that the above is consistent with the topological uniqueness result of
[10]. Indeed, by Proposition 5.6 of [10], if
$\lambda[\mathcal{C}]<\lambda[\mathbb{S}^{n-1}\times\mathbb{R}]$, the flow
produced by 1.5 is smooth for all time. On the other hand, any such double
cone will not have a connected self-expander.

###### Proof.

Let $\sigma=\mathcal{L}(\mathcal{C})$, and let $W$ be the connected component
of $\mathbb{S}^{n}$ lying between the two connected components of $\sigma$. By
Corollary 1.2 of [12], the set of generic cones (in the sense that there is no
$C^{2}$-asymptotically self-expander with nontrivial Jacobi fields that fix
the infinity) whose link lie in $W$, is dense near $\mathcal{C}$. These facts
allow us to take a sequence of $C^{2,\alpha}$-hypersurfaces $\sigma_{i}$ in
$\mathbb{S}^{2}$ such that

  * •

$\sigma_{i}\to\sigma$ in $C^{2,\alpha}(\mathbb{S}^{n})$ as $i\to\infty$;

  * •

$\mathcal{C}_{i}$ is a generic, smooth double cone for all $i$, where
$\mathcal{C}_{i}$ is the cone over $\sigma_{i}$;

  * •

$\lambda[\mathcal{C}_{i}]<2$ for sufficiently large $i$, by Lemma 6.2 of [10].

From the above we immediately see that there exists a unique disconnected,
stable self-expander $\Gamma_{i}$ $C^{2,\alpha}$-asymptotic to
$\mathcal{C}_{i}$ (by evolution of entire graph [22]). We also see that
$\mathcal{C}_{i}\subset\Omega$, where $\Omega$ is the connected component of
$\mathbb{R}^{n+1}\setminus\mathcal{C}$ that contains $W$. Denote by
$\Sigma_{0}$ the connected self-expander asymptotic to $\mathcal{C}$. Using a
direct method with $\Sigma_{0}$ as the barrier, similar to Lemma 8.2 of [8],
we can find a connected self-expander asymptotic to $\mathcal{C}_{i}$ in
$\Omega^{\prime}$, where $\Omega^{\prime}$ is the connected component of
$\mathbb{R}^{n+1}\setminus\Sigma_{0}$ such that the outward unit normal of
$\mathcal{C}$ points into $\Omega^{\prime}$.

Since there exists a unique disconnected self-expander $\Gamma_{i}$ asymptotic
to $\mathcal{C}_{i}$, by the partial ordering of self-expanders asymptotic to
a fixed cone (Theorem 4.1 of [10]), we can pick an innermost connected self-
expander $\Sigma_{i}$ $C^{2,\alpha}$-asymptotic to $\mathcal{C}_{i}$ (i.e.
pick any $\Sigma_{i}$ such that the only self-expander lying on the inside of
$\Sigma_{i}$ is the disconnected $\Gamma_{i}$ \- note that $\Sigma_{i}$ might
not be unique). We claim that $\Sigma_{i}$ is unstable. If not, the mountain
pass theorem (Corollary 1.2 [5], this requires $2\leq n\leq 6$) and the
genericity of $\mathcal{C}_{i}$ imply the existence of an unstable self-
expander $\Sigma^{\prime}$ lying between $\Sigma_{i}$ and $\Gamma_{i}$.
$\Sigma^{\prime}$ must then be connected, but this contradicts the partial
ordering.

Since $\Sigma_{i}$ is unstable and $\lambda[\mathcal{C}_{i}]<2$, we can
produce using 1.5 an integral Brakke flow
$\mathcal{M}^{i}=\\{\mu^{i}_{t}\\}_{t\in(0,\infty)}$ that moves inwards
initially (by expander mean convexity) and satisfies $\lim_{t\to
0}\mu^{i}_{t}=\mathcal{H}^{n}\llcorner\mathcal{C}_{i}$. Suppose for a
contradiction that $\mathcal{M}^{i}$ is smooth for all $t$, then the flow is
expander mean convex (in the classical sense) and moves inwards for all time.
Moreover, using an almost identical argument as in Proposition 5.1(3) of [10],
the rescaled flow (rescaling as in the proof of 1.5) $\tilde{\mathcal{M}}^{i}$
converges as $s\to\infty$ to a smooth, stable self-expander asymptotic to
$\mathcal{C}_{i}$, which must lie inside $\Sigma_{i}$. Since $\Sigma_{i}$ is
an innermost connected self-expander, the stable limit must be $\Gamma_{i}$
which is disconnected, a contradiction.

Now let $s^{i}$ denote the first singular time of the rescaled flows
$\tilde{\mathcal{M}}^{i}$, and time translate $\tilde{\mathcal{M}}^{i}$ by
$-s^{i}$ to obtain rescaled flows $\tilde{\mathcal{M}}^{i,s^{i}}$ with a
singularity at time $0$. By compactness of MCF we obtain a rescaled flow
$\tilde{\mathcal{M}}$ such that
$\tilde{\mathcal{M}}^{i,s^{i}}\to\tilde{\mathcal{M}}$ subsequentially.
Rescaling back we see that, by upper semicontinuity of the Gaussian density,
$\mathcal{M}$ has its first singularity at time $t=1$. Finally, we claim that
$\mathcal{M}$ indeed comes out of the cone $\mathcal{C}$. As $\mathcal{M}$ is
smooth on $(0,1)$, it is enough to show that

$\displaystyle\lim_{t\to
0}\operatorname{supp}{\mu}_{t}\cap\mathbb{S}^{n}=\sigma\text{ in
}C^{2,\alpha}(\mathbb{S}^{n}).$ (5.2)

Since each $\mathcal{M}^{i}$ attains $\mathcal{C}_{i}$ as initial data and
$\mathcal{C}_{i}$ is $C^{2,\alpha}$-regular, we have

$\displaystyle\lim_{t\to
0}\operatorname{supp}\mu_{t}^{i}\cap\mathbb{S}^{n}=\sigma_{i}\text{ in
}C^{2,\alpha}(\mathbb{S}^{n}).$

As $\sigma_{i}\to\sigma$ in $C^{2,\alpha}(\mathbb{S}^{n})$, by a
diagonalization argument, we see that Equation 5.2 holds. This completes the
proof. ∎

Assuming 7.1, the assumptions in 5.7 are satisfied by cones of the type
Equation 1.2, given that the parameter $m$ is sufficiently small. In fact,
numerical computations do confirm that the cones
$x_{1}^{2}=m^{2}(x_{2}^{2}+x_{3}^{2})$ for $m\leq 1$ have entropy less than 2.
Moreover, by [1], there exists a connected self-expander for sufficiently
small $m$. In these cases, combining the above with 1.3, we have the much
stronger conclusion that any such cone has a potential evolution which
disconnects at a cylindrical singularity.

##  6\. Self expanders with triple junctions

An important question in the study of self-expander is to determine the number
of self-expanders coming out of a given cone. A classical result of
Ecker–Huisken [22] shows that there exists a unique self-expander coming out
of a graphical cone. In general, however, Angenent–Ilmanen–Chopp [1] showed
numerically that uniqueness fails for double cones. It is proved rigorously by
Helmensdorfer [27] that there are at least three distinct smooth self-
expanders asymptotic to a rotationally symmetric double cone of the form
Equation 1.2 provided the cone angle is sufficiently large (when the cone
angle is small, a barrier argument shows that uniqueness indeed holds — see
Lemma 8.1 of [8]).

In this section we prove a simple ODE result on the existence of two self-
expanders with triple junctions for rotationally symmetric double cones with
sufficiently large cone angle. The proof roughly follows the setup of
Helmensdorfer [27], although we do not need the clearing out lemma for MCF in
the following analysis. This provides an example of a singular self-expander,
and also illustrates that the cyclicity assumption in 1.1 is essential as
tameness (3.2) clearly fails for self-expanders with triple junction
singularities. However, the examples constructed below are in fact still
rotationally symmetric.

We consider cones $\mathcal{C}_{m}$ of the form Equation 1.2, where $m$ is the
parameter therein. Observe that $\mathcal{C}_{m}$ has a rotational symmetry
across the $x_{1}$-axis as well as a reflection symmetry across the
$\\{x_{1}=0\\}$ hyperplane.

Assume the expander $\Sigma$ has a triple junction singularity at a point
$(0,x_{0})$. Imposing rotational symmetry on $\Sigma$ across the $x_{1}$-axis
we may assume $x_{0}=(a,0,\ldots,0)$. Note also at a triple junction
singularity, the tangent cone is stationary and is therefore the union of
three half-lines meeting at an angle of $2\pi/3$. These observations reduce
the problem to finding a function $u:\mathbb{R}^{+}\to\mathbb{R}$ satisfying
the following ODE (written in spherical coordinates):

$\displaystyle\frac{u_{rr}}{1+u_{r}^{2}}-\frac{n-1}{u}+\frac{1}{2}ru_{r}-\frac{1}{2}u=0.$
(6.1)

with initial data $u(0)=a$ and $u^{\prime}(0)=\frac{\sqrt{3}}{3}$. The
solution is asymptotic to $\mathcal{C}_{m}$ if

$\displaystyle\lim_{r\to\infty}\frac{u(r)}{r}=\frac{1}{m}.$ (6.2)

Of course, by the usual ODE existence and uniqueness theorem, the solution to
the problem Equation 6.1 is unique with a given initial data $a$. First we
show that any solution to Equation 6.1 is asymptotically conical.

######  Proposition 6.1.

For any $a>0$, there is a (unique) $m=m(a)$ such that Equation 6.2 holds.

###### Proof.

First we prove $u>0$ for all $r>0$. Suppose for a contradiction that there is
$r_{0}$ such that $u(r_{0})<0$. Since initially $u$ and $u_{r}$ are both
positive, by the mean value theorem there must be a local maximum
$r_{1}\in(0,r_{0})$ with $u(r_{1})>0$, but at such an $r_{1}$ we can use
Equation 6.1 to get

$\displaystyle u_{rr}(r_{1})=\frac{n-1}{u(r_{1})}+\frac{1}{2}u(r_{1})>0,$

a contradiction. So $u$ is indeed positive. By the above calculation, this
implies that all critical points of $u$ are local minima.

To continue, let $\alpha(r)=\arctan(u/r)$. Differentiating, we obtain

$\displaystyle\alpha^{\prime}(r)=\frac{ru_{r}-u}{u^{2}+r^{2}}\text{ and
}\alpha^{\prime\prime}(r)=\frac{ru_{rr}}{u^{2}+r^{2}}-\frac{(ru_{r}-u)(2uu_{r}+2r)}{(u^{2}+r^{2})^{2}}.$

At a critical point $r_{0}$ of $\alpha(r)$, we have
$r_{0}u_{r}(r_{0})-u(r_{0})=0$ and Equation 6.1 implies that
$u_{rr}(r_{0})=\frac{n-1}{u(r_{0})}>0$. Hence
$\alpha^{\prime\prime}(r_{0})=\frac{ru_{rr}}{u^{2}+r^{2}}>0$. Therefore all
critical points of $\alpha(r)$ are local minima as well. Since $\alpha(r)$ is
bounded and only has local minima, monotone convergence theorem shows that
$\lim_{r\to\infty}\alpha(r)$ exists.

It remains to show that the $\lim_{r\to\infty}\alpha(r)\in(0,\frac{\pi}{2})$.
If the limit is 0, then $m=0$ and the MCF of $\Sigma$ is contained in the
level set flow of the $x_{1}$-axis, which disappears immediately. This is
impossible. If the limit is $\frac{\pi}{2}$, the MCF of $\Sigma$ is contained
in the level set flow of the hyperplane $\\{x_{1}=0\\}$, which is static. This
is again impossible by, say, the initial condition
$u_{r}(0)=\frac{\sqrt{3}}{3}$. ∎

Knowing the above, our problem becomes essentially a shooting problem: Given
$m$, we wish to find the appropriate initial condition $a$ so that the
solution to Equation 6.1 satisfies Equation 6.2. Let $u^{a}(r)$ be the
solution to Equation 6.1 with initial condition $u^{a}(0)=a$. Consider the
asymptotic cone angle parameter $m$ as a function of $a$:

$\displaystyle
m(a)=\lim_{r\to\infty}\frac{u^{a}(r)}{r}=\lim_{r\to\infty}u_{r}^{a}(r)\in(0,\infty).$

######  Proposition 6.2.

$m(a)$ is a continuous function on $(0,\infty)$.

###### Proof.

First we record that

$\displaystyle
u_{rr}(0)=\frac{4}{3}\left(\frac{n-1}{a}+\frac{1}{2}a\right)>0.$

From the proof of 6.1, we see that a critical point of $u$ must be a local
minimum, but since $u$ is initially increasing and smooth, there cannot be any
critical point at all. So $u$ is strictly increasing and we deduce that
$u_{r}>0$ for all $r>0$. On the other hand, l’Hôpital’s rule on Equation 6.2
yields $\lim_{r\to\infty}u_{r}=m$. Going back to Equation 6.1 and taking the
limit as $r\to\infty$ yield also $\lim_{r\to\infty}u_{rr}=0$.

Fix an $a\in(0,\infty)$. Clearly $u^{a}\geq a$ and $u_{r}^{a}$ is bounded
above. Therefore we may fix a constant $N$ with $\frac{1}{N}<a$ such that
$\left|a^{\prime}-a\right|<\frac{1}{N}$ implies that

$\displaystyle\frac{1}{u^{a^{\prime}}}\leq c$ (6.3)

for some constant $c$ depending on $a$ and $N$. We compute

$\displaystyle\left(\frac{u}{r}\right)_{r}=\frac{1}{r}\left(u_{r}-\frac{u}{r}\right)=\frac{2}{r^{2}}\left(\frac{n-1}{u}-\frac{u_{rr}}{1+(u_{r})^{2}}\right).$
(6.4)

For $\left|a-a^{\prime}\right|<\frac{1}{N}$ there are two cases. If
$u_{rr}^{a^{\prime}}$ is never zero, then it is always positive. This
immediately gives the bound:

$\displaystyle\left(\frac{u^{a^{\prime}}}{r}\right)_{r}\leq\frac{c}{r^{2}}.$
(6.5)

If $u_{rr}^{a^{\prime}}(r_{0})=0$ at some point $r_{0}$, differentiating
Equation 6.1 once we get

$\displaystyle\frac{1}{1+u_{r}^{2}}\left(u_{rrr}-\frac{2u_{r}(u_{rr})^{2}}{1+u_{r}^{2}}\right)+\frac{n-1}{u^{2}}u_{r}+\frac{1}{2}ru_{rr}=0.$
(6.6)

From this we immediately see that when $u_{rr}^{a^{\prime}}(r_{0})=0$,

$\displaystyle u_{rrr}^{a}(r_{0})=-(1+u_{r}^{2})\frac{n-1}{u^{2}}<0,$

so every critical point of $u_{r}^{a^{\prime}}$ is a local maximum, for which
there can be at most one of them. Therefore $r_{0}$ is the only zero of
$u_{rr}^{a^{\prime}}$. In this case, we see from Equation 6.6 that, at any
negative local minimum of $u_{rr}^{a^{\prime}}$, we have

$\displaystyle\frac{2u_{r}^{a^{\prime}}}{(1+(u_{r}^{a^{\prime}})^{2})^{2}}(u_{rr}^{a^{\prime}})^{2}\leq\frac{(n-1)u_{r}^{a^{\prime}}}{(u^{a^{\prime}})^{2}}\implies\frac{(u_{rr}^{a^{\prime}})^{2}}{(1+(u_{r}^{a^{\prime}})^{2})^{2}}\leq\frac{n-1}{2(u^{a^{\prime}})^{2}}\leq
c,$

where we used Equation 6.3. Going back to Equation 6.4, this yields the same
type of uniform upper bound as Equation 6.5 (up to increase $c$). Therefore we
conclude Equation 6.5 holds for all $\left|a-a^{\prime}\right|<\frac{1}{N}$.

Integrating Equation 6.5 from $r$ to $\infty$, we obtain the estimate:

$\displaystyle
m(a^{\prime})-\frac{u^{a^{\prime}}(r)}{r}<\frac{c}{r},\;\;\left|a^{\prime}-a\right|<\frac{1}{N}.$

Now given $\varepsilon>0$ we can pick $r_{0}>0$ such that
$c/r_{0}<\varepsilon/3$ and $\delta$ so small that
$\left|a^{\prime}-a\right|<\delta$ implies that
$\left|u^{a^{\prime}}-u^{a}\right|<\varepsilon/3$ on $(0,r_{0}]$ (this follows
from continuous dependence on initial data as we are now in a compact set).
Using the triangle inequality we get that

$\displaystyle\left|m(a)-m(a^{\prime})\right|\leq\left|m(a)-\frac{u^{a}(r_{0})}{r_{0}}\right|+\left|\frac{u^{a}(r_{0})}{r_{0}}-\frac{u^{a^{\prime}}(r_{0})}{r_{0}}\right|+\left|m(a^{\prime})-\frac{u^{a^{\prime}}(r_{0})}{r_{0}}\right|<\varepsilon.$

This finishes the proof of continuity. ∎

We are now in the position to prove the existence theorem.

######  Theorem 6.3.

There is an $M_{0}>0$ such that for all $M>M_{0}$, there exists at least two
distinct values $a_{1},a_{2}\in(0,\infty)$ depending on $M$ such that
$m(a_{1})=m(a_{2})=M$.

###### Proof.

We will show that $m(a)\to\infty$ both as $a\to 0$ and as $a\to\infty$. In
view of 6.2 this will prove the theorem.

Let us show that $m(a)\to\infty$ as $a\to\infty$. First of all, we show that
$u_{r}^{a}$ cannot be uniformly bounded above. Indeed if $u_{r}^{a}$ is
uniformly bounded above by some constant $C$, then $u-ru_{r}\geq a-C$ for
$r\in[0,1]$ and so $u_{rr}\geq\frac{1}{2}(a-C)$ on $[0,1]$ by Equation 6.1.
But then

$\displaystyle
u^{a}_{r}(1)=\frac{\sqrt{3}}{3}+\int_{0}^{1}u^{a}_{rr}(r)dr\geq\frac{\sqrt{3}}{3}+\frac{1}{2}(a-C)\to\infty$

as $a\to\infty$, a contradiction.

Recall from the proof of 6.2 that $u_{rr}^{a}$ can have at most one zero. If
there is a sequence $a_{i}\to\infty$ such that $u_{rr}^{a_{i}}$ has one zero
$r_{i}$, then Equation 6.4 immediately implies that

$\displaystyle\left(\frac{u^{a_{i}}}{r}\right)_{r}>0,r>r_{i}.$

Integrating the above from $r_{i}$ to $\infty$ we get

$\displaystyle m(a_{i})>\frac{u^{a_{i}}(r_{i})}{r_{i}}.$ (6.7)

On the other hand, at a zero of $u_{rr}$, Equation 6.1 gives that

$\displaystyle\frac{u^{a_{i}}(r_{i})}{r_{i}}=u^{a_{i}}_{r}(r_{i})-\frac{2(n-1)}{u^{a_{i}}(r_{i})r_{i}}$
(6.8)

Observe that $u_{r}^{a_{i}}(r_{i})=\sup_{r}u_{r}^{a_{i}}(r)$ because $r_{i}$
is a local maximum of $u^{a_{i}}_{r}$ and $u^{a_{i}}_{rr}(r)<0$ for all
$r>r_{i}$. Since $u_{r}^{a_{i}}$ is not uniformly bounded, we have that

$\displaystyle\lim_{i\to\infty}\frac{u^{a_{i}}(r_{i})}{r_{i}}=\lim_{i\to\infty}u^{a_{i}}_{r}(r_{i})-\frac{2(n-1)}{u^{a_{i}}(r_{i})r_{i}}=\infty,$

where we also used $u^{a_{i}}(r_{i})>a_{i}$. Recalling Equation 6.7, we see
that $\lim_{i\to\infty}m(a_{i})=\infty$.

Otherwise there is $a_{0}>0$ such that $u_{rr}^{a}$ has no zero for all
$a>a_{0}$. This means that $u_{r}^{a}$ is strictly increasing for all
$a>a_{0}$. Since $u_{r}^{a}$ is not uniformly bounded we can find a sequence
$a_{i}\to\infty$ and $\\{r_{i}\\}\subset[0,\infty)$ such that
$u_{r}^{a_{i}}(r_{i})>i$. Of course monotonicity implies that $m(a_{i})\geq
u_{r}^{a_{i}}(r_{i})>i$. This shows that $m(a)\to\infty$ as $a\to\infty$.

Next we will show that $m(a)\to\infty$ as $a\to 0$ as well. Again we will
first argue that $u_{r}^{a}$ cannot be uniformly bounded near $0$. Suppose for
a contradiction that there is $a_{0}>0$ such that $\left|u_{r}^{a}\right|\leq
C$ for all $0<a<a_{0}$, then since $u_{r}^{a}>0$ we get

$\displaystyle u^{a}(r)=a+\int_{0}^{r}u_{r}^{a}(t)dt\leq a+Cr,\;\;a<a_{0}.$
(6.9)

On the other hand, Equation 6.1 gives that

$\displaystyle
u_{rr}^{a}\geq\frac{n-1}{u^{a}}+\frac{1}{2}(u^{a}-ru_{r}^{a})\geq\frac{n-1}{2u^{a}}+\sqrt{n-1}-Cr,\;\;a<a_{0}.$

In particular for sufficiently small $a$ we can ensure $\sqrt{n-1}\geq Cr$ and
so that $u_{rr}^{a}(r)\geq\frac{n-1}{2u^{a}(r)}$ for $r<\sqrt{a}$. Hence,
using Equation 6.9, we may estimate

$\displaystyle u_{r}^{a}(\sqrt{a})$
$\displaystyle\geq\frac{\sqrt{3}}{3}+\int_{0}^{\sqrt{a}}\frac{n-1}{2u^{a}(r)}dr$
$\displaystyle\geq\frac{\sqrt{3}}{3}+\frac{n-1}{2}\int_{0}^{\sqrt{a}}\frac{1}{a+Cr}dr=\frac{\sqrt{3}}{3}+\frac{n-1}{2C}\log(1+Ca^{-\frac{1}{2}})\to\infty$

as $a\to 0$, a contradiction. This shows that $u_{r}^{a}$ is not uniformly
bounded near $0$.

Suppose there is a sequence $a_{i}\to 0$ such that $u_{rr}^{a_{i}}$ has one
zero $r_{i}$. In view of Equation 6.8, if $r_{i}$ is bounded away from 0, then
taking $i\to\infty$ will give

$\displaystyle\lim_{i\to\infty}\frac{u^{a_{i}}(r_{i})}{r_{i}}=\lim_{i\to\infty}u^{a_{i}}_{r}(r_{i})-\frac{2(n-1)}{u^{a_{i}}(r_{i})r_{i}}=\infty,$

where we used the fact that $u^{a_{i}}(r_{i})\geq
a_{i}+\frac{\sqrt{3}}{3}r_{i}$ and that $u^{a}_{r}$ is not uniformly bounded
near 0. By Equation 6.7, we can conclude $m(a_{i})\to\infty$ as $i\to\infty$
as before. So we henceforth assume that $r_{i}\to 0$ and assume for a
contradiction that there is a constant $C$ such that $r^{-1}u^{a_{i}}(r)\leq
C$ uniformly for $r\geq r_{i}$. Since $u_{r}$ is decreasing on
$[r_{i},\infty)$ we get

$\displaystyle
u^{a_{i}}(2r_{i})\geq\int_{r_{i}}^{2r_{i}}u^{a_{i}}_{r}(t)dt\geq
r_{i}u^{a_{i}}_{r}(2r_{i})\implies
u_{r}^{a_{i}}(2r_{i})\leq\frac{u^{a_{i}}(2r_{i})}{r_{i}}\leq 2C.$

Since $u^{a_{i}}_{rr}(2r_{i})<0$, Equation 6.1 gives that

$\displaystyle
0<\frac{u^{a_{i}}(2r_{i})}{2r_{i}}\leq\frac{-2(n-1)}{2r_{i}u^{a_{i}}(2r_{i})}+u^{a_{i}}_{r}(2r_{i}).$

Rearranging, we deduce that
$r_{i}u^{a_{i}}(2r_{i})\geq(n-1)u_{r}^{a_{i}}(2r_{i})^{-1}\geq\frac{1}{2}(n-1)C^{-1}$,
and so

$\displaystyle 2Cr_{i}^{2}\geq
r_{i}u^{a_{i}}(2r_{i})\geq\frac{1}{2}(n-1)C^{-1}\implies
r_{i}^{2}\geq\frac{1}{4}(n-1)C^{-2}$

a contradiction as $r_{i}\to 0$. Hence $r^{-1}u^{a_{i}}(r)$ is not uniformly
bounded on $(r_{i},\infty)$, so for each $i$ we may find
$r^{\prime}_{i}\in(r_{i},\infty)$ such that
$(r_{i}^{\prime})^{-1}u^{a_{i}}(r^{\prime}_{i})>i$. Equation 6.7 then implies
$m(a_{i})\to\infty$ as $i\to\infty$.

Otherwise $u_{rr}^{a}$ has no zero for sufficiently small $a$ and monotonicity
as before implies that $m(a)\to\infty$ as $a\to 0$. This completes the proof.
∎

##  7\. Further remarks

We conclude our article with some open questions and conjectures, some of
which we have already alluded to before.

The most natural question to ask is what happens if $\mathcal{C}$ is
rotationally symmetric and the link $\mathcal{L}(\mathcal{C})$ has three (or
more) connected components. It is not expected that self-expanders asymptotic
to $\mathcal{C}$ will be rotationally symmetric. This should be compared to
the case of minimal surfaces - the Costa surface, which is not rotationally
symmetric, has two catenoidal ends and one planar end. It is therefore natural
to expect that something similar happens if the cone is given by the union of
a rotationally symmetric double cone and a hyperplane. We suspect that gluing
method by desingularizing the connected expander (asymptotic to the cone) with
the hyperplane will produce a counterexample.

Another important problem is to determine the entropy of rotationally
symmetric double cones. In fact, we conjecture that the assumption
$\lambda[\mathcal{C}]<2$ in 1.2 is redundant (where as the same assumption in
1.1 is essential). More precisely we conjecture:

######  Conjecture 7.1.

Let $\mathcal{C}$ be a rotationally symmetric double cone of the form:

$\displaystyle
x_{1}^{2}=\begin{cases}m_{1}(x_{2}^{2}+x_{3}^{2}+\cdots+x_{n}^{2}),&x_{1}\geq
0\\\ m_{2}(x_{2}^{2}+x_{3}^{2}+\cdots+x_{n}^{2}),&x_{1}<0\end{cases},$

where $m_{1},m_{2}>0$. Then $\lambda[\mathcal{C}]<2$.

When $m_{1}=m_{2}=m$, observe that when $m\to 0$, $\mathcal{C}$ converges to a
multiplicity 2 plane which has entropy 2, and that when $m\to\infty$,
$\mathcal{C}$ converges after suitable translations to a cylinder
$\mathbb{R}\times\mathbb{S}^{n-1}$ which has entropy strictly less than 2.
This also explains why the two connected components of
$\mathcal{L}(\mathcal{C})$ need to be in different half-spaces, for otherwise
when $m$ is small the double cone could be close to a multiplicity two
cylinder which has entropy strictly larger than 2.

It is not so hard to calculate the Gaussian area of $\mathcal{C}$ at the
origin, but unlike self-shrinkers, the maximum in our case needs not to happen
at the origin (the argument for self-shrinkers can be found in eg. Section 7
of [20]). In fact when $m$ is large, the entropy is achieved far away from the
origin (in a region where the cone looks more like a cylinder). Although we
have strong numerical evidence that the conjecture is true, the analysis of
the Gaussian area functional on the cone centered away from the origin is
complicated to handle.

Less clear is the entropy of cones with $O(p+1)\times O(n-p+1)$ symmetry.
Ilmanen and White [32] give an exact formula for the Gaussian density of
$\mathcal{C}_{n,p}$ at the origin:

$\displaystyle\Theta_{\mathcal{C}_{n,p}}(0)=\frac{\sigma_{p}\sigma_{n-p}}{\sigma_{n}}\left(\frac{p}{n}\right)^{p/2}\left(\frac{n-p}{n}\right)^{(n-p)/2},$

where $\sigma_{p}$ is the volume of the unit sphere in $\mathbb{R}^{p+1}$. It
can be checked this is less than 2 for all $n,p$ (the proof is by rather
tedious computation so we omit it here, but one can easily verify by numerics
as well). Since $\mathcal{C}_{n,p}$ are minimal, they are also self-shrinkers.
It follows from a theorem of Colding–Minicozzi [20] that the entropy of
$\mathcal{C}_{n,p}$ is achieved at the origin. Hence, in fact,
$\lambda[\mathcal{C}_{n,p}]=\Theta_{\mathcal{C}_{n,p}}(0)<2$. For example, the
cone $\mathcal{C}_{2,1}\subset\mathbb{R}^{4}$ has entropy $\frac{3}{2}$. It is
therefore reasonable to make the following conjecture, partly due to Solomon:

######  Conjecture 7.2 (cf. Section 4 of [32]).

Any double cone with $O(p+1)\times O(n-p+1)$ symmetry has entropy at least
that of $\mathcal{C}_{n,p}$ and at most 2.

Finally we discuss the rotational symmetry of singular self-expanders. As we
remarked before, the Hopf lemma B.2 cannot deal with triple junction
singularities which are a possibility as we have seen in Section 6.

######  Question 7.3.

What can we say about the symmetry of a singular self-expander coming out of a
rotationally symmetric double cone?

In particular, if $\Sigma$ is a self-expander asymptotic to $\mathcal{C}_{m}$
(notation as in Section 6) that is smooth away from $\\{x_{1}=0\\}$ and only
has triple junction singularities on $\\{x_{1}=0\\}$, is $\Sigma$ rotationally
symmetric?

It is possible that one can use a parabolic variant of the argument of
Bernstein–Maggi [3] for singular Plateau surfaces (in particular Lemma 2.4
therein) to prove the desired rotational symmetry.

##  Appendix A Construction of the Smooth Flow

Here we give a more detailed overview of the construction of smooth Morse flow
lines from Section 5.4. The construction is an adaptation of the work of
Bernstein and Wang in a series of papers, including [5], [11], [10] and [7].

Throughout the section, we assume $\mathcal{C}\subset\mathbb{R}^{n+1}$ is a
smooth cone. The first proposition was originally proven for self-shrinkers in
[7] and we give a modified proof for self-expanders.

######  Proposition A.1 (cf. Lemma 4.3 of [7]).

Suppose $\Sigma$ is an hypersurface $C^{2,\alpha}$-asymptotic to
$\mathcal{C}$, and there is $N>0$ such that $\Sigma\setminus
B_{NR}(0)\subset\mathcal{T}_{R^{-1}}(\mathcal{C})$ for $R>1$. If
$\\{\Sigma_{t}\\}_{t\in[1,T]}$ is an integral Brakke flow starting from
$\Sigma$ in $\mathbb{R}^{n+1}$, then there is a constant $N^{\prime}>0$ such
that

$\displaystyle\Sigma_{t}\setminus
B_{N^{\prime}R\sqrt{t}}(0)\subset\mathcal{T}_{R^{-1}\sqrt{t}}(\mathcal{C})$

for $R>1$.

###### Proof.

For any
$x\in\mathbb{R}^{n+1}\setminus(B_{NR}(0)\cup\mathcal{T}_{R^{-1}}(\mathcal{C}))$
let

$\displaystyle\rho(x)=\inf\\{\rho^{\prime}\geq 0\mid
B_{\rho^{\prime}}(x)\cap(B_{NR}(0)\cup\mathcal{T}_{R^{-1}}(\mathcal{C}))\neq\emptyset\\}$

and let

$\displaystyle\rho_{t}(x)=\begin{cases}\sqrt{\rho(x)^{2}-2n(t-1)}&\rho(x)^{2}\geq
2n(t-1)\\\ 0&\rho(x)^{2}<2n(t-1)\end{cases}$

be the corresponding MCF starting from $B_{\rho}(x)$. Let

$\displaystyle U_{t}=\bigcup_{\rho_{t}(x)>0}B_{\rho_{t}(x)}(x).$

be the time $t$ slice of the above MCF. Since initially $\Sigma\cap
U_{1}=\emptyset$ maximum principle implies that
$U_{t}\cap\Sigma_{t}=\emptyset$ for all $t\in[1,T)$. This proves that

$\displaystyle\Sigma_{t}\setminus
B_{NR+\sqrt{2n(t-1)}}(0)\subset\mathcal{T}_{R^{-1}+\sqrt{2n(t-1)}}(\mathcal{C})$
(A.1)

since
$\mathbb{R}^{n+1}\setminus(B_{NR+\sqrt{2n(t-1)}}(0)\cup\mathcal{T}_{R^{-1}+\sqrt{2n(t-1)}}(\mathcal{C}))\subset
U_{t}$.

Next we consider the map
$\Phi:\mathcal{C}\setminus\\{0\\}\times\mathbb{R}\to\mathbb{R}^{n+1}$ given by
$\Phi(x,\lambda)=x+\lambda\nu_{\mathcal{C}}$. Choose $\lambda_{0}<1/2$
depending on $\mathcal{C}$ small enough so that
$\Phi|_{\mathcal{L}(\mathcal{C})\times(-2\lambda_{0},2\lambda_{0})}$ is a
diffeomorphism onto its image. It follows, since $\mathcal{C}$ is a cone, that
$\Phi$ is a diffeomorphism on the set
$\\{(x,\lambda)\mid\left|\lambda\right|<2\lambda_{0}\left|x\right|\\}$.

Consider, for some $N^{\prime}$ to be chosen later,

$\displaystyle V_{t}=\mathbb{R}^{n+1}\setminus(U_{t}\cup
B_{N^{\prime}R\sqrt{t}}(0)).$

We first claim that $N^{\prime}$ can be chosen so that $y\in V_{t}$ can be
written as $x+\lambda^{\prime}\left|x\right|\nu_{\mathcal{C}}(x)$ for some
$\left|\lambda^{\prime}\right|<\lambda_{0}$. Indeed, since $y\not\in U_{t}$ we
have that $\operatorname{dist}(y,\mathcal{C})\leq R^{-1}+\sqrt{2n(t-1)}$
provided we choose $N^{\prime}$ large enough so that

$\displaystyle N^{\prime}R\sqrt{t}>NR+\sqrt{2n(t-1)}.$

Let $x$ be the nearest point projection of $y$ onto $\mathcal{C}$, then using
$R>1$ we have that

$\displaystyle\frac{\left|y-x\right|}{\left|x\right|}$
$\displaystyle\leq\frac{1+\sqrt{2nt}}{N^{\prime}\sqrt{t}-1-\sqrt{2nt}}=\frac{(\sqrt{t})^{-1}+\sqrt{2n}}{N^{\prime}-(\sqrt{t})^{-1}-\sqrt{2n}}<\frac{1+\sqrt{2n}}{N^{\prime}-1-\sqrt{2n}}<\lambda_{0}$

provided we choose $N^{\prime}$ large depending on $n$ only. Hence the claim
holds.

Let $y\in\Sigma_{t}\setminus B_{N^{\prime}R\sqrt{t}}(0)\subset V_{t}$. We
claim that, up to further increasing $N^{\prime}$,
$\operatorname{dist}(y,\mathcal{C})<R^{-1}\sqrt{t}$ and this will finish the
proof. To this end consider
$y_{0}=x+\lambda_{0}\left|x\right|\nu_{\mathcal{C}}$. We have that

$\displaystyle\left|y_{0}\right|=\left|x+\lambda_{0}\left|x\right|\nu_{\mathcal{C}}\right|=\sqrt{1+\lambda_{0}^{2}}\left|x\right|>NR+\lambda_{0}\left|x\right|-\frac{1}{3}R^{-1},$

if $N^{\prime}$ is chosen large enough so that

$\displaystyle\left|x\right|\geq(N^{\prime}R-\sqrt{2n})\sqrt{t}-R^{-1}>4NR-R^{-1},$

where we used the fact that
$\sqrt{1+\lambda_{0}^{2}}-\lambda_{0}>\frac{1}{3}$. This implies that
$B_{\lambda_{0}\left|x\right|-R^{-1}}(y_{0})\cap B_{NR}(0)=\emptyset$. Since
$\operatorname{dist}(y_{0},\mathcal{C})=\lambda_{0}\left|x\right|$ we conclude
that

$\displaystyle\rho(y_{0})=\lambda_{0}\left|x\right|-R^{-1}.$

Increasing $N^{\prime}$ if necessary we can also ensure that

$\displaystyle\lambda_{0}\left|x\right|-R^{-1}>\sqrt{2n(t-1)}.$

Since $y\not\in B_{\rho_{t}(y_{0})}(y_{0})\subset U_{t}$ we can estimate

$\displaystyle\operatorname{dist}(y,\mathcal{C})$
$\displaystyle\leq\operatorname{dist}(y_{0},\mathcal{C})-\rho_{t}(y_{0})=\lambda_{0}\left|x\right|-\sqrt{(\lambda_{0}\left|x\right|-R^{-1})^{2}-2n(t-1)}.$

To finish the proof we compute, for large $N^{\prime}$,

$\displaystyle\phantom{{}={}}(\lambda_{0}\left|x\right|-R^{-1}\sqrt{t})^{2}-((\lambda_{0}\left|x\right|-R^{-1})^{2}-2n(t-1))$
$\displaystyle=2R^{-1}\lambda_{0}\left|x\right|(1-\sqrt{t})+R^{-2}(t-1)+2n(t-1)$
$\displaystyle=R^{-1}(\sqrt{t}-1)(-2\lambda_{0}\left|x\right|+(R^{-1}+2nR)(\sqrt{t}+1))\leq
0.$

which is equivalent to

$\displaystyle\lambda_{0}\left|x\right|-R^{-1}\sqrt{t}\leq\sqrt{(\lambda_{0}\left|x\right|-R^{-1})^{2}-2n(t-1)}\implies\operatorname{dist}(y,\mathcal{C})\leq
R^{-1}\sqrt{t}$

provided $N^{\prime}$ is chosen large enough so that

$\displaystyle\lambda_{0}\left|x\right|\geq(N^{\prime}R-\sqrt{2n})\sqrt{t}-R^{-1}\geq(R^{-1}+2nR)\sqrt{t}.$

The next proposition shows the desired regularity for an asymptotically
conical MCF. Since the proof is used repeatedly in our presentation, we have
also included a proof for the sake of completeness. In the proof we will use
the notation $[f]_{\alpha;\Omega}$ to denote the $\alpha$-Hölder seminorm of
$f$ on $\Omega$ (note $\Omega$ could be a subset of $\mathbb{R}^{n+1}$ or a
time interval), i.e.

$\displaystyle[f]_{\alpha;\Omega}=\sup_{x,y\in\Omega,x\neq
y}\frac{\left|f(x)-f(y)\right|}{\left|x-y\right|^{\alpha}}.$

######  Proposition A.2 (Lemma 5.3(2) of [10]).

Suppose $\Sigma$ is an hypersurface $C^{2,\alpha}$-asymptotic to $\mathcal{C}$
and let $\\{\Sigma_{t}\\}_{t\in[1,T)}$ be a MCF starting from $\Sigma$. If
there is $N>0$ such that $\Sigma_{1}\setminus
B_{NR}(0)\subset\mathcal{T}_{R^{-1}}(\mathcal{C})$ for all $R>1$, then there
is $N^{\prime}>0$ such that $\Sigma_{t}\setminus B_{N^{\prime}\sqrt{t}}(0)$
can be written as a (smooth) normal graph over $\mathcal{C}\setminus
B_{N^{\prime}\sqrt{t}}(0)$. In particular we have the uniform curvature bound

$\displaystyle\sup_{t\in[1,T)}\sup_{\Sigma_{t}\setminus
B_{N^{\prime}\sqrt{t}}}\left|A_{\Sigma_{t}}\right|<\infty.$

###### Proof.

Fix $t\in[1,T)$ and let $\delta>0$. Since $\Sigma$ is asymptotically conical,
by 5.2 there is $N_{1}$ and $\varepsilon>0$ depending on $\delta$ such that
for any $x_{0}\in\mathcal{C}\setminus B_{N_{1}}(0)$, $\Sigma_{1}\cap
C_{\eta}(x_{0})$ can be written as a graph $\tilde{f}_{x_{0}}(x)$ over some
neighborhood of $x_{0}$ in $T_{x}\mathcal{C}$ containing
$B^{n}_{\eta}(x_{0})$. Here $\eta=\varepsilon\left|x\right|$. Moreover up to
increasing $N_{1}$ we can ensure A.1 holds, and that $\tilde{f}_{x_{0}}$
satisfies the estimates

$\displaystyle\sum_{i=0}^{2}\eta^{-1+i}\sup_{B^{n}_{\eta}(x_{0})}|\nabla^{i}\tilde{f}_{x_{0}}|+r^{1+\alpha}[\nabla^{2}\tilde{f}_{x_{0}}]_{\alpha}<\delta$
(A.2)

By pseudolocality of MCF (Theorem 1.5 in [31]), given $\varepsilon>0$, there
is $N_{1}>0$ such that for every $x_{0}\in\Sigma_{1}\setminus B_{N_{1}}(0)$
and $s\in[1,t]$, $\Sigma_{s}\cap C_{\eta/2}(x_{0})$ can be written as a normal
graph over $\Sigma_{1}\cap B^{n}_{\eta/2}(x_{0})$. Combining the above two
facts, we see that for sufficiently small $\delta$ and $\varepsilon$,
$\Sigma_{s}\cap C_{\eta/2}(x_{0})$ can be written as the graph of a function
$f_{x_{0}}(s,x)$ over some neighborhood of $T_{x_{0}}\mathcal{C}$ for all
$s\in[1,t]$ and $x_{0}\in\mathcal{C}\setminus B_{N_{1}}(0)$. Moreover,
$f_{x_{0}}$ satisfies the pointwise estimates

$\displaystyle(\eta/2)^{-1}\sup_{B^{n}_{\eta/2}(x_{0})}\left|f_{x_{0}}(s,\cdot)\right|+\sup_{B^{n}_{\eta/2}(x_{0})}\left|\nabla_{x}f_{x_{0}}(s,\cdot)\right|<1.$

For the rest of the proof we fix an $x_{0}$ and put $f=f_{x_{0}}$. Since
$\\{\Sigma_{s}\\}_{s\in[1,t]}$ is a graphical MCF near $x_{0}$, $f$ satisfies
the evolution equation:

$\displaystyle\frac{\partial f}{\partial
s}=\sqrt{1+\left|\nabla_{x}f\right|^{2}}\operatorname{div}\left(\frac{\nabla_{x}f}{\sqrt{1+\left|\nabla_{x}f\right|^{2}}}\right).$

This is a quasilinear parabolic equation, so we may use Hölder estimates (eg.
Theorem 1.1 in Chapter 4 of [35]) to get that

$\displaystyle\sup_{s\in[1,t]}[\nabla_{x}f(s,\cdot)]_{\alpha;B_{\eta/4}(x_{0})}+\sup_{B^{n}_{\eta/4}(x_{0})}[\nabla_{x}f(\cdot,x)]_{\alpha/2;[1,t]}\leq
C(\eta/4)^{-\alpha}.$

for any $\alpha\in(0,1)$. Standard Schauder estimates (see eg. Chapter 5 of
[34] or Theorem 5.1 in Chapter 4 of [35]) yield higher order estimates of the
form

$\displaystyle\sum_{i=0}^{2}(\eta/8)^{i-1}\sup_{B^{n}_{\eta/8}(x_{0})}\left|\nabla^{i}f(s,\cdot)\right|+(\eta/8)^{1+\alpha}[\nabla^{2}f(s,\cdot)]_{\alpha;B_{\eta/8}(x_{0})}\leq
C$

for $s\in[1,t]$ and

$\displaystyle\sup_{x\in
B^{n}_{\eta/8}(x_{0})}[\nabla_{x}f_{x_{0}}(s,x)]_{\frac{1}{2};[1,t]}\leq
C(\eta/8)^{-1}.$

From the above we may estimate

$\displaystyle\left|f_{x_{0}}(s,x)-f_{x_{0}}(1,x_{0})\right|\leq
C(s-1)(\eta/8)^{-1}+\delta\left|x-x_{0}\right|+C(\eta/8)^{-1}\left|x-x_{0}\right|^{2}$

where we also used the evolution equation and the fact that
$\left|\nabla_{x}f(1,x_{0})\right|<\delta$ from Equation A.2. These implies
that, for $\rho<1/8$, a fixed $s\in[1,t]$ and $x_{0}\in\mathcal{C}\setminus
B_{\tilde{N}\sqrt{s}}(0)$,

$\displaystyle(\rho\eta)^{-1}\sup_{x\in
B_{\rho\eta}^{n}(x_{0})}\left|f(s,x)\right|$
$\displaystyle\leq(\rho\eta)^{-1}N_{1}\left|x_{0}\right|^{-1}+C(s-1)\rho^{-1}\eta^{-2}+\delta+C\rho$
$\displaystyle\leq\frac{(\rho\varepsilon)^{-1}N_{1}}{\tilde{N}^{2}s}+\frac{C(s-1)\rho^{-1}}{\tilde{N}^{2}s}+\delta+C\rho$

where we used that $\left|f(1,x_{0})\right|<N_{1}\left|x_{0}\right|^{-1}$ by
A.1. The right hand side of the above equation can be made arbitrarily small
provided we choose $\delta$ and $\rho$ small enough and $\tilde{N}$ large
enough. Similarly we can estimate the derivative

$\displaystyle\left|\nabla_{x_{0}}f_{x_{0}}(s,x)-f_{x_{0}}(1,x_{0})\right|\leq
C(\eta/8)^{-1}\left|x-x_{0}\right|+C(\eta/8)^{-1}\sqrt{s-1}$

and

$\displaystyle\sup_{x\in
B^{n}_{\rho\eta}(x_{0})}\left|\nabla_{x}f(s,x)\right|\leq\delta+C\rho+C\frac{\sqrt{s-1}}{\tilde{N}\sqrt{s}}$

which can be made arbitrarily small as well. These two decay estimates
together with Schauder estimates above with $\eta/8$ replaced by $\rho\eta$
give

$\displaystyle\sum_{i=0}^{2}(\eta/8)^{i-1}\sup_{B^{n}_{\eta/8}(x_{0})}\left|\nabla^{i}f(s,\cdot)\right|+(\eta/8)^{1+\alpha}[\nabla^{2}f(s,\cdot)]_{\alpha;B_{\eta/8}(x_{0})}\leq\frac{1}{2}+C(\rho+\rho^{1+\alpha})$

which can be made to be less than 1 provided $\rho$ is chosen small enough.
This proves that $\Sigma_{t}\cap C_{\rho\eta}(x_{0})$ is a graph over (a
neighborhood of) $T_{x_{0}}\mathcal{C}$ for $x_{0}\in\mathcal{C}\setminus
B_{\tilde{N}\sqrt{t}}(0)$ with derivative bounds up to the second order. The
curvature bounds follow easily, and higher order bounds follow similarly using
Schauder estimates. ∎

Given a MCF $\\{\Sigma_{t}\\}_{t\in I}$, the expander mean curvature of
$\Sigma_{t}$ is

$\displaystyle E_{\Sigma_{t}}(x)=2tH_{\Sigma_{t}}+\left\langle
x,\nu_{\Sigma_{t}}\right\rangle.$

We say $\\{\Sigma_{t}\\}$ is expander mean convex if the $E_{\Sigma_{t}}(x)>0$
along the flow. For a fixed time $t$ and a hypersurface $\Sigma$, the relative
expander mean curvature of $\Sigma$ is

$\displaystyle E_{\Sigma}^{t}(x)=2tH_{\Sigma}+x^{\perp}.$

For $\beta>0$ define the auxiliary function
$g_{\beta}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ by

$\displaystyle g_{\beta}(s)=s^{-\beta}e^{-\beta s}.$

We now prove the main existence theorem for expander mean convex hypersurfaces
without entropy bound.

######  Theorem A.3 (Existence Theorem, cf. Proposition 5.1 of [10]).

Let $\Sigma$ be a hypersurface $C^{2,\alpha}$-asymptotic to $\mathcal{C}$ with
no closed components. Suppose that there is $N$ such that $\Sigma\setminus
B_{NR}(0)\subset\mathcal{T}_{R^{-1}}(\mathcal{C})$ and that there is
$c,\beta>0$ such that

$\displaystyle E_{\Sigma}(x)\geq
cg_{\beta}(1+\left|x\right|^{2})>0,\;\;x\in\Sigma.$

Then there exists a unique MCF $\\{\Sigma_{t}\\}_{t\in[1,T)}$ with
$\Sigma_{1}=\Sigma$, where $T$ is the first singular time (possibly $\infty$).
Moreover the MCF satisfies

  1. (1)

$\Sigma_{t}$ is $C^{2,\alpha}$-asymptotic to $\mathcal{C}$ for all
$t\in[1,T)$.

  2. (2)

$E_{\Sigma_{t}}(x)>cg_{\beta}(1+\left|x\right|^{2}+2n(t-1))$ for all
$t\in[1,T)$ and $x\in\Sigma_{t}$.

  3. (3)

If $T<\infty$, we have

$\displaystyle\lim_{t\to T}\sup_{\Sigma_{t}\cap
B_{N^{\prime}\sqrt{t}}}\left|A_{\Sigma_{t}}\right|=\infty.$

###### Proof.

Consider the map
$\Phi:\Sigma\times(-\varepsilon,\varepsilon)\to\mathbb{R}^{n+1}$ given by

$\displaystyle\Phi(x,\lambda)=x+\lambda\nu_{\Sigma}.$

Since $\Sigma$ is asymptotically conical, we can choose $\varepsilon$
sufficiently small so that the above map is a diffeomorphism onto its image
for every $\lambda\in(-\varepsilon,\varepsilon)$. Using this parametrization
we can invoke standard existence theorem for MCF to conclude that there exists
a unique MCF starting from $\Sigma_{1}=\Sigma$. For the properties, Item (1)
and (3) follow from A.2, and Item (2) is Lemma 5.4 of [10]. ∎

Given an unstable self-expander $\Sigma$ asymptotic to $\mathcal{C}$, we wish
to apply A.3 to the perturbed self-expander $\Sigma^{\varepsilon}$ defined by

$\displaystyle\Sigma^{\varepsilon}=\Psi^{\varepsilon}(\Sigma)\text{ where
}\Psi^{\varepsilon}(x)=x+\varepsilon f(x)\nu_{\Sigma}.$

It remains to check that $\Sigma^{\varepsilon}$ satisfies the assumption of
A.3. We need the following $C^{0}$-estimate on the first eigenfunction $f$.

######  Lemma A.4 (Proposition 3.2 of [5]).

Let $\Sigma$ be an unstable connected self-expander. Let $f$ be the unique
positive eigenfunction of $L_{\Sigma}$ with eigenvalue $\mu_{1}<0$ and
$\left\lVert f\right\rVert_{W^{0}_{1/4}(\Sigma)}=1$. Then $f$ satisfies the
following $C^{0}$ estimate:

$\displaystyle
C^{-1}(1+\left|x\right|^{2})^{-\frac{1}{2}(n+1-2\mu_{1})}e^{-\frac{\left|x\right|^{2}}{4}}\leq
f\leq
C(1+\left|x\right|^{2})^{-\frac{1}{2}(n+1-2\mu_{1})}e^{-\frac{\left|x\right|^{2}}{4}}.$

where $C=C(\Sigma)$.

A.4 and 5.2 imply that there exists $N>0$ such that
$\Sigma^{\varepsilon}\setminus
B_{NR}(0)\subset\mathcal{T}_{R^{-1}}(\mathcal{C})$ for $R>1$, so the first
condition to apply A.3 is satisfied. Finally we need to show that perturbing
by the first eigenfunction produces a hypersurface with positive expander mean
curvature. This relies on the fact that $-L_{\Sigma}$ is the linearization of
the self-expander equation.

######  Lemma A.5.

Let $\Sigma$ be a connected self-expander $C^{2,\alpha}$-asymptotic to
$\mathcal{C}$. Let $f$ be the unique positive eigenfunction corresponding to
$\mu_{1}$ of $L_{\Sigma}$ with $\left\lVert
f\right\rVert_{W^{0}_{1/4}(\Sigma)}=1$. Then there exists $\varepsilon_{0}>0$
such that for all $\left|\varepsilon\right|<\varepsilon_{0}$ there is
$\beta=\beta(\varepsilon)$ such that

$\displaystyle E_{\Sigma^{\varepsilon}}(x)\geq
cg_{\beta}(1+\left|x\right|^{2}).$

Here $\Sigma^{\varepsilon}$ is the image of $\Sigma$ under the map
$\Phi(x)=x+\varepsilon f(x)\nu_{\Sigma}$.

###### Proof.

By Lemma A.2 of [11] we have (the computation is long, but the result should
be standard),

$\displaystyle E_{\Sigma^{\varepsilon}}(x)=-\varepsilon
L_{\Sigma}f+\varepsilon^{2}Q(f,\left\langle
x,\nabla_{\Sigma}f\right\rangle,\nabla_{\Sigma}f,\nabla^{2}_{\Sigma}f)$

for some homogeneous quadratic polynomial $Q$ with bounded coefficients. When
$\varepsilon>0$, it follows from A.4, that, up to further shrinking
$\varepsilon$,

$\displaystyle
E_{\Sigma^{\varepsilon}}(x)\geq\varepsilon\mu_{1}C^{-1}(1+\left|x\right|^{2})^{-\frac{1}{2}(n+1-2\mu_{1})}e^{-\frac{1+\left|x\right|^{2}}{4}}\geq
cg_{\beta}(1+\left|x\right|^{2})$

where $\beta=\frac{1}{2}(n+1-2\mu_{1})>0$. The case $\varepsilon<0$ can be
handled similarly. ∎

A.5 shows that $\Sigma^{\varepsilon}$ satisfies the second condition of A.3
for sufficiently small $\varepsilon$. As such, A.3 can be applied to conclude
the short-time existence of an expander mean-convex MCF starting from
$\Sigma^{\varepsilon}$.

##  Appendix B Maximum principles

Here we record the maximum principles from Section 3 of [18] for Brakke flows.
These are the essential tools in applying the moving plane method without
smoothness. If $\mathcal{M}$ is an integral Brakke flow and
$X=(x_{0},t_{0})\in\operatorname{supp}\mathcal{M}$, the Gaussian density at
$X$ is

$\displaystyle\Theta_{\mathcal{M}}(X)=\lim_{\rho\to
0}\frac{1}{(4\pi\rho^{2})^{n/2}}\int_{\mathcal{M}_{t_{0}-\rho^{2}}}e^{-\frac{\left|x-x_{0}\right|^{2}}{4\rho^{2}}}d\mathcal{H}^{n}.$

$\Theta_{\mathcal{M}}(X)$ is well-defined by Huisken’s monotonicity formula.
Observe that an entropy upper bound automatically gives upper bounds on all
Gaussian densities.

######  Theorem B.1 (Maximum principle for Brakke flows, Theorem 3.4 of [18]).

Let $\mathcal{M}$ be a smooth MCF defined in a parabolic ball $P(X,r)$, where
$X=(x_{0},t_{0})\in\operatorname{supp}\mathcal{M}$ and $r>0$ is sufficiently
small such that $\operatorname{supp}\mathcal{M}$ separates $P(x,r)$ into two
open connected components $U$ and $U^{\prime}$. Let $\mathcal{M}^{\prime}$ be
an integral Brakke flow in $P(X,r)$ with
$X\in\operatorname{supp}\mathcal{M}^{\prime}$ and Gaussian density
$\Theta_{X}(\mathcal{M}^{\prime})<2$. If
$\operatorname{supp}\mathcal{M}^{\prime}\subset
U\cup\operatorname{supp}\mathcal{M}$, then $X$ is a smooth point for
$\mathcal{M}^{\prime}$, and $\mathcal{M}^{\prime}$ agrees with $\mathcal{M}$
in a small parabolic ball.

######  Theorem B.2 (Hopf lemma for tame Brakke flows, Theorem 3.19 of [18]).

Let $\mathcal{M}$ and $\mathcal{M}^{\prime}$ be two integral Brakke flows
defined in a parabolic ball $P(X,r)$ where
$X=(x_{0},t_{0})\in\operatorname{supp}\mathcal{M}\cap\mathcal{M}^{\prime}$.
Suppose $X$ is a tame point (see 3.1) for both $\mathcal{M}$ and
$\mathcal{M}^{\prime}$ and let $\mathbb{H}\subset\mathbb{R}^{n+1}$ be an open
half space with $x_{0}\in\partial\mathbb{H}$. If in addition
$\partial\mathbb{H}$ is not the tangent flow to either $\mathcal{M}$ or
$\mathcal{M}^{\prime}$, and $\operatorname{reg}\mathcal{M}_{t}\cap\mathbb{H}$
and $\operatorname{reg}\mathcal{M}^{\prime}_{t}\cap\mathbb{H}$ are disjoint
for $t\in(t_{0}-r^{2},t_{0})$, then $\mathcal{M}$ and $\mathcal{M}^{\prime}$
are smooth at $(0,0)$ with distinct tangents.

We remark that in contrast to the usual smooth maximum principle and Hopf
lemma, the smoothness is in fact a conclusion in both of the statements above.

## References

  * AIC [95] Sigurd B. Angenent, Tom Ilmanen, and David L. Chopp.  A computed example of nonuniqueness of mean curvature flow in r3.  Commun. in Partial Differential Equations, 20:1937–1958, 1995.
  * [2] Jacob Bernstein, Letian Chen, and Lu Wang.  Existence of monotone morse flow lines of the expander functional.  In preparation.
  * BM [21] Jacob Bernstein and Francesco Maggi.  Symmetry and rigidity of minimal surfaces with plateau-like singularities.  Arch. Ration. Mech. Anal., 239(3):1177–1210, 2021.
  * Bra [78] Kenneth A. Brakke.  The Motion of a Surface by its Mean Curvature, volume 20 of Mathematical Notes.  Princeton University Press, Princeton, NJ, 1978.
  * [5] Jacob Bernstein and Lu Wang.  A mountain-pass theorem for asymptotically conical self-expanders.  Peking Math. J. To Appear.
  * BW [16] Jacob Bernstein and Lu Wang.  A sharp lower bound for the entropy of closed hypersurfaces up to dimension six.  Invent. Math., 206(3):601–627, 2016.
  * BW [17] Jacob Bernstein and Lu Wang.  A topological property of asymptotically conical self-shrinkers of small entropy.  Duke Math. J., 166(3):403–435, 2017.
  * BW [18] Jacob Bernstein and Lu Wang.  An integer degree for asymptotically conical self-expanders.  https://arxiv.org/abs/1807.06494, 2018.  Preprint.
  * [9] Jacob Bernstein and Lu Wang.  Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow.  Int. Math. Res. Not. IMRN, 2021(12):9016–9044, 2019.
  * [10] Jacob Bernstein and Lu Wang.  Topological uniqueness for self-expanders of small entropy.  https://arxiv.org/abs/1902.02642, 2019.  Preprint.
  * BW [20] Jacob Bernstein and Lu Wang.  Relative expander entropy in the presence of a two-sided obstacle and applications.  https://arxiv.org/abs/1906.07863, 2020.  Preprint.
  * BW [21] Jacob Bernstein and Lu Wang.  The space of asymptotically conical self-expanders of mean curvature flow.  Math. Ann., 380:175–230, 2021.
  * CCMS [20] Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, and Felix Schulze.  Mean curvature flow with generic initial data.  https://arxiv.org/abs/2003.14344, 2020.  Preprint.
  * CF [16] Otis Chodosh and Frederick Tsz-Ho Fong.  Rotational symmetry of conical kähler-ricci solitons.  Math. Ann., 364(3-4):777–692, 2016.
  * CGG [91] Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations.  J. Differential Geom., 33(3):749–786, 1991.
  * Che [21] Letian Chen.  Rotational symmetry of solutions of mean curvature flow coming out of a double cone.  https://arxiv.org/abs/2101.09801, 2021.  Preprint.
  * [17] Kyeongsu Choi, Robert Haslhofer, and Or Hershkovits.  Ancient low entropy flows, mean convex neighborhoods, and uniqueness.  Acta. Math. To Appear.
  * CHHW [19] Kyeongsu Choi, Robert Haslhofer, Or Hershkovits, and Brian White.  Ancient asymptotically cylindrical flows and applications.  https://arxiv.org/abs/1910.00639, 2019.  Preprint.
  * Cho [14] Otis Chodosh.  Expanding ricci solitons asymptotic to cones.  Calc. Var. Partial Differential Equations, 51:1–15, 2014.
  * CM [12] Tobias Holck Colding and William P. Minicozzi II.  Generic mean curvature flow i: generic singularities.  Ann. of Math. (2), 175(2):755–833, 2012.
  * CM [15] Tobias Holck Colding and William P. Minicozzi II.  Uniqueness of blowups and Łojasiewicz inequalities.  Ann. of Math. (2), 182(1):221–285, 2015.
  * EH [89] Klaus Ecker and Gerhard Huisken.  Mean curvature evolution of entire graphs.  Ann. of Math., 130(3):453–471, 1989.
  * EH [91] Klaus Ecker and Gerhard Huisken.  Interior estimates for hypersurfaces moving by mean curvature.  Invent. Math., 105:547–569, 1991.
  * ES [91] Lawrence C. Evans and Joel Spruck.  Motion of level sets by mean curvature. i.  J. Differential Geom., 33(3):635–681, 1991.
  * FM [19] Frederick Tsz-Ho Fong and Peter McGrath.  Rotational symmetry of asymptotically conical mean curvature flow self-expanders.  Comm. Anal. Geom., 27(3):599–618, 2019.
  * GNN [79] Basilis Gidas, Wei-Ming Ni, and Louis Nirenberg.  Symmetry and related properties via the maximum principle.  Comm. Math. Phys., 68(3):209–243, 1979.
  * Hel [12] Sebastian Helmensdorfer.  A model for the behavior of fluid droplets based on mean curvature flow.  SIAM J. Math. Anal., 44(3):1359–1371, 2012.
  * HHW [20] Robert Haslhofer, Or Hershkovits, and Brian White.  Moving plane method for varifolds and applications.  https://arxiv.org/abs/2003.01505, 2020.  Preprint.
  * [29] Or Hershkovits and Brian White.  Avoidance for set-theoretic solutions of mean-curvature-type flows.  Commun. Anal. Geom. To appear.
  * Ilm [94] Tom Ilmanen.  Elliptic Regularization and Partial Regularity for Motion by Mean Curvature.  Number 520 in Mem. Amer. Math. Soc. American Mathematical Society, Providence, RI, 1994.
  * INS [19] Tom Ilmanen, André Neves, and Felix Schulze.  On short time existence for the planar network flow.  J. Differential Geom., 111(1):39–89, 2019.
  * IW [15] Tom Ilmanen and Brian White.  Sharp lower bounds on density for area-minimizing cones.  Camb. J. Math., 3(1–2):1–18, 2015.
  * KM [14] Stephen Kleene and Niels Martin Møller.  Self-shrinkers with a rotational symmetry.  Trans. Amer. Math. Soc., 366(8):3943–3963, 2014.
  * Lie [96] Gary M. Lieberman.  Second Order Parabolic Differential Equations.  World Scientific Publishing Co., 1996.
  * LSU [68] O.A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’ceva.  Linear and quasi-linear equations of parabolic type, volume 23 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1968.
  * Maz [17] Laurent Mazet.  Minimal hypersurfaces asymptotic to simons cones.  J. Inst. Math. Jussieu, 16(1):39–58, 2017.
  * MSS [15] Francisco Martín, Andreas Savas-Halilaj, and Knut Smoczyk.  On the topology of translating solitons of the mean curvature flow.  Calc. Var. Partial Differential Equations, 18(54):2853–2882, 2015\.
  * Sch [83] Richard M. Schoen.  Uniqueness, symmetry, and embeddedness of minimal surfaces.  J. Differential Geom., 18(4):791–809, 1983.
  * Sim [84] Leon Simon.  Lectures on Geometric Measure Theory.  Proceedings of the Centre of Mathematical Analysis, Australian National University. Australian National University, Centre of Mathematical Analysis, Canberra, 1984.
  * SW [89] Bruce Solomon and Brian White.  A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals.  Indiana Univ. Math. J., 38(3):683–691, 1989.
  * Ton [19] Yoshihiro Tonegawa.  Brakke’s Mean Curvature Flow: An Introduction.  Springer Briefs in Mathematics. Springer, Singapore, 2019.
  * Wan [20] Shengwen Wang.  Round spheres are hausdorff stable under small perturbation of entropy.  J. Reine Angew. Math., 758:261–280, 2020.
  * Whi [95] Brian White.  The topology of hypersurfaces moving by mean curvature.  Comm. Anal. Geom., 3(2):317–333, 1995.
  * Whi [09] Brian White.  Currents and flat chains associated to varifolds, with an application to mean curvature flow.  Duke Math. J., 148(1):41–62, 2009.
  * Whi [21] Brian White.  Mean curvature flow with boundary.  Ars Inveniendi Analytica, 1(4):1–43, 2021.