[title=Index of Notation]

# Mean curvature flow with generic low-entropy initial data II

Otis Chodosh  OC: Department of Mathematics, Bldg. 380, Stanford University,
Stanford, CA 94305, USA<EMAIL_ADDRESS>,  Christos Mantoulidis  CM:
Department of Mathematics, Rice University, Houston, TX 77005, USA
<EMAIL_ADDRESS>and  Felix Schulze  FS: Department of
Mathematics, Zeeman Building, University of Warwick, Gibbet Hill Road,
Coventry CV4 7AL, UK<EMAIL_ADDRESS>
###### Abstract.

We prove that the mean curvature flow of a generic closed embedded
hypersurface in $\mathbb{R}^{4}$ or $\mathbb{R}^{5}$ with entropy $\leq 2$, or
with entropy $\leq\lambda(\SS^{1})$ if in $\mathbb{R}^{6}$, encounters only
generic singularities.

##  1\. Introduction

Mean curvature flow is the gradient flow of area. A family of hypersurfaces
$M(t)\subset\mathbb{R}^{n+1}$ is flowing by mean curvature flow if the
following equation is satisfied

(1.1) $\left(\tfrac{\partial}{\partial
t}\mathbf{x}\right)^{\perp}=\mathbf{H}_{M(t)}(\mathbf{x}).$

Here, $\mathbf{H}_{M(t)}(\mathbf{x})$ denotes the mean curvature vector of
$M(t)$ at $\mathbf{x}$. When the initial data $M(0)$ is compact, mean
curvature flow is guaranteed to become singular in finite time, so
understanding the nature of such singularities is a fundamental problem.

A well-known conjecture of Huisken suggests that the singularities of a
generic mean curvature flow should be as simple as possible: spherical and
cylindrical [Ilm03, #8]. One approach to this issue is to study only the
singularities that persist under a generic perturbation of the initial data
$M(0)$. In this article we study this problem under a low-entropy condition
(see Definition 1.6 for the definition of entropy). In particular, we obtain
the following result (see Corollary 1.18 for the precise statement)

######  Theorem 1.1 (Low entropy generic flow in $\mathbb{R}^{4}$, informal).

If $M^{3}\subset\mathbb{R}^{4}$ is a closed embedded hypersurface with entropy
$\lambda(M)\leq 2$ then there exist arbitrarily small $C^{\infty}$ graphs
$M^{\prime}$ over $M$ so that the mean curvature flow starting from
$M^{\prime}$ has only multiplicity-one singularities of
$\SS^{3},\SS^{2}\times\mathbb{R}$, and $\SS^{1}\times\mathbb{R}^{2}$-type.

######  Remark 1.2.

This implies that any mean curvature flow of $M^{\prime}$ will be smooth for
a.e. time $t$, see Corollary 1.23 below.

Together with K. Choi, we proved Theorem 1.1 for $M^{2}\subset\mathbb{R}^{3}$
with $\lambda(M)\leq 2$ in [CCMS21, Theorem 1.1]. The generalization to
$\mathbb{R}^{4}$ presents significant new challenges. In particular, the flow
of a closed embedded $M^{3}\subset\mathbb{R}^{4}$ with $\lambda(M)\leq 2$ may
have singularities modeled on singular self-shrinkers, such as the cone over
the Clifford torus.111Or more generally, this includes the cone over any of
the infinitely many minimal submanifolds of $\SS^{3}$ with area less than
$8\pi$. Singular shrinkers were a major obstacle in many previous works (cf.
$(\star_{n,\Lambda})$ in [BW18, BW22a] and $(\dagger_{n,\Lambda})$ in
[CCMS21]), and precisely informed our previous strong entropy
$\leq\lambda(\SS^{1})$ assumption for $M^{3}\subset\mathbb{R}^{4}$ in [CCMS21,
Theorem 1.2]).

We also have results analogous to Theorem 1.1 in higher dimensions (see
Corollaries 1.19 and 1.20 for the precise statements):

######  Theorem 1.3 (Low entropy generic flow in $\mathbb{R}^{5}$, informal).

If $M^{4}\subset\mathbb{R}^{5}$ is a closed embedded hypersurface with entropy
$\lambda(M)\leq 2$ then there exist arbitrarily small $C^{\infty}$ graphs
$M^{\prime}$ over $M$ so that the mean curvature flow starting from
$M^{\prime}$ has only multiplicity-one singularities of
$\SS^{4},\SS^{3}\times\mathbb{R}$, $\SS^{2}\times\mathbb{R}^{2}$, and
$\SS^{1}\times\mathbb{R}^{3}$-type.

######  Theorem 1.4 (Low entropy generic flow in $\mathbb{R}^{6}$, informal).

If $M^{5}\subset\mathbb{R}^{6}$ is a closed embedded hypersurface with entropy
$\lambda(M)\leq\lambda(\SS^{1})$ then there exist arbitrarily small
$C^{\infty}$ graphs $M^{\prime}$ over $M$ so that the mean curvature flow
starting from $M^{\prime}$ has only multiplicity-one singularities of
$\SS^{5},\SS^{4}\times\mathbb{R}$, $\SS^{3}\times\mathbb{R}^{2}$,
$\SS^{2}\times\mathbb{R}^{3}$-type.

Our results also imply a topological classification of low-entropy
hypersurfaces (see Corollary 1.22 for a more general statement and proof).

######  Corollary 1.5 (Mean curvature flow as a smooth isotopy in low
entropy).

Let $n\in\\{2,3,4,5\\}$ and $M^{n}\subset\mathbb{R}^{n+1}$ be a closed
connected embedded hypersurface.

  1. (a)

If $\lambda(M)\leq\lambda(\SS^{n-1})$ then perhaps after a small initial
$C^{\infty}$ perturbation, the mean curvature flow provides a smooth isotopy
from $M$ to a standard $\SS^{n}$ in $\mathbb{R}^{n+1}$.

  2. (b)

If $\lambda(M)\leq\lambda(\SS^{n-2})$ then perhaps after a small initial
$C^{\infty}$ perturbation, the mean curvature flow with surgery of [DH22] (cf.
[HS09, HK17b]) provides a smooth isotopy from $M$ to the boundary of a
standard handlebody that is either a standard ball $B^{n}$ or a boundary
connect sum of finitely many $B^{n-1}\times\SS^{1}$’s.

This result is only new when $n\in\\{4,5\\}$. When $n\in\\{2,3\\}$, a slightly
different version of (a) was first proven by Bernstein–Wang [BW22a, BW17], and
as presented (a) and (b) both follow from our work with K. Choi [CCMS20,
CCMS21]. Our work relied on various insights from the Bernstein–Wang program
(see also [CIMW13, BW16, Zhu20, BW17, BW22b, BW21, BW23, BW22c, HW19]).

At the moment, serious obstacles remain to extend the isotopy construction
(Corollary 1.5) to the case $\lambda(M)\leq 2$ using Theorem 1.1, even
assuming a (hypothetical) construction of a flow with surgery (cf. [DH21,
CHH21, CHH23, DH23, CDD+22]).

We refer the reader to [HS99a, HS99b, HS09, Bre15, BH16, HK17a, HK17b, BHH16,
BHH19, ADS19, ADS20, BC19, BC21, CHH22, CHHW22, SX21a, SX21b, SX22, CS23] for
other results on mean curvature flow.

###  1.1. The low entropy assumption and a strong multiplicity-one conjecture

As a consequence of our work with K. Choi [CCMS20, CCS23] (building on [Ilm95,
CM12, CIMW13, Bre16, Wan16, BW17]), the “only” remaining obstacle to a theory
of generic mean curvature flows in $\mathbb{R}^{3}$ without any entropy
assumptions is the possible occurrence of smooth tangent flows with integer
multiplicity—which the well-known “multiplicity-one conjecture” of Ilmanen
[Ilm03] posits should not occur.

In Appendix A’s Theorem A.5, we show how Theorems 1.1 and 1.3 can generalize
to give generic mean curvature flows in $\mathbb{R}^{4}$, $\mathbb{R}^{5}$
without assuming $\lambda(M)\leq 2$ but instead a stronger form of the
multiplicity-one conjecture.

###  1.2. Generic regularity of minimizing hypersurfaces

In [CMS23a] we proved generic regularity for solutions to the Plateau problem
(for hypersurfaces) in ambient $9$\- and $10$-dimensions (see also [CMS23b]).
The present paper can be considered as a parabolic analogue of [CMS23a], but
there are several new serious issues in this paper that were not present
there.

In the elliptic problem, a crucial feature is that singular points cannot
limit to regular points (by density considerations). In the parabolic problem,
we do not know in general whether non-generic singularities can limit to
generic ones (but see [CIM15] for a related result); it is only known that
this cannot happen if the limiting singularity is $\SS^{n}(\sqrt{2n})$ (then
apply [Hui84]) or $\SS^{n-1}(\sqrt{2(n-1)})\times\mathbb{R}$ (then apply
[BW17, CHH22, CHHW22]). This creates serious issues when working with an
entropy bound $\lambda(M)\leq 2$ (as we do in this paper) rather than
$\lambda(M)\leq\lambda(\SS^{n-1})$ (as in [CCMS21]). We expect the techniques
we have developed here to handle this (cf. Section 6) to be applicable in
other situations.

There are two more serious issues in going from the elliptic to the parabolic
problem:

  1. (1)

The dimension of the spine of a singularity versus the eigenvalue estimate are
_coupled_ (cf. Propositions 3.6 and 5.2), necessitating a more complicated
covering argument to prove the final result.

  2. (2)

The parabolic Harnack inequality requires one to “go back in time,”
significantly complicating the proof of the separation estimates (cf.
Proposition 4.12).

###  1.3. Related work in other settings

We also mention here some other related work. Figalli–Ros-Oton–Serra [FROS20]
have proven generic regularity of solutions to the obstacle problem (see also
[FS19]). They have subsequently applied these techniques to study the Stefan
problem [FROS23]. These papers have some features in common with the present
paper (and [CMS23a, CMS23b]), particularly the general strategy of (super-
linear) space-time Hölder estimates versus estimates for the space-time
singular set, although the technical details are very different. In
particular, the serious issue of “generic” versus “non-generic” singularities
faced here does not appear to have an analog in the previously cited works.

Finally, we also note the recent work of Fernández-Real–Yu [XFR23]
establishing improved generic regularity for free-boundary problems by using a
strategy that seems to have some relation with the technique used in [CMS23a,
CMS23b].

###  1.4. Definitions

Before precisely stating our main results we first recall several definitions,
some of which are now standard.

######  Definition 1.6 (Colding–Minicozzi entropy).

Following Colding–Minicozzi [CM12], we define the entropy of
$M^{n}\subset\mathbb{R}^{n+1}$ to be:

$\lambda(M):=\sup_{\begin{subarray}{c}\mathbf{x}_{0}\in\mathbb{R}^{n+1}\\\
t_{0}>0\end{subarray}}\int_{M}(4\pi
t_{0})^{-\frac{n}{2}}e^{-\frac{1}{4t_{0}}|\mathbf{x}-\mathbf{x}_{0}|^{2}}\,d\mathcal{H}^{n}(\mathbf{x})\,.$

By Huisken’s monotonicity of Gaussian area, $t\mapsto\lambda(M(t))$ is non-
increasing when $M(t)$ is a family of closed hypersurfaces flowing smoothly by
mean curvature flow, or more generally a Brakke flow with bounded area ratios
(which holds, including for blowups, assuming that it holds for the initial
data).

######  Remark 1.7.

It is not hard to see that if $M^{n}\subset\mathbb{R}^{n+1}$ is a closed
hypersurface, then
$\lambda(M)=\lambda(\mathbf{z}_{0}+O(M\times\mathbb{R}^{\ell}))$ for all
$\ell\geq 1$, $O\in O(n+\ell+1)$, $\mathbf{z}_{0}\in\mathbb{R}^{n+\ell+1}$.

######  Definition 1.8 (Shrinkers).

A (possibly incomplete) smooth hypersurface $\Sigma\subset\mathbb{R}^{n+1}$ is
said to be a shrinker if it satisfies
$\mathbf{H}+\tfrac{1}{2}\mathbf{x}^{\perp}=\mathbf{0}$. By Huisken’s
monotonicity formula, these model the smooth parts of singularity models in
mean curvature flow. We consider the following sets of shrinkers:

  1. (1)

$\mathcal{S}_{n}$, the set of complete smooth self-shrinkers in
$\mathbb{R}^{n+1}$ with $\lambda(\Sigma)<\infty$.

  2. (2)

$\mathcal{S}_{n}^{*}$, the set of non-flat elements of $\mathcal{S}_{n}$.

  3. (3)

$\mathcal{S}^{\textrm{gen}}_{n}$, the set of “generic” shrinkers, i.e., round
self-shrinking spheres and cylinders over them in $\mathbb{R}^{n+1}$:

$\mathcal{S}^{\textrm{gen}}_{n}:=\left\\{O(\SS^{j}(\sqrt{2j})\times\mathbb{R}^{n-j})\in\mathcal{S}_{n}:j=1,\dots,n,\,O\in
O(n+1)\right\\}.$

When working with entropy upper bounds, we need only consider low-entropy
shrinkers. For $\Lambda>0$, we define:

$\displaystyle\mathcal{S}_{n}(\Lambda)$
$\displaystyle:=\\{\Sigma\in\mathcal{S}_{n}:\lambda(\Sigma)<\Lambda\\},$
$\displaystyle\mathcal{S}_{n}^{*}(\Lambda)$
$\displaystyle:=\mathcal{S}_{n}(\Lambda)\cap\mathcal{S}_{n}^{*}.$

######  Remark 1.9 (Generic shrinker entropies).

Note that Stone [Sto94] computed the entropies of spheres—and thus all generic
shrinkers by Remark 1.7—to be

$2>\lambda(\SS^{1})=\sqrt{\frac{2\pi}{e}}\approx
1.52>\frac{3}{2}>\lambda(\SS^{2})=\frac{4}{e}\approx
1.47>\dots>\lambda(\SS^{n}).$

######  Definition 1.10 (Forwards-backwards parabolic balls).

Throughout the paper, for spacetime points
$X=(\mathbf{x},t)\in\mathbb{R}^{n+1}\times\mathbb{R}$ and radii $r>0$,

$P_{r}(X)=B_{r}(\mathbf{x})\times(t-r^{2},t+r^{2})$

denotes the forwards-backwards parabolic ball of radius $r$ about $X$.

######  Definition 1.11 (Regular vs singular and generic vs non-generic
points).

Let $\mathcal{M}$ be an integral $n$-Brakke flow in $\mathbb{R}^{n+1}$ with
support $\operatorname{supp}\mathcal{M}$.

  1. (1)

Let $\operatorname{reg}\mathcal{M}$ be the set of all _regular points_
$X\in\operatorname{supp}\mathcal{M}$, i.e., those for which there exists $r>0$
so that $\mathcal{M}\lfloor P_{r}(X)$ is the multiplicity one Brakke flow
associated to a smooth mean curvature flow.

  2. (2)

Let $\operatorname{sing}\mathcal{M}$ be the set of all _singular points_
$X\in\operatorname{supp}\mathcal{M}\setminus\operatorname{reg}\mathcal{M}$ so
that $\mathcal{M}$ is defined for times slightly before $t$.

  3. (3)

Decompose $\operatorname{sing}\mathcal{M}$ into “generic” and “non-generic”
singular points:

$\operatorname{sing}\mathcal{M}=\operatorname{sing}_{\textrm{gen}}\mathcal{M}\cup\operatorname{sing}_{\textrm{non-
gen}}\mathcal{M},$

where a point $X\in\operatorname{sing}\mathcal{M}$ is declared to be in
$\operatorname{sing}_{\textrm{gen}}\mathcal{M}$ if some (and thus any by
[CIM15, CM15]) tangent flow is the self-similar flow associated to
$\Sigma\in\mathcal{S}_{n}^{\textrm{gen}}$ with multiplicity one; otherwise
$X\in\operatorname{sing}_{\textrm{non-gen}}\mathcal{M}$.

Note that, by this definition, if $\mathcal{M}$ is a Brakke flow starting at
$t=0$, then we have the disjoint union decomposition

$\operatorname{supp}\mathcal{M}=\operatorname{supp}\mathcal{M}(0)\cup\operatorname{reg}\mathcal{M}\cup\operatorname{sing}\mathcal{M}.$

We now refer the reader to [CCMS20, Section 2] for the complete definitions of
the terms used below.

######  Definition 1.12.

Let $M^{n}\subset\mathbb{R}^{n+1}$ be a closed embedded hypersurface.

  1. (a)

We denote by $\hat{\mathfrak{F}}(M)$ the set of unit-regular cyclic integral
Brakke flows $\mathcal{M}$ with $\mathcal{M}(0)=\mathcal{H}^{n}\lfloor M$.

  2. (b)

We denote by $\mathfrak{F}(M)\subset\hat{\mathfrak{F}}(M)$ the set of
$\mathcal{M}\in\hat{\mathfrak{F}}(M)$ with the following property: there exist
closed embedded hypersurfaces $M_{i}\subset\mathbb{R}^{n+1}$, each disjoint
from $M$, but converging smoothly to $M$, and
$\mathcal{M}_{i}\rightharpoonup\mathcal{M}$ for some
$\mathcal{M}_{i}\in\hat{\mathfrak{F}}(M_{i})$.

Note that $\mathfrak{F}(M)\neq\emptyset$ by [Ilm93, Whi09] (see also [HW20,
Appendix B]).

###  1.5. Main theorem and discussion of main hypotheses

######  Theorem 1.13.

Let $n\in\\{2,3,\ldots\\}$, $\Lambda\in(0,2]$, and assume that our two main
hypotheses, ($\diamondsuit_{n,\Lambda}$) and ($\heartsuit_{n,\Lambda}$)
(discussed below), both hold.

Then, for every closed embedded hypersurface $M^{n}\subset\mathbb{R}^{n+1}$
with $\lambda(M)\leq\Lambda$ there exist arbitrarily small $C^{\infty}$ graphs
$M^{\prime}$ over $M$ so that $\operatorname{sing}_{\textnormal{non-
gen}}\mathcal{M}^{\prime}=\emptyset$ for all
$\mathcal{M}^{\prime}\in\mathfrak{F}(M^{\prime})$.

######  Remark 1.14.

The case $n=2$ already follows from [CCMS21, Theorem 1.1], but it is recovered
by our more general techniques, so we include it for generality.

First main hypothesis, ($\diamondsuit_{n,\Lambda}$). Not every shrinker one
encounters along a weak mean curvature flow is a complete smooth hypersurface,
as considered in $\mathcal{S}_{n}$. Instead, it may itself have singular
points, which (after a blow-up argument) are modeled by minimal cones. In
certain occasions (see Proposition 3.1) these cones are, in fact, stable.
Together with a dimension reduction argument in geometric measure theory due
to Federer, this naturally leads us to consider the following class of
objects:

######  Definition 1.15 (Regular stable cones).

We define $\mathcal{R}\mathcal{S}\mathcal{C}_{k}^{*}$ to be the set of regular
stable stationary cones $\mathcal{C}^{k}\subset\mathbb{R}^{k+1}$, i.e., those
cones that are stationary with
$\operatorname{sing}\mathcal{C}=\\{\mathbf{0}\\}$ (in particular,
$\mathcal{C}$ is non-flat), and so that $\operatorname{reg}\mathcal{C}$ is
stable as a minimal surface.

By [Sim68], $\mathcal{R}\mathcal{S}\mathcal{C}_{k}^{*}=\emptyset$ for $2\leq
k\leq 6$. On the other hand,
$\mathcal{R}\mathcal{S}\mathcal{C}_{k}^{*}\neq\emptyset$ for $k\geq 7$. As
with shrinkers, we will only be interested in cones with entropy below our
upper bound $\Lambda>0$:

$\mathcal{R}\mathcal{S}\mathcal{C}^{*}_{k}(\Lambda):=\\{\mathcal{C}\in\mathcal{R}\mathcal{S}\mathcal{C}^{*}_{k}:\lambda(\mathcal{C})<\Lambda\\}.$

Our first main hypothesis is that there are no such cones below our entropy
threshold:

($\diamondsuit_{n,\Lambda}$)
$\mathcal{R}\mathcal{S}\mathcal{C}_{k}^{*}(\Lambda)=\emptyset\qquad\textrm{
for }k=7,\ldots,n.$

######  Remark 1.16.

Note that ($\diamondsuit_{n,\Lambda}$) is vacuously true for $n\leq 6$ and
$\Lambda>0$. It is a well-known open problem to estimate $\Lambda>0$ so that
($\diamondsuit_{n,\Lambda}$) holds if $n\geq 7$ (cf. [IW15]).

Second main hypothesis, ($\heartsuit_{n,\Lambda}$). For a smooth but possibly
incomplete shrinker $\Sigma$, we define $\mu(\Sigma)$ to be the first
eigenvalue of the $L$-operator, namely

(1.2) $\mu(\Sigma):=\inf\left\\{\int_{\Sigma}(|\nabla
f|^{2}-|A|^{2}f^{2}-\tfrac{1}{2}f^{2})e^{-\frac{1}{4}|\cdot|^{2}}:f\in
C^{\infty}_{c}(\Sigma),\int_{\Sigma}f^{2}e^{-\frac{1}{4}|\cdot|^{2}}=1\right\\},$

We then define

$\mathcal{S}_{n}^{*}(\Lambda,\mu):=\\{\Sigma\in\mathcal{S}_{n}^{*}(\Lambda):\mu(\Sigma)\geq\mu\\}.$

The second main hypothesis is that low-entropy non-generic shrinkers are
“sufficiently” unstable:

($\heartsuit_{n,\Lambda}$)
$\mathcal{S}_{k}^{*}(\Lambda,-\tfrac{n-k}{2})\subset\mathcal{S}_{k}^{\textrm{gen}}\qquad\textrm{
for }k=2,\ldots,n-3.$

######  Remark 1.17.

Note that ($\heartsuit_{n,\Lambda}$) does hold for:

  1. (a)

$n\leq 4$ and all $\Lambda$, vacuously;

  2. (b)

$n=5$ and $\Lambda\leq\lambda(\SS^{1})$, since
$\mathcal{S}_{2}^{*}(\lambda(\SS^{1}))\subset\mathcal{S}_{2}^{\textrm{gen}}$
by [BW17, Corollary 1.2];

###  1.6. Applications of main theorem

The following are all immediate corollaries of our main theorem and Remarks
1.16, 1.17. We skip the low dimensional case $n=2$ since it was already stated
in [CCMS21]. Everywhere below, $M$ denotes a closed embedded hypersurface.

######  Corollary 1.18.

For $M^{3}\subset\mathbb{R}^{4}$ with $\lambda(M)\leq 2$ there exist
arbitrarily small $C^{\infty}$ graphs $M^{\prime}$ over $M$ so that
$\operatorname{sing}_{\textnormal{non-gen}}\mathcal{M}^{\prime}=\emptyset$ for
all $\mathcal{M}^{\prime}\in\mathfrak{F}(M^{\prime})$.

######  Corollary 1.19.

For $M^{4}\subset\mathbb{R}^{5}$ with $\lambda(M)\leq 2$ there exist
arbitrarily small $C^{\infty}$ graphs $M^{\prime}$ over $M$ so that
$\operatorname{sing}_{\textnormal{non-gen}}\mathcal{M}^{\prime}=\emptyset$ for
all $\mathcal{M}^{\prime}\in\mathfrak{F}(M^{\prime})$.

######  Corollary 1.20.

For $M^{5}\subset\mathbb{R}^{6}$ with $\lambda(M)\leq\lambda(\SS^{1})$ there
exist arbitrarily small $C^{\infty}$ graphs $M^{\prime}$ over $M$ so that
$\operatorname{sing}_{\textnormal{non-gen}}\mathcal{M}^{\prime}=\emptyset$ for
all $\mathcal{M}^{\prime}\in\mathfrak{F}(M^{\prime})$.

######  Remark 1.21.

To relax the entropy bound $\lambda(\SS^{1})$ to $2$ in Corollary 1.20, one
could try to see whether or not $(\heartsuit_{5,2})$ holds. This is an
independently interesting question: we are asking if all
$\Sigma\in\mathcal{S}_{2}^{*}\setminus\mathcal{S}_{2}^{\textrm{gen}}$ with
entropy $\lambda(\Sigma)<2$ have $\mu(\Sigma)<-\tfrac{3}{2}$. Note that
numerics [BK21a, BK21b, BK22] suggest that Angenent’s torus has $\lambda\sim
1.85$, $\mu\sim-3.74$.

Here is a topological application of our theorem:

######  Corollary 1.22.

Take $n\in\\{2,3,\ldots\\}$, $\Lambda\in(0,2]$ such that
($\diamondsuit_{n,\Lambda}$) and ($\heartsuit_{n,\Lambda}$) hold. Let
$M^{n}\subset\mathbb{R}^{n+1}$ be a closed connected embedded hypersurface
with $\lambda(M)\leq\Lambda$.

  1. (a)

If $\Lambda\leq\lambda(\SS^{n-1})$ then perhaps after a small initial
$C^{\infty}$ perturbation, the mean curvature flow provides a smooth isotopy
from $M$ to a standard $\SS^{n}$ in $\mathbb{R}^{n+1}$.

  2. (b)

If $\Lambda\leq\lambda(\SS^{n-2})$ then perhaps after a small initial
$C^{\infty}$ perturbation, the mean curvature flow with surgery of [DH22]
provides a smooth isotopy from $M$ to the boundary of a standard handlebody
that is either a standard ball $B^{n}$ or a boundary connect sum of finitely
many $B^{n-1}\times\SS^{1}$’s.

By Remarks 1.16 and 1.17, Corollary 1.22 holds unconditionally (cf. Corollary
1.5) when $n\leq 5$.

###### Proof.

Combining the argument used in the beginning of Section 6 with Theorem 1.13,
either $M$ is a round sphere (in which case we are done), or we can find a
small $C^{\infty}$ graph $M^{\prime}$ over $M$ so that $\lambda(M)<\Lambda$
and so that there is $\mathcal{M}^{\prime}\in\mathfrak{F}(M^{\prime})$ with
$\operatorname{sing_{\textnormal{non-gen}}}\mathcal{M}^{\prime}=\emptyset$.

Then, in case (a), we have $\lambda(\mathcal{M}^{\prime})<\lambda(\SS^{n-1})$
so $\mathcal{M}^{\prime}$ can only have multiplicity-one $\SS^{n}$-type
singularities. Thus, the mean curvature flow is completely smooth until it
becomes a round sphere. In case (b), we additionally use the surgery of
[DH22]. ∎

Here is an application to regularity of the flow (by convention,
$\lambda(\SS^{-1})=\lambda(\SS^{0})=2$):

######  Corollary 1.23.

Take $n\in\\{2,\dots\\}$ and $\Lambda\in(0,\lambda(\SS^{n-3})]$ such that
($\diamondsuit_{n,\Lambda}$) and ($\heartsuit_{n,\Lambda}$) hold. If Let
$M^{n}\subset\mathbb{R}^{n+1}$ be a closed connected embedded hypersurface
with $\lambda(M)\leq\Lambda$. There is an arbitrarily small graph $M^{\prime}$
over $M$ so that any mean curvature flow
$\mathcal{M}^{\prime}\in\mathfrak{F}(M^{\prime})$ is completely smooth for
a.e. time $t$ and each connected component of
$\operatorname{sing}\mathcal{M}^{\prime}$ is contained in some time-slice.

###### Proof.

Combine Theorem 1.13 with [CM16]. ∎

By Remarks 1.16 and 1.17, Corollary 1.23 holds unconditionally when $n\leq 5$.

###  1.7. Organization

Section 2 contains some preliminary discussions about rescaled flows,
regularity, smoothly crossing flows, and parabolic covering. Section 3
contains regularity and eigenvalue estimates for low-entropy $F$-stationary
varifolds. In Section 4 we establish the main local estimates for nearly
ancient one-sided flows near nearly-self-similar flows. We then use this in
Section 5 to prove the central density drop result. We then show how to handle
the issue of non-generic singularities limiting to bubble-sheet generic
singularities in Section 6, completing the proof of the main result. Appendix
A discusses a form of the multiplicity-one conjecture as it relates to our
results here. In Appendix B we estimate the first eigenvalue of the
$L$-operator on non-flat stable cones (this is not used elsewhere in the
paper).

###  1.8. Acknowledgements

We are grateful to Kyeongsu Choi and Brian White for helpful discussions. O.C.
was supported by a Terman Fellowship and an NSF grant (DMS-2304432). C.M. was
supported by an NSF grant (DMS-2147521).

##  2\. Preliminaries and notation

###  2.1. $F$-functional and density

Let us begin by recalling a few basic notions in the smooth setting. First,
for smooth hypersurfaces $M^{n}\subset\mathbb{R}^{n+1}$ one has the
$F$-functional,

(2.1)
$F(M)=(4\pi)^{-\frac{n}{2}}\int_{M}e^{-\tfrac{1}{4}|\cdot|^{2}}\,d\mathcal{H}^{n},$

which in turn gives rise to the Colding–Minicozzi entropy (Definition 1.6) via

(2.2)
$\lambda(M)=\sup_{\begin{subarray}{c}\mathbf{x}_{0}\in\mathbb{R}^{n+1}\\\
t_{0}>0\end{subarray}}F\left(\frac{1}{\sqrt{t_{0}}}(M-\mathbf{x}_{0})\right).$

Moreover, for a smooth mean curvature flow with bounded area ratios
$\mathcal{M}:t\mapsto M(t)$, the $F$-functional also gives rise to the density
function, defined as

(2.3)
$\Theta_{\mathcal{M}}(X,r)=F\left(\frac{1}{r}(M(t-r^{2})-\mathbf{x})\right),\;X\in\mathbb{R}^{n+1}\times\mathbb{R},\;r>0,$

which is nondecreasing in $r$ by Huisken’s well-known monotonicity formula. In
particular,

(2.4) $\Theta_{\mathcal{M}}(X)=\lim_{r\to
0}\Theta_{\mathcal{M}}(X,r),\;X\in\mathbb{R}^{n+1}\times\mathbb{R},$

is well-defined.

For our proofs, we will need to work with weaker objects than smooth
hypersurfaces, namely $n$-varifolds in $\mathbb{R}^{n+1}$ (see [Sim83]) and
Brakke flows (see [Ilm94] and [CCMS20, Section 2]). In this setting, (2.1),
(2.2) extend to $n$-varifolds in $\mathbb{R}^{n+1}$ (by looking at their
induced Radon measures), and the monotonicity underlying (2.3), (2.4) extends
to $n$-Brakke flows in $\mathbb{R}^{n+1}$ with bounded area ratios (cf.
[Ilm95, Lemma 7]).

###  2.2. $F$-stationary varifolds

It is a simple computation that all elements of $\mathcal{S}_{n}$ are
stationary for the $F$-functional (cf. [CM12, §3]) or equivalently for the
area functional with respect to the conformal metric
$e^{-|\cdot|^{2}/(2n)}g_{\mathbb{R}^{n+1}}$ on $\mathbb{R}^{n+1}$. Thus,
Definition 2.1 below precisely generalizes the notion of shrinkers to the
varifold setting.

######  Definition 2.1.

An $n$-varifold $V$ in $\mathbb{R}^{n+1}$ is called $F$-stationary if is
stationary with respect to the conformally metric
$e^{-|\cdot|^{2}/(2n)}g_{\mathbb{R}^{n+1}}$ on $\mathbb{R}^{n+1}$.

######  Definition 2.2.

Let $V$ be an $F$-stationary $n$-varifold in $\mathbb{R}^{n+1}$. We define an
associated $n$-Brakke flow $\mathcal{M}_{V}$ as follows: if $V$ is a cone we
define $\mathcal{M}_{V}$ as a static flow

$\mathcal{M}_{V}(t)=V\text{ for all }t\in\mathbb{R},$

while if $V$ is not a cone, we define $\mathcal{M}_{V}$ as a shrinking flow
that disappears at $t=0$:

$\mathcal{M}_{V}(t)=\begin{cases}\sqrt{-t}V&\text{ for }t<0,\\\ 0&\text{ for
}t\geq 0.\end{cases}$

In the latter case $\mathcal{M}_{V}$ may only be unit-regular for times $t<0$;
e.g., if $V$ is a non-conical self-shrinker with asymptotically conical ends,
then $\mathcal{M}_{V}$ vanishes “unexpectedly” at $t=0$. This will not pose
any issue in the sequel; cf. the $T$ parameter in Lemma 2.10.

It is not hard to see, using the monotonicity formula on $\mathcal{M}_{V}$,
that

$\lambda(V)=F(V)$

for all $F$-stationary $n$-varifolds $V$ in $\mathbb{R}^{n+1}$.

###  2.3. Regularity scale of Brakke flows

######  Definition 2.3 (Regularity scale).

Let $\mathcal{M}$ be an integral $n$-Brakke flow in $\mathbb{R}^{n+1}$ and
$X=(\mathbf{x},t)\in\operatorname{reg}\mathcal{M}$. We define the _regularity
scale_ of $\mathcal{M}$ at $X$ by

$r_{\mathcal{M}}(X):=\sup\\{r>0:P_{r}(X)\cap\operatorname{supp}\mathcal{M}\subset\operatorname{reg}\mathcal{M},\;|A|\leq
r^{-1}\textrm{ on }P_{r}(X)\\}.$

For $\rho,r>0$ and $(\mathbf{x},t)\in\mathbb{R}^{n+1}\times\mathbb{R}$ we set

$\mathcal{R}_{\rho,r}\mathcal{M}(\mathbf{x},t):=\left\\{\mathbf{y}\in\operatorname{supp}\mathcal{M}(t)\cap
B_{r}(\mathbf{x}):r_{\mathcal{M}}(\mathbf{y},t)>\rho\right\\}\subset\operatorname{reg}\mathcal{M}(t).$

For an integral $F$-stationary varifold we abuse notation and write
$r_{V}(\mathbf{x}):=r_{\mathcal{M}_{V}}(\mathbf{x},-1)$ and
$\mathcal{R}_{\rho,r}V(\mathbf{x}):=\mathcal{R}_{\rho,r}\mathcal{M}_{V}(\mathbf{x},-1)$.

######  Remark 2.4.

For $\lambda>0$, the rescaled flows
$\mathcal{M}_{\lambda}(t)=\lambda\mathcal{M}(\lambda^{-2}t)$ satisfy

$\mathcal{R}_{\lambda\rho,\lambda
r}\mathcal{M}_{\lambda}(\lambda^{-1}\mathbf{x},\lambda^{-2}t)=\lambda\mathcal{R}_{\rho,r}\mathcal{M}(\mathbf{x},t).$

Furthermore, for $\rho^{\prime}\leq\rho,r\leq r^{\prime}$ we have

$\mathcal{R}_{\rho,r}\mathcal{M}(\mathbf{x},t)\subset\mathcal{R}_{\rho^{\prime},r^{\prime}+|\mathbf{x}-\mathbf{x}^{\prime}|}\mathcal{M}(\mathbf{x}^{\prime},t).$

######  Remark 2.5 (Continuity of regularity scale).

It is standard to show, under the assumption that the entropy is uniformly
bounded above by $2-\varepsilon$ with $\varepsilon>0$, that
$r_{\mathcal{M}}(X)$ is continuous with respect to convergence of the point
$X$ and the $n$-Brakke flow $\mathcal{M}$ if, e.g., the initial conditions
converge smoothly. See [CMS23a, Lemma 2.4] for the analogous result for
minimizing hypersurfaces (note that the upper bound on the entropy prevents
issues with multiplicity, which is not a concern for minimizing
hypersurfaces).

###  2.4. Rescaled Brakke flows

######  Definition 2.6.

Given a Brakke flow $\mathcal{N}$ and a spacetime point
$X_{0}=(\mathbf{x}_{0},t_{0})$ we can define the _rescaled Brakke flow around
$X_{0}$ at scale $\beta$_ by the usual recipe

$\tilde{\mathcal{N}}(\tau):=\beta^{-1}e^{\tau/2}(\mathcal{N}(t_{0}-\beta^{2}e^{-\tau})-\mathbf{x}_{0}).$

Note that $\tau=0$ corresponds to $t=t_{0}-\beta^{2}$ (if $\mathcal{N}$ is
defined then).

######  Definition 2.7.

Consider a rescaled Brakke flow $\tilde{\mathcal{N}}$ at scale $\beta=1$ for
simplicity. We define the regularity scale
$r_{\tilde{\mathcal{N}}}(\mathbf{x},\tau)$ as follows. Define a non-rescaled
flow

$\mathcal{N}(t)=\sqrt{-t}\tilde{\mathcal{N}}(\tau-\log(-t))$

and then set
$r_{\tilde{\mathcal{N}}}(\mathbf{x},\tau):=r_{\mathcal{N}}(\mathbf{x},-1)$.

###  2.5. Regularity of low-entropy $F$-stationary varifolds

######  Lemma 2.8.

Let $V$ be a cyclic integral $F$-stationary $n$-varifold in $\mathbb{R}^{n+1}$
with $\lambda(V)<2$. Then:

  1. (a)

We have $\dim_{H}\operatorname{sing}V\leq n-3$.

  2. (b)

If $\operatorname{sing}V\neq\emptyset$, then $V$ has an iterated tangent cone
of the form (up to rotation)

$\mathbb{R}^{n-k}\times\mathcal{C}^{k},\;k\geq 3,$

with $\mathcal{C}^{k}\subset\mathbb{R}^{k+1}$ a smooth multiplicity-one non-
flat minimal cone.

###### Proof.

The assumption $\lambda(V)<2$ rules out tangent cones given by higher
multiplicity planes or unions four or more half-planes. The union of three
half-planes is ruled out by the cyclic assumption thanks to [Whi09]. The
assertion then follows from standard dimension reduction arguments (cf.
[Whi97]). ∎

###  2.6. Smoothly crossing Brakke flows

We recall here

######  Definition 2.9 ([CCMS21, Definition B.1]).

Consider integral $n$-Brakke flows $\mathcal{M}$, $\mathcal{M}^{\prime}$ in
$\mathbb{R}^{n+1}$. We say that $\mathcal{M}$, $\mathcal{M}^{\prime}$ _cross
smoothly_ at
$X=(\mathbf{x},t)\in\operatorname{reg}\mathcal{M}\cap\operatorname{reg}\mathcal{M}^{\prime}$
if there is $r>0$ so that $\mathcal{M}\lfloor
P(X,r),\mathcal{M}^{\prime}\lfloor P(X,r)$ are equal to smooth connected
multiplicity-one flows $\Gamma(s),\Gamma^{\prime}(s)$ so that $\Gamma(t)$ has
points on both sides of $\Gamma^{\prime}(t)$ in any small neighborhood of
$\mathbf{x}$. We also say that $\mathcal{M}$ and $\mathcal{M}^{\prime}$ cross
smoothly (with no reference to a point) if they cross smoothly at some
$X\in\operatorname{reg}\mathcal{M}\cap\operatorname{reg}\mathcal{M}^{\prime}$.

######  Lemma 2.10 ([CCMS21, Lemma B.2]).

Consider integral $n$-Brakke flows $\mathcal{M}_{i}$,
$\mathcal{M}_{i}^{\prime}$ in $\mathbb{R}^{n+1}$ that are unit-regular for
times $t<T$ (possibly $T=\infty$). Assume that
$\mathcal{M}_{i}\rightharpoonup\mathcal{M}$,
$\mathcal{M}^{\prime}_{i}\rightharpoonup\mathcal{M}^{\prime}$, and that
$\mathcal{M}$, $\mathcal{M}^{\prime}$ cross smoothly at $X=(\mathbf{x},t)$
with $t<T$. Then, there are $X_{j}\to X$ so that $\mathcal{M}_{j}$,
$\mathcal{M}_{j}^{\prime}$ cross smoothly at $X_{j}$ for all sufficiently
large $j$.

###### Proof.

The proof in the reference (where no $T$ parameter was present) still applies.
∎

######  Lemma 2.11.

Let $M,M^{\prime}\subset\mathbb{R}^{n+1}$ be closed embedded hypersurfaces
which are disjoint or coincide. Then, no pair $\mathcal{M}\in\mathfrak{F}(M)$,
$\mathcal{M}^{\prime}\in\mathfrak{F}(M^{\prime})$ can cross smoothly.

###### Proof.

If $M\neq M^{\prime}$, then $M\cap M^{\prime}=\emptyset$ implies
$\operatorname{supp}\mathcal{M}\cap\operatorname{supp}\mathcal{M}^{\prime}=\emptyset$
by the avoidance principle (cf. [Ilm94, Lemma 10.6]).

It remains to consider $M=M^{\prime}$. Using the definition of
$\mathfrak{F}(M)$, pick any $M_{i}$ converging smoothly to $M$ with $M_{i}\cap
M=\emptyset$, and its corresponding
$\mathcal{M}_{i}\in\hat{\mathfrak{F}}(M_{i})$ converging weakly to
$\mathcal{M}$. Do the same with $M^{\prime}$, $M^{\prime}_{i}$,
$\mathcal{M}^{\prime}_{i}$. Passing to a subsequence of
$\\{M^{\prime}_{i}\\}$, we may assume that $M_{i}\cap
M^{\prime}_{i}=\emptyset$, and thus
$\operatorname{supp}\mathcal{M}_{i}\cap\operatorname{supp}\mathcal{M}^{\prime}_{i}=\emptyset$
by avoidance. The result follows from Lemma 2.10. ∎

Abusing notation, we’ll say that two $F$-stationary $n$-varifolds $V$,
$V^{\prime}$ in $\mathbb{R}^{n+1}$ cross smoothly if their associated
shrinking Brakke flows $\mathcal{M}_{V}$, $\mathcal{M}_{V^{\prime}}$ cross
smoothly in the sense of Definition 2.9. Equivalently, $V$, $V^{\prime}$ cross
smoothly at $\mathbf{x}$ if there exists $r>0$ so that $V\lfloor
B_{r}(\mathbf{x})$, $V^{\prime}\lfloor B_{r}(\mathbf{x})$ are smooth connected
multiplicity-one self-shrinkers $\Sigma$, $\Sigma^{\prime}$ so that $\Sigma$
has points on both sides of $\Sigma^{\prime}$ in any small neighborhood of
$\mathbf{x}$.

######  Lemma 2.12.

If $V$, $V^{\prime}$ are cyclic integral $F$-stationary $n$-varifolds in
$\mathbb{R}^{n+1}$ with $\lambda(V)$, $\lambda(V^{\prime})<2$, then either
$V=V^{\prime}$ or $V$, $V^{\prime}$ cross smoothly.

###### Proof.

By the Frankel property for self-shrinkers (cf. [CCMS20, Corollary C.4]) we
have $\operatorname{supp}V\cap\operatorname{supp}V^{\prime}\not=\emptyset$.
Suppose that $\operatorname{reg}V\cap\operatorname{reg}V^{\prime}=\emptyset$.
Since

(2.5)
$\dim_{H}\operatorname{sing}V,\;\dim_{H}\operatorname{sing}V^{\prime}\leq n-3$

by Lemma 2.8, this would imply that
$\dim_{H}(\operatorname{supp}V\cap\operatorname{supp}V^{\prime})\leq n-3$.
This contradicts the strong maximum principle for varifolds [Ilm96, Theorem
A(i)] (cf. [Wic14b]).

We may thus consider
$\mathbf{x}\in\operatorname{reg}V\cap\operatorname{reg}V^{\prime}$. If $V$,
$V^{\prime}$ cross smoothly at $\mathbf{x}$, we are done. If they do not, then
$V$ lies (weakly) to one-side of $V^{\prime}$ near $\mathbf{x}$. The strong
maximum principle then implies that $V=V^{\prime}$ near $\mathbf{x}$. Since
$\operatorname{reg}V,\operatorname{reg}V^{\prime}$ are connected (this holds
by combining the Frankel property for self-shrinkers with [Ilm96, Theorem
A(ii)]) we thus see that $\operatorname{reg}V=\operatorname{reg}V^{\prime}$.
Combined with (2.5) and $\lambda(V)$, $\lambda(V^{\prime})<2$, we thus see
that $V=V^{\prime}$. ∎

######  Corollary 2.13.

Let $M\subset\mathbb{R}^{n+1}$ be a closed embedded hypersurface with
$\lambda(M)<2$. Suppose that
$\mathcal{M},\mathcal{M}^{\prime}\in\mathfrak{F}(M)$ and
$X\in\operatorname{supp}\mathcal{M}\cap\operatorname{supp}\mathcal{M}^{\prime}\cap\\{t>0\\}$.
Then:

  1. (a)

$\Theta_{\mathcal{M}}(X)=\Theta_{\mathcal{M}^{\prime}}(X)$, and

  2. (b)

all tangent flows at $X$ for $\mathcal{M}$, $\mathcal{M}^{\prime}$ coincide
for $t<0$.

###### Proof.

Note that (b) implies (a), so we can just prove (b). To that end, it suffices
to show that any tangent flow $\tilde{\mathcal{M}}$ to $\mathcal{M}$ at $X$ is
also a tangent flow to $\mathcal{M}^{\prime}$ at $X$. The result then follows
by swapping $\mathcal{M}$ and $\mathcal{M}^{\prime}$.

Let $\tilde{\mathcal{M}}$ be a tangent flow to $\mathcal{M}$ at $X$ obtained
by parabolically dilating by $\lambda_{i}\to\infty$. Let the corresponding
dilated flows of $\mathcal{M}$ be $\tilde{\mathcal{M}}_{j}$, and those of
$\mathcal{M}^{\prime}$ be $\tilde{\mathcal{M}}^{\prime}_{j}$, so that

$\tilde{\mathcal{M}}_{j}\rightharpoonup\tilde{\mathcal{M}}$

and, after passing to a subsequence,

$\tilde{\mathcal{M}}^{\prime}_{j}\rightharpoonup\tilde{\mathcal{M}}^{\prime},$

for a tangent flow $\tilde{\mathcal{M}}^{\prime}$ of $\mathcal{M}^{\prime}$ at
$X$. By Lemma 2.11, $\mathcal{M}$ and $\mathcal{M}^{\prime}$ cannot cross
smoothly, and thus neither can their dilations $\tilde{\mathcal{M}}_{j}$,
$\tilde{\mathcal{M}}^{\prime}_{j}$. Thus, by Lemma 2.10,
$\tilde{\mathcal{M}}$, $\tilde{\mathcal{M}}^{\prime}$ cannot cross smoothly
either.

If $V$ and $V^{\prime}$ are the $F$-stationary $n$-varifolds associated with
$\tilde{\mathcal{M}}$ and $\tilde{\mathcal{M}}^{\prime}$ then $\lambda(M)<2$
implies that $\lambda(V)$ and $\lambda(V^{\prime})<2$, so in particular
$V=V^{\prime}$ by Lemma 2.12. Thus, the set of tangent flows to $\mathcal{M}$
at $X$ is the same as the set of tangent flows to $\mathcal{M}^{\prime}$ at
$X$, at least in their $t<0$ portions that are uniquely determined by $V$,
$V^{\prime}$. ∎

###  2.7. Parabolic planes and a covering lemma

We will be relying on estimates on the parabolic Hausdorff dimension of the
singular set of a mean curvature flow obtained in [Whi97, CHN13]. These
estimates, as in the case of minimizing hypersurfaces, rely on the notion of
the spine of a tangent flow (see Definition 3.3 below). To that end, we
introduce some notation for the set of possible spines.

######  Definition 2.14.

We denote $\mathscr{S}$ by the set of subspaces $\Pi$ of
$\mathbb{R}^{n+1}\times\mathbb{R}$ that are either of the form
$\Pi=\Pi^{\prime}\times\\{0\\}$ or $\Pi^{\prime}\times\mathbb{R}$ for
$\Pi^{\prime}\subset\mathbb{R}^{n+1}$ a subspace. Write
$D(\Pi)=\dim\Pi^{\prime}$ in the first case and $D(\Pi)=\dim\Pi^{\prime}+2$ in
the second (this is just the _parabolic_ dimension of
$\Pi\subset\mathbb{R}^{n+1}\times\mathbb{R}$).

######  Remark 2.15.

In the general parabolic setting spines need not be elements of $\mathscr{S}$
due to the possibility of quasi-static cones arising as tangent flows.
However, unit-regular flows with entropy $<2$ do not have quasi-static cones
arising as tangent flows, so Definition 2.14 precisely suffices for our
consideration of spines.

######  Lemma 2.16 (cf. [CHN13, (3.14)]).

There is $C=C(n)\in[1,\infty)$ with the following property.

For any $\Pi\in\mathscr{S}$ and $\gamma\in(0,1)$, if $\mathfrak{X}\subset
U_{\gamma}(\Pi)\cap P((\mathbf{0},0),1)$ is an arbitrary nonempty subset, then
there are points $X_{1},\dots,X_{K}\in\mathfrak{X}$ so that

$\mathfrak{X}\subset\cup_{k=1}^{K}P(X_{k},\gamma)\text{ with }K\leq
C\gamma^{-D(\Pi)}.$

###### Proof.

This follows in a standard way from the Vitali covering lemma. ∎

##  3\. Low-entropy $F$-stationary varifolds

In this section we consider a cyclic integral $F$-stationary $n$-varifold $V$
in $\mathbb{R}^{n+1}$ with $\lambda(V)<2$. We will write
$\mu(V)=\mu(\operatorname{reg}V)$ (see (1.2)).

######  Proposition 3.1.

If $\mu(V)>-\infty$ then any blow-up limit of $V$ has stable regular part.

###### Proof.

Consider a blow-up limit
$\tilde{V}_{i}=\lambda_{i}(V-\mathbf{x}_{i})\rightharpoonup\tilde{V}$ with
$\lambda_{i}\to\infty$, $\mathbf{x}_{i}\to\mathbf{x}_{\infty}$. Assume, for
contradiction, that $\tilde{V}$ has unstable regular part. Let $U$ be an open
set such that

$\operatorname{sing}\tilde{V}\cap U=\emptyset\text{ and
}\operatorname{reg}\tilde{V}\cap U\text{ is unstable}.$

Fix some $\delta>0$ and a nonzero $\tilde{f}\in
C^{\infty}_{c}(\operatorname{reg}\tilde{V}\cap U)$ with

$\int_{\operatorname{reg}\tilde{V}}\big{(}|\nabla\tilde{f}(\mathbf{y})|^{2}-|A(\mathbf{y})|^{2}\tilde{f}(\mathbf{y})^{2}\big{)}\,d\mathbf{y}\leq-2\delta\int_{\operatorname{reg}\tilde{V}}\tilde{f}(\mathbf{y})^{2}\,d\mathbf{y}.$

Since $\operatorname{sing}\tilde{V}\cap U=\emptyset$ we can find non-zero
$\tilde{f}_{i}\in C^{\infty}_{c}(\operatorname{reg}\tilde{V}_{i}\cap U)$
converging to $f$ smoothly. Observing that as $i\to\infty$,

$e^{\tfrac{1}{4}|\mathbf{x}_{i}|^{2}}e^{-\tfrac{1}{4}|\mathbf{x}_{i}+\lambda_{i}^{-1}\mathbf{y}|^{2}}=e^{-\tfrac{1}{2}\lambda_{i}^{-1}\mathbf{x}_{i}\cdot\mathbf{y}-\tfrac{1}{4}\lambda_{i}^{-2}|\mathbf{y}|^{2}}\to
1\text{ uniformly for $|\mathbf{y}|$ bounded},$

we thus conclude that, for $i$ sufficiently large,

$\int_{\operatorname{reg}\tilde{V}_{i}}\big{(}|\nabla\tilde{f}_{i}|(\mathbf{y})^{2}-|A(\mathbf{y})|^{2}\tilde{f}_{i}(\mathbf{y})^{2}\big{)}e^{-\tfrac{1}{4}|\mathbf{x}_{i}+\lambda_{i}^{-1}\mathbf{y}|^{2}}\,d\mathbf{y}\leq-\delta\int_{\operatorname{reg}\tilde{V}_{i}}\tilde{f}_{i}(\mathbf{y})^{2}e^{-\tfrac{1}{4}|\mathbf{x}_{i}+\lambda_{i}^{-1}\mathbf{y}|^{2}}\,d\mathbf{y}.$

Define $f_{i}\in C^{\infty}_{c}(\operatorname{reg}V)$ by pulling
$\tilde{f}_{i}$ back to the original scale, i.e.
$f_{i}(\mathbf{x})=\tilde{f}_{i}(\lambda_{i}(\mathbf{x}-\mathbf{x}_{i}))$. We
have:

$\int_{\operatorname{reg}V}\big{(}|\nabla
f_{i}|(\mathbf{x})^{2}-|A(\mathbf{x})|^{2}f_{i}(\mathbf{x})^{2}-\tfrac{1}{2}f_{i}(\mathbf{x})^{2}\big{)}e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}\,d\mathbf{x}\leq(-\delta\lambda_{i}^{2}-\tfrac{1}{2})\int_{\operatorname{reg}V}f_{i}(\mathbf{x})^{2}e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}\,d\mathbf{x}.$

Since $\lambda_{i}\to\infty$ we find that $\mu(V)=-\infty$. This completes the
proof. ∎

Arguing as in the proof of Lemma 2.8 we have the following result.

######  Corollary 3.2.

Suppose that ($\diamondsuit_{n,\Lambda}$) holds. If $\lambda(V)<\Lambda$ and
$\mu(V)>-\infty$ then $\operatorname{sing}V=\emptyset$, i.e., $V$ is the
varifold associated to some smooth self-shrinker $\Sigma\in\mathcal{S}_{n}$.

We recall Definition 2.2 for the shrinking flow $\mathcal{M}_{V}$ associated
to $V$.

######  Definition 3.3.

For any $F$-stationary varifold $V$ we set

(3.1)
$\operatorname{spine}V:=\\{X\in\mathbb{R}^{n+1}\times\mathbb{R}:\mathcal{M}_{V}+X=\mathcal{M}_{V}\\}.$

It follows from [Whi97, §8] and Remark 2.15 that
$\operatorname{spine}V\in\mathscr{S}$ and $\Theta_{\mathcal{M}_{V}}(X)=F(V)$
if and only if $X\in\operatorname{spine}V$. By abuse of notation we will write

$D(V)=D(\operatorname{spine}V),$

where $D(\operatorname{spine}V)$ is as in Definition 2.14.

######  Lemma 3.4.

Suppose that $V_{j}\rightharpoonup V$ is a convergent sequence of
$F$-stationary integral varifolds. Then:

  1. (a)

$\limsup_{j}D(V_{j})\leq D(V)$, and

  2. (b)

$\limsup_{j}\mu(V_{j})\leq\mu(V)$.

###### Proof.

Suppose that $\mathcal{M}_{V_{j}}+X_{j}=\mathcal{M}_{V_{j}}$ for
$X_{j}\in\operatorname{spine}V_{j}$. Without loss of generality, $|X_{j}|=1$
for all $j$ and thus $X_{j}\to X$ after passing to a subsequence. Passing
$\mathcal{M}_{V_{j}}+X_{j}=\mathcal{M}_{V_{j}}$ to the limit, too, we get
$\mathcal{M}_{V}+X=\mathcal{M}_{V}$, so $X\in\operatorname{spine}V$. This
proves (a).

Assertion (b) follows from (1.2) since on every open set
$U\Subset\mathbb{R}^{n+1}\setminus\operatorname{sing}V$ we have smooth
convergence of $V_{j}\to V$. ∎

The following estimate222With some extra effort, it should be possible to show
that there is equality in the above lemma, but we will not need this fact
here.  is straightforward to prove by cutting off in the
$\mathbb{R}^{\ell}$-factors and using the finiteness of Gaussian area.

######  Lemma 3.5.

For $\Sigma\in\mathcal{S}_{k}$ we have
$\mu(\mathbb{R}^{\ell}\times\Sigma)\leq\mu(\Sigma)$.

The following result is a crucial ingredient in this paper. We estimate the
size of the spine versus the first eigenvalue of the $L$ operator. It is
crucial that we prove a uniform bound, and this necessarily complicates the
statement and proof.

######  Proposition 3.6.

Fix $n\in\\{2,3,\ldots\\}$, $\Lambda\in(0,2]$ so that
($\heartsuit_{n,\Lambda}$) and ($\diamondsuit_{n,\Lambda}$) hold and let
$\varepsilon>0$. There are constants
$\kappa=\kappa(n,\Lambda,\varepsilon)>0,\rho_{0}=\rho_{0}(n,\Lambda,\varepsilon)>0,r_{0}=r_{0}(n,\Lambda,\varepsilon)>2$
so that the following holds.

Let $V$ be a non-flat and non-generic $F$-stationary cyclic integral
$n$-varifold in $\mathbb{R}^{n+1}$ with $\lambda(V)\leq\Lambda-\varepsilon$.
Then:

  1. (a)

$\mathcal{R}_{3\rho_{0},r_{0}/3}V(\mathbf{0})\neq\emptyset$.

  2. (b)

For any
$\mathcal{R}_{3\rho_{0},r_{0}/3}V(\mathbf{0})\subset\Omega\subset\operatorname{reg}V$,
we have

$2\mu(\Omega)+D(V)<-2\kappa.$

###### Proof.

Suppose for contradiction that (a) fails, i.e., there is a sequence of
$F$-stationary cyclic integral $n$-varifolds in $\mathbb{R}^{n+1}$, $V_{j}$,
with

$\mathcal{R}_{3j^{-1},j/3}V_{j}(\mathbf{0})=\emptyset.$

Pass to a subsequential limit $V_{j}\rightharpoonup V$. Note that $V$ is a
non-flat $F$-stationary cyclic integral $n$-varifold with
$\lambda(V)\leq\Lambda-\varepsilon$ (by a cutoff argument) and
$V\not\in\mathcal{S}_{n}^{\textrm{gen}}$ (by [CIM15]). In particular,
$\mathcal{R}_{\rho,r}V\neq\emptyset$ for $\rho$ sufficiently small and $r$
sufficiently large. This is a contradiction, proving (a).

Note that (a) remains valid if we decrease $\rho_{0}$ and increase $r_{0}$. As
such, to prove (b) it suffices to consider $V_{j}$ with
$\mathcal{R}_{3j^{-1},j/3}V_{j}(\mathbf{0})\subset\Omega_{j}\subset\operatorname{reg}V_{j}$
so that

(3.2) $0\leq\limsup_{j}(2\mu(\Omega_{j})+D(V_{j}))$

Pass to a subsequential limit $V_{j}\rightharpoonup V$ as above.

######  Claim 3.7.

We have $2\mu(V)+D(V)<0$.

Assume this claim for now. Choose $\rho>0$, $r>2$,
$\Omega^{\prime}\subset\mathcal{R}_{4\rho,r/4}V(\mathbf{0})\subset\operatorname{reg}V$
so that

$2\mu(\Omega^{\prime})+D(V)<0.$

Allard’s theorem yields for sufficiently large $j$ regions
$\Omega_{j}^{\prime}\subset\mathcal{R}_{3\rho,r/3}V_{j}(\mathbf{0})\subset\Omega_{j}\subset\operatorname{reg}V$
converging as smooth graphs over $\Omega^{\prime}$. Transplanting test
functions on $\Omega^{\prime}$ to $\Omega_{j}^{\prime}$ we find that
$\limsup_{j}\mu(\Omega_{j}^{\prime})\leq\mu(\Omega^{\prime})$. Combined with
$\mu(\Omega_{j})\leq\mu(\Omega_{j}^{\prime})$ and Lemma 3.4 we find

$\limsup_{j}(2\mu(\Omega_{j})+D(V_{j}))\leq\limsup_{j}(2\mu(\Omega_{j}^{\prime})+D(V_{j}))\leq
2\mu(\Omega^{\prime})+D(V)<0.$

This contradicts (3.2), and proves (b).

It remains to prove Claim 3.7. Note it trivially holds if $\mu(V)=-\infty$, so
by Corollary 3.2 we can take $V$ to be the varifold associated to a smooth
self-shrinker
$\Sigma\in\mathcal{S}_{n}^{*}\setminus\mathcal{S}_{n}^{\textrm{gen}}$. Then,
$\mu(\Sigma)<-1$ by [CM12, Theorems 9.36 and 0.17], so Claim 3.7 automatically
holds if $D(V)\leq 2$. We can thus consider $D(V)\geq 3$. We know that

$\Sigma=\Pi^{\prime}\times\Sigma^{\prime}\text{ for
}\Sigma^{\prime}\in\mathcal{S}_{n-D(V)}^{*}\setminus\mathcal{S}_{n-D(V)}^{\textrm{gen}}.$

Set $k:=n-D(V)$ and note that $2\leq k\leq n-3$ (the first inequality follows
from $\mathcal{S}_{1}^{*}=\mathcal{S}_{1}^{\textrm{gen}}$ and the second from
$D(V)\geq 3$). Since
$\Sigma^{\prime}\in\mathcal{S}_{k}^{*}(\Lambda)\setminus\mathcal{S}_{k}^{\textrm{gen}}$,
($\heartsuit_{n,\Lambda}$) implies

$2\mu(\Sigma^{\prime})<-n+k=-D(V).$

Lemma 3.5 implies that $\mu(\Sigma)\leq\mu(\Sigma^{\prime})$, yielding Claim
3.7. ∎

##  4\. One-sided flows

###  4.1. Crossing between non-generic shrinkers and their translates

The following result is [CCMS21, Proposition 2.2] generalized to the current
setting of singular tangent flows. Recall Definition 2.2 for the notion of
associated shrinking flows.

######  Proposition 4.1.

Take $n\in\\{2,3,\ldots\\}$, $\Lambda\in(0,2]$ such that
($\diamondsuit_{n,\Lambda}$) holds.

Let $V$, $V^{\prime}$ be cyclic integral $F$-stationary $n$-varifolds in
$\mathbb{R}^{n+1}$ with $\lambda(V)$, $\lambda(V^{\prime})<\Lambda$ and
associated shrinking flows $\mathcal{M}_{V},\mathcal{M}_{V^{\prime}}$. If:

  * •

$V$ is non-flat and non-generic, and

  * •

there is $(\mathbf{0},0)\neq X^{\prime}\in\mathbb{R}^{n+1}\times\mathbb{R}$ so
that $\mathcal{M}_{V}$, $\mathcal{M}_{V^{\prime}}+X^{\prime}$ do not cross
smoothly,

then:

  1. (a)

$V^{\prime}=V$, and

  2. (b)

$X^{\prime}\in\operatorname{spine}V$.

###### Proof.

Note that the tangent flow at $-\infty$ to $\mathcal{M}_{V}$ is
$\mathcal{M}_{V}$ and the tangent flow at $-\infty$ to
$\mathcal{M}_{V^{\prime}}+X^{\prime}$ is $\mathcal{M}_{V^{\prime}}$. By Lemma
2.10 and definition of smoothly crossing $F$-stationary varifolds, $V$ and
$V^{\prime}$ do not cross smoothly. Lemma 2.12 thus implies that
$V=V^{\prime}$.

It remains to prove that $X^{\prime}\in\operatorname{spine}V$. Write
$X^{\prime}=(\mathbf{x}^{\prime},t^{\prime})$. Since $\mathcal{M}_{V}$ is
invariant under parabolic dilation, we note that $\mathcal{M}_{V}$ and
$\mathcal{M}_{V}+(\lambda\mathbf{x},\lambda^{2}t^{\prime})$ do not cross
smoothly for any $\lambda>0$. By considering $t=-1$, we see that the speed of
the family

$\lambda\mapsto\sqrt{1+\lambda^{2}t^{\prime}}\operatorname{reg}V+\lambda\mathbf{x}^{\prime}$

at $\lambda=0$ must not change sign. Up to changing the sign of the unit
normal $\nu$ on $\operatorname{reg}V$ (recalling $\operatorname{reg}V$ is
connected) we thus find that $\mathbf{x}^{\prime}\cdot\nu\geq 0$ on
$\operatorname{reg}V$. Since $\mathbf{x}^{\prime}\cdot\nu$ satisfies the
elliptic PDE
$L(\mathbf{x}^{\prime}\cdot\nu)=\frac{1}{2}\mathbf{x}^{\prime}\cdot\nu$ along
$\operatorname{reg}V$ (cf. [CM12, Theorem 5.2]), the strong maximum principle
implies that either $\mathbf{x}^{\prime}\cdot\nu>0$ on $\operatorname{reg}V$
or $\mathbf{x}^{\prime}\cdot\nu\equiv 0$ on $\operatorname{reg}V$.

We begin by considering the first case $\mathbf{x}^{\prime}\cdot\nu>0$ on
$\operatorname{reg}V$. By the argument in [Wan11, Lemma 2.1] we find that
$\mu(V)\geq-\tfrac{1}{2}$ (recalling that $\mu(V)$ is the first eigenvalue of
the $L$ operator on $\operatorname{reg}V$). Thus Corollary 3.2 implies that
$\operatorname{sing}V=\emptyset$. Hence, by [Wan11, Theorem 1.1], $V$ is a
hyperplane. This contradicts that $V$ is non-flat.

Thus, we find that $\mathbf{x}^{\prime}\cdot\nu\equiv 0$. We first assume that
$\mathbf{x}^{\prime}\neq 0$. This implies that $\operatorname{reg}V$ splits a
line in the $\mathbf{x}^{\prime}$ direction. By Lemma 2.8, we see that $V$
splits a line in the $\mathbf{x}^{\prime}$ direction. If $t^{\prime}=0$, we
see that $X^{\prime}\in\operatorname{spine}V$. If $t^{\prime}\neq 0$ then
since $V$ splits a line in the $\mathbf{x}^{\prime}$ direction, we see that
$\mathcal{M}_{V}+(\mathbf{0},t^{\prime})=\mathcal{M}_{V}+(\mathbf{x}^{\prime},t^{\prime})$
does not cross $\mathcal{M}_{V}$ smoothly.

As such, it remains to consider the case that $\mathbf{x}^{\prime}=0$ and
$t^{\prime}\neq 0$. In this case, we can argue as above (by considering
$t=-1$) see that the speed of

$\lambda\mapsto(1+\lambda t^{\prime})\operatorname{reg}V$

does not change sign on $\operatorname{reg}V$. This implies that (up to
changing the sign of the unit normal) that the mean curvature $H\geq 0$ on
$\operatorname{reg}V$. As above, we recall that

$LH=H$

along $\operatorname{reg}V$, so the strong maximum principle (and
connectedness of $\operatorname{reg}V$) yields either $H>0$ or $H\equiv 0$. As
above, if $H>0$ then $\operatorname{sing}V=\emptyset$ (by arguing as in
[Wan11, Lemma 2.1] and using Lemma 2.8). In this case, by [CM12, Theorem 0.17]
(cf. [Hui90]) $V\in\mathcal{S}_{n}^{\textrm{gen}}$, a contradiction. Thus,
$H\equiv 0$ along $\operatorname{reg}V$. By Lemma 2.8 we thus see that $V$ is
a stationary cone. Thus, $(0,t^{\prime})\in\operatorname{spine}V$. This
completes the proof. ∎

###  4.2. Nearly self-similar non-generic flows

Fix $n\in\\{2,3,\ldots\\}$, $\Lambda\in(0,2]$, $\varepsilon>0$, and consider a
class of nearly self-similar flows about a non-generic singular point:

######  Definition 4.2.

For $\eta\in(0,1]$ define $\mathscr{P}(\eta)$ to be the set of pairs
$(X,\mathcal{M})$ so that:

  1. (1)

$X\in P((\mathbf{0},0),1)$,

  2. (2)

$\mathcal{M}$ is a unit-regular cyclic integral $n$-Brakke flow in
$\mathbb{R}^{n+1}$ for $t\in[-\eta^{-2},\infty)$,

  3. (3)

$\lambda(\mathcal{M})\leq\Lambda-\varepsilon$,

  4. (4)

$X\in\operatorname{sing}_{\textnormal{non-gen}}\mathcal{M}$,

  5. (5)

$\Theta_{\mathcal{M}}(X)\geq\Theta_{\mathcal{M}}(X,\eta^{-1})-4\eta$.

######  Remark 4.3.

It is useful to observe that for all $(X,\mathcal{M})\in\mathscr{P}(\eta)$,
$r\in(0,\eta^{-1}]$,

$\Theta_{\mathcal{M}}(X)\geq\Theta_{\mathcal{M}}(X,r)-4\eta.$

This follows from monotonicity and (5).

######  Lemma 4.4.

Fix $\Lambda\in(0,2]$, $\varepsilon>0$. Suppose that
$((\mathbf{0},0),\mathcal{M}_{j})\in\mathscr{P}(j^{-1})$ and
$\mathcal{M}_{j}\rightharpoonup\mathcal{M}$. Then, there exists a non-flat and
non-generic $F$-stationary cyclic integral $n$-varifold $V$ with
$\lambda(V)\leq\Lambda-\varepsilon$, and $\mathcal{M}=\mathcal{M}_{V}$
whenever the latter is nonvanishing (see Definition 2.2).

###### Proof.

It is clear that $\mathcal{M}$ is a unit-regular cyclic integral $n$-Brakke
flow in $\mathbb{R}^{n+1}$ defined for $(-\infty,\infty)$. Due to Remark 4.3
and upper-semicontinuity of density

$\Theta_{\mathcal{M}}((\mathbf{0},0))\geq\Theta_{\mathcal{M}}((\mathbf{0},0),r)$

for all $r>0$. Thus, by monotonicity,

$\mathcal{M}(t)=\mathcal{M}_{V}(t)\text{ for all }t<0,$

where $V$ is an $F$-stationary varifold with
$\lambda(V)\leq\Lambda-\varepsilon$. Note that if $F$ is a stationary cone,
this holds for all $t\in\mathbb{R}$ by unit-regularity and
$\Lambda-\varepsilon<2$, and that $V$ is non-flat since
$(\mathbf{0},0)\in\operatorname{sing}\mathcal{M}_{j}$ for all $j$.

It remains to prove that $V\not\in\mathcal{S}_{n}^{\textrm{gen}}$. This
follows from [CIM15] as we now explain (cf. [CIM15, Theorem 0.2]). We now
recall [CIM15, (2.11)]. Fix a countable dense subset $\\{f_{k}\\}$ of the unit
ball in $C^{0}_{c}(\mathbb{R}^{n+1})$ and set

$d_{*}(\mu_{1},\mu_{2}):=\sum_{k}2^{-k}\left|\int
f_{k}e^{-\tfrac{1}{4}|\cdot|^{2}}d\mu_{1}-\int
f_{k}e^{-\tfrac{1}{4}|\cdot|^{2}}d\mu_{2}\right|$

for $\mu_{1},\mu_{2}$ Radon measures with $F(\mu_{i})<\infty$. (This is
denoted $d_{V}$ in [CIM15].) One can check that $d_{*}$ metrizes the weak-$*$
topology on the space of Radon measures $\mu$ with $F(\mu)<\infty$. By [CIM15,
Corollary 2.12] there is $c_{0}$ so that if $V$ is an $F$-stationary varifold
with $F(V)\leq 2-\varepsilon$ and

(4.1) $d_{*}(V,\mathcal{S}_{n}^{\textrm{gen}})\leq 2c_{0}\qquad\implies\qquad
V\in\mathcal{S}_{n}^{\textrm{gen}},$

where the $d_{*}$-metric on a varifold $V$ is the metric with respect to the
induced Radon measure $\|V\|$.

Returning to the above setup, assume, for contradiction, that
$V\in\mathcal{S}_{n}^{\textrm{gen}}$. For each $j$, let
$\tilde{\mathcal{M}}_{j}$ denote the rescaled Brakke flow around
$(\mathbf{0},0)$ at scale $1$ corresponding to $\mathcal{M}_{j}$. We will use
a few times that, by Remark 4.3, we have for any sequence
$\tau_{j}\in[0,\infty)$,

(4.2) $\tilde{\mathcal{M}}_{j}(\cdot+\tau_{j})\rightharpoonup V^{\prime},$

where $V^{\prime}$ is a $F$-stationary $n$-varifold. By assumption,
$V^{\prime}=V$ if $\tau_{j}\equiv 0$. Since
$(\mathbf{0},0)\not\in\operatorname{sing}_{\textrm{gen}}\mathcal{M}_{j}$ for
each $j$, there is a sequence $\tau_{k,j}^{\prime}\to\infty$ so that (4.2)
gives

$\tilde{\mathcal{M}}_{j}(\cdot+\tau_{k,j}^{\prime})\rightharpoonup
V_{j}^{\prime}\not\in\mathcal{S}_{n}^{\textrm{gen}}\text{ as }k\to\infty,$

so, by (4.1),

(4.3)
$d_{*}(\tilde{\mathcal{M}}_{j}(\cdot+\tau_{k,j}^{\prime}),\mathcal{S}_{n}^{\textrm{gen}})\geq
2c_{0}-o(1)\text{ as }k\to\infty.$

Take $\bar{\tau}_{j}\in[0,\infty]$ to be as large as possible so that

$d_{*}(\tilde{\mathcal{M}}_{j}(\tau),\mathcal{S}_{n}^{\textrm{gen}})\leq
c_{0}\text{ for all }\tau\in[0,\bar{\tau}_{j}).$

By (4.3), $\bar{\tau}_{j}<\infty$ for all $j$. Moreover, (4.2) with
$\tau_{j}\equiv 0$ gives that $\bar{\tau}_{j}\to\infty$ as $j\to\infty$. Up to
a subsequence, we can assume from (4.2) that
$\tilde{\mathcal{M}}_{j}(\cdot+\bar{\tau}_{j})\rightharpoonup\bar{V}$, where
$\bar{V}$ is a $F$-stationary $n$-varifold. Then, by our choice of
$\bar{\tau}_{j}$,

$d_{*}(\bar{V},\mathcal{S}_{n}^{\textrm{gen}})\leq
d_{*}(\tilde{\mathcal{M}}_{j}(\bar{\tau}_{j}-1),\mathcal{S}_{n}^{\textrm{gen}})+o(1)\leq
c_{0}+o(1),$

so $\bar{V}\in\mathcal{S}_{n}^{\textrm{gen}}$ by (4.1). However, this yields

$c_{0}<\sup_{\tau\in[0,1]}d_{*}(\tilde{\mathcal{M}}_{j}(\bar{\tau}_{j}+\tau),\mathcal{S}_{n}^{\textrm{gen}})\leq\sup_{\tau\in[0,1]}d_{*}(\tilde{\mathcal{M}}_{j}(\bar{\tau}_{j}+\tau),\bar{V})=o(1).$

This is a contradiction, completing the proof. ∎

######  Lemma 4.5.

Assume that $(\diamondsuit_{n,\Lambda})$ holds and take any
$((\mathbf{0},0),\mathcal{M}_{j})$,
$(X_{j}^{\prime},\mathcal{M}_{j}^{\prime})\in\mathscr{P}(j^{-1})$ such that:

  * •

$\mathcal{M}_{j}$, $\mathcal{M}^{\prime}_{j}$ don’t cross smoothly,

  * •

$\mathcal{M}_{j}\rightharpoonup\mathcal{M}$,

  * •

$\mathcal{M}_{j}^{\prime}\rightharpoonup\mathcal{M}^{\prime}$,

  * •

$X_{j}^{\prime}\to X^{\prime}\neq(\mathbf{0},0)$,

and $V$ be as in Lemma 4.4 (applied with $\mathcal{M}_{j}$). Then,

  1. (a)

$\mathcal{M}^{\prime}=\mathcal{M}_{V}$ whenever the latter is nonvanishing,
and

  2. (b)

$X^{\prime}\in\operatorname{spine}V$.

###### Proof.

Denote $X^{\prime}=(\mathbf{x}^{\prime},t^{\prime})$.

By Lemma 4.4 (applied to
$\mathcal{M}_{j},\mathcal{M}_{j}^{\prime}-X^{\prime}_{j}$), there exist two
non-flat and non-generic $F$-stationary cyclic integral $n$-varifolds $V$,
$V^{\prime}$ with $\lambda(V),\lambda(V^{\prime})\leq\Lambda-\varepsilon$ so
that

$\mathcal{M}=\mathcal{M}_{V}\text{ and
}\mathcal{M}^{\prime}=\mathcal{M}_{V^{\prime}}(\cdot+t^{\prime})+\mathbf{x}^{\prime},$

whenever the right hand sides are nonvanishing.

By Lemma 2.10, $\mathcal{M}$ and $\mathcal{M}^{\prime}$ do not cross smoothly
(otherwise some $\mathcal{M}_{j}$, $\mathcal{M}_{j}^{\prime}$ would). Hence,
Proposition 4.1 implies that $V=V^{\prime}$ and
$X^{\prime}\in\operatorname{spine}V$, as desired. ∎

###  4.3. Graphical distance estimate for nearby rescaled flows

Going forward we fix $n\in\\{2,3,\dots\\}$, $\Lambda\in(0,2]$,
$\varepsilon>0$, so that ($\heartsuit_{n,\Lambda}$) and
($\diamondsuit_{n,\Lambda}$) both hold. Recall that constants
$\rho_{0},r_{0}>0$ were then chosen in Proposition 3.6 (depending only on
$n,\Lambda,\varepsilon$).

######  Lemma 4.6.

Let $\tilde{\mathcal{M}}_{j}^{\prime}$,
$\tilde{\mathcal{M}}_{j}^{\prime\prime}$ be sequences of unit-regular cyclic
integral rescaled Brakke flows in $\mathbb{R}^{n+1}$, defined for
$\tau\in[T_{j},\infty)$ with $T_{j}\to-\infty$, which satisfy

  * •

$\lambda(\tilde{\mathcal{M}}_{j}^{\prime})$,
$\lambda(\tilde{\mathcal{M}}_{j}^{\prime\prime})\leq\Lambda-\varepsilon$,

  * •

$\tilde{\mathcal{M}}_{j}^{\prime}$, $\tilde{\mathcal{M}}_{j}^{\prime\prime}$
do not cross smoothly,

  * •

$\tilde{\mathcal{M}}_{j}^{\prime},\tilde{\mathcal{M}}_{j}^{\prime\prime}\rightharpoonup\tilde{\mathcal{M}}_{V}$
for some $F$-stationary cyclic integral $n$-varifold $V$, where
$\tilde{\mathcal{M}}_{V}$ denotes the rescaling about $(\mathbf{0},0)$ (see
Definition 2.6) of the flow $\mathcal{M}_{V}$ associated to $V$ (see
Definition 2.2).

Then, for sufficiently large $j$:

  1. (a)

$\mathcal{R}_{\rho_{0},r_{0}}\tilde{\mathcal{M}}_{j}^{\prime}(\mathbf{0},0)\neq\emptyset$.

  2. (b)

There is an increasing sequence of relatively open sets $\tilde{W}_{j}$
exhausting $\operatorname{reg}\tilde{\mathcal{M}}_{V}$, with the property that
the portions of $\tilde{\mathcal{M}}_{j}^{\prime}$,
$\tilde{\mathcal{M}}_{j}^{\prime\prime}$ within a fixed vertical distance from
$\tilde{W}_{j}$ are normal graphs of $\tilde{w}_{j}^{\prime}$,
$\tilde{w}_{j}^{\prime\prime}\in C^{\infty}(\tilde{W}_{j})$, and

$\tilde{w}_{j}^{\prime},\;\tilde{w}_{j}^{\prime\prime}\to 0\text{ in
}C^{\infty}_{\textnormal{loc}}(\operatorname{reg}\tilde{\mathcal{M}}_{V}).$

Up to switching the unit normal of $\operatorname{reg}\tilde{\mathcal{M}}_{V}$
we can assume that $\tilde{w}_{j}^{\prime}\geq\tilde{w}_{j}^{\prime\prime}$.

  3. (c)

There is a sequence of second order elliptic operators $L_{j}$ on
$\tilde{W}_{j}$ so that

$(\partial_{\tau}-L_{j})(\tilde{w}_{j}^{\prime}-\tilde{w}_{j}^{\prime\prime})=0\text{
on }\tilde{W}_{j},$ $L_{j}\to L\text{ in
}C^{\infty}_{\textrm{loc}}(\operatorname{reg}\tilde{\mathcal{M}}_{V}),$

where $L$ is the operator from (1.2).

######  Remark 4.7.

If $\tilde{\mathcal{M}}_{V}$ is as above, then
$\operatorname{reg}\tilde{\mathcal{M}}_{V}=\operatorname{reg}V\times\mathbb{R}$.

###### Proof of Lemma 4.6.

By Proposition 3.6,
$\mathcal{R}_{2\rho_{0},r_{0}/2}\tilde{\mathcal{M}}_{V}(\mathbf{0},0)\neq\emptyset$.
This yields the first assertion. The remaining claims follow from unit-
regularity and standard geometric arguments (recall that
$\operatorname{reg}\tilde{\mathcal{M}}_{V}$ is connected). ∎

Consider the follow geometric variant of the infimum of vertical distances:

######  Definition 4.8.

For nonempty subsets $A,B\subset\mathbb{R}^{n+1}$ we write

$d(A,B)=\inf\\{|\mathbf{a}-\mathbf{b}|:\mathbf{a}\in A,\mathbf{b}\in B\\}.$

Note that $d(A,B)=0$ does not mean that $A=B$.

######  Lemma 4.9.

There is $H=H(n,\Lambda,\varepsilon)\in[1,\infty)$ with the following
property.

Consider
$\tilde{\mathcal{M}}^{\prime}_{j},\tilde{\mathcal{M}}_{j}^{\prime\prime},\tilde{\mathcal{M}}_{V},\tilde{W}_{j},\tilde{w}_{j}^{\prime},\tilde{w}_{j}^{\prime\prime}$
as in Lemma 4.6. Assume that

$\tilde{d}_{j}:=d(\mathcal{R}_{\rho_{0},r_{0}}\tilde{\mathcal{M}}_{j}^{\prime}(\mathbf{0},0),\operatorname{supp}\tilde{\mathcal{M}}_{j}^{\prime\prime}(0))>0$

for $j=1,2,\dots$, and define

$\tilde{u}_{j}:=\tilde{d}_{j}^{-1}(\tilde{w}_{j}^{\prime}-\tilde{w}_{j}^{\prime\prime}).$

Then, after perhaps passing to a subsequence:

  1. (a)

$\tilde{u}_{j}\to\tilde{u}$ in
$C^{\infty}_{\textnormal{loc}}(\operatorname{reg}V\times(-\infty,0))$ with
$\partial_{\tau}\tilde{u}=L\tilde{u}$ and $\tilde{u}\geq 0$.

  2. (b)

At rescaled time $\tau=-1$,

$\sup_{\mathcal{R}_{\rho_{0},r_{0}}V(\mathbf{0})}\tilde{u}(\cdot,-1)\leq H.$

  3. (c)

For each rescaled time $\tau<-1$,

$\sup_{\mathcal{R}_{3\rho_{0},r_{0}/3}V(\mathbf{0})}\tilde{u}(\cdot,\tau_{0})\leq
H\inf_{\mathcal{R}_{\rho_{0},r_{0}}V(\mathbf{0})}\tilde{u}(\cdot,\tau_{0}+1).$

###### Proof.

By Lemma 4.6, the functions $\tilde{u}_{j}$ satisfy a second order parabolic
PDE

$(\partial_{t}-L_{j})\tilde{u}_{j}=0\text{ on
}\tilde{W}_{j}\to\operatorname{reg}\tilde{\mathcal{M}}_{V},$ $L_{j}\to L\text{
in }C^{\infty}_{\textrm{loc}}(\operatorname{reg}\tilde{\mathcal{M}}_{V}).$

Recall, per Remark 4.7, that
$\operatorname{reg}\tilde{\mathcal{M}}_{V}=\operatorname{reg}V\times\mathbb{R}$.
By a standard geometric argument relating graphical separation to distance we
find

(4.4)
$\inf_{\mathcal{R}_{\rho_{0}/2,2r_{0}}V(\mathbf{0})}\tilde{u}_{j}(\cdot,0)\leq
1+o(1).$

We can now apply parabolic Harnack and Schauder theory in a standard way to
conclude (a). The proofs of (b) and (c) will follow similarly except we must
argue that the constant $H$ can be chosen to only depend on
$n,\Lambda,\varepsilon$.

We first consider (b). Suppose for contradiction that for each $k=1,2,\dots,$
there is a sequence
$\tilde{\mathcal{M}}^{\prime}_{j,k},\tilde{\mathcal{M}}_{j,k}^{\prime\prime},\tilde{\mathcal{M}}_{V_{k}},\tilde{W}_{j,k},\tilde{w}_{j,k}^{\prime},\tilde{w}_{j,k}^{\prime\prime}$,
with $j=1,2,\dots$, as in Lemma 4.6 so that

$\tilde{u}_{j,k}=\tilde{d}_{j,k}^{-1}(\tilde{w}_{j,k}^{\prime}-\tilde{w}_{j,k}^{\prime\prime})$

converges as $j\to\infty$ in
$C^{\infty}_{\textrm{loc}}(\operatorname{reg}V_{k}\times(-\infty,0))$ to
$\tilde{u}_{k}$ with $\partial_{\tau}\tilde{u}_{k}=L_{V_{k}}\tilde{u}_{k}$ but
so that there are
$\mathbf{x}_{k}\in\mathcal{R}_{\rho_{0},r_{0}}V_{k}(\mathbf{0})$ with

(4.5) $\tilde{u}_{k}(\mathbf{x}_{k},-1)\geq 2k.$

We can pass to a subsequence so that $V_{k}\rightharpoonup V$. Choose $j=j(k)$
so that

(4.6) $\tilde{u}_{j(k),k}(\mathbf{x}_{k},-1)\geq k.$

Using (4.4) we also choose
$\mathbf{y}_{k}\in\mathcal{R}_{\rho_{0}/2,2r_{0}}V_{k}$ with

(4.7) $\tilde{u}_{j(k),k}(\mathbf{y}_{k},0)\leq 2.$

Up to passing to a subsequence,
$\mathbf{x}_{k}\to\mathbf{x},\mathbf{y}_{k}\to\mathbf{y}\in\operatorname{reg}V$.
Writing $L$ for the $L$-operator on $\operatorname{reg}V$, we can choose a
finite chain of overlapping coordinate balls (from $\mathbf{x}$ to
$\mathbf{y}$ in $\operatorname{reg}V$ so that in each ball, the $L$-operator
has controlled ellipticity and coefficients. Since the $L_{j}$ operator is
smoothly converging to the $L$ operator on
$\operatorname{reg}\tilde{\mathcal{M}}_{V}$, the same thing holds for $L_{j}$.
Now, we can apply parabolic Harnack on the chain of balls in the usual way to
bound

$\tilde{u}_{j(k),k}(\mathbf{x}_{k},-1)\leq\tilde{H}\tilde{u}_{j(k),k}(\mathbf{y}_{k},0),$

where $\tilde{H}$ depends only on $V$ and the chosen chain of balls, but not
on $k$. However, this contradicts (4.6) and (4.7). This proves (b).

Finally, to prove (c) we argue in the same way, except we replace (4.5) by

$\tilde{u}_{k}(\mathbf{x}_{k},\tau_{k})\geq(2k)\tilde{u}_{k}(\mathbf{y}_{k},\tau_{k}+1).$

for some $\tau_{k}<-1$. The argument is the same as before except we shift
rescaled time by $+\tau_{k}$ to account for the possibility that
$\tau_{k}\to-\infty$. Since the rescaled flow of a shrinker is stationary,
this does not change the above analysis. This completes the proof. ∎

###  4.4. Separation and packing estimates

We first need an elementary growth lemma about nonnegative solutions of
$(\partial_{\tau}-L)\tilde{u}=0$ on compact subdomains of shrinkers, which we
will then “de-linearize.” Below is a parabolic analogue of the linear
estimates obtained in [Sim08] and [Wan22, Appendix A].

######  Lemma 4.10.

Let $\Sigma\subset\mathbb{R}^{n+1}$ be a compact self-shrinker with (possibly
empty) boundary. Let $\varphi>0$ denote the first eigenfunction of the
$L$-operator on $\Sigma$ with Dirichlet boundary conditions. Suppose that
$\tilde{u}\in C^{\infty}(\Sigma\times[\tau_{0},\tau_{1}])$ satisfies

$\partial_{\tau}\tilde{u}=L\tilde{u},\;\tilde{u}\geq 0.$

Then, for all $\tau\in[\tau_{0},\tau_{1}]$ we have

$\int_{\Sigma}\tilde{u}(\mathbf{x},\tau)\varphi(\mathbf{x})e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}\,d\mathbf{x}\geq
e^{-\mu(\Sigma)\cdot(\tau-\tau_{0})}\int_{\Sigma}\tilde{u}(\mathbf{x},\tau_{0})\varphi(\mathbf{x})e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}\,d\mathbf{x}.$

###### Proof.

Set

$\tilde{U}(\tau):=\int_{\Sigma}\tilde{u}(\mathbf{x},\tau)\varphi(\mathbf{x})e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}\,d\mathbf{x}.$

Differentiating under the integral sign and then integrating by parts we find

$\displaystyle\tilde{U}^{\prime}(\tau)$
$\displaystyle=\int_{\Sigma}(L\tilde{u}(\mathbf{x},\tau))\varphi(\mathbf{x})e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}$
$\displaystyle=\int_{\Sigma}\tilde{u}(\mathbf{x},\tau)(L\varphi(\mathbf{x}))e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}-\int_{\partial\Sigma}\tilde{u}(\mathbf{x},\tau)(\partial_{\zeta}\varphi(\mathbf{x}))e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}$
$\displaystyle\geq-\mu(\Sigma)\int_{\Sigma}\tilde{u}(\mathbf{x},\tau)\varphi(\mathbf{x})e^{-\tfrac{1}{4}|\mathbf{x}|^{2}}=-\mu(\Sigma)\tilde{U}(\tau).$

Above, $\zeta$ is the outward pointing unit conormal and we used that
$\partial_{\zeta}\varphi\leq 0$ due to the Dirichlet boundary conditions and
$\varphi\geq 0$. Integrating this proves the assertion. ∎

Now continue to fix $n\in\\{2,3,\dots,\\}$, $\Lambda\in(0,2]$, $\varepsilon>0$
so that ($\heartsuit_{n,\Lambda}$) and ($\diamondsuit_{n,\Lambda}$) hold.

######  Remark 4.11.

We have chosen several other constants depending only on
$n,\Lambda,\varepsilon$, as we recall here:

  1. (1)

The covering constant $C$ was fixed in Lemma 2.16.

  2. (2)

The gain of decay constant $\kappa$ and regularity scale constants
$\rho_{0},r_{0}$ were fixed in Proposition 3.6.

  3. (3)

The Harnack constant $H$ was fixed in Lemma 4.9.

We now fix the scale size constant $\gamma\in(0,1)$ by

(4.8)
$\gamma=\min\left\\{\left(\tfrac{1}{2}H^{-2}(\tfrac{5}{4e})^{2\kappa+n+2}\right)^{1/\kappa},(2C)^{-1/\kappa},\tfrac{\sqrt{3}}{2e}\right\\}$

Note that $\gamma=\gamma(n,\Lambda,\varepsilon)<\tfrac{1}{2}$.

With these preparations, we can now discuss the key covering/separation
result. To state it, it is convenient to consider the projection operator:

$\pi:(X,\mathcal{M})\mapsto X.$

######  Proposition 4.12.

There exists $\eta_{0}\in(0,1)$ with the following property.

Let $\mathcal{Q}\subset\mathscr{P}(\eta_{0})$ be such that:

  * •

$(\mathbf{0},0)\in\pi(\mathcal{Q})$, and

  * •

if $(X^{\prime},\mathcal{M}^{\prime})$ and
$(X^{\prime\prime},\mathcal{M}^{\prime\prime})\in\mathcal{Q}$ then
$\mathcal{M}^{\prime}$ and $\mathcal{M}^{\prime\prime}$ don’t cross smoothly.

Then there exist $D\in\mathbb{N}$ and points
$X_{1}=(\mathbf{x}_{1},t_{1}),\ldots,X_{K}=(\mathbf{x}_{K},t_{K})\in\pi(\mathcal{Q})$
such that:

  1. (a)

$\pi(\mathcal{Q})\subset\cup_{i=1}^{K}P(X_{i},\gamma)$ with $K\leq
C\gamma^{-D}$;

  2. (b)

for
$(X^{\prime},\mathcal{M}^{\prime})\in\mathcal{Q}\cap\pi^{-1}(P(X_{i},\gamma))$
it holds that

$\displaystyle\mathcal{R}_{2\rho_{0},2r_{0}}\mathcal{M}^{\prime}(\mathbf{0},-4)$
$\displaystyle\neq\emptyset,$
$\displaystyle\mathcal{R}_{2\gamma\rho_{0},2\gamma
r_{0}}\mathcal{M}^{\prime}(\mathbf{x}_{i},t_{i}-4\gamma^{2})$
$\displaystyle\neq\emptyset;\text{ and,}$

  3. (c)

for
$(X^{\prime},\mathcal{M}^{\prime}),(X^{\prime\prime},\mathcal{M}^{\prime\prime})\in\mathcal{Q}\cap\pi^{-1}(P(X_{i},\gamma))$
we have:

$\displaystyle\gamma^{-(\kappa+D)}\cdot
2^{-1}d(\mathcal{R}_{2\rho_{0},2r_{0}}\mathcal{M}^{\prime}(\mathbf{0},-4),\operatorname{supp}\mathcal{M}^{\prime\prime}(-4))$
$\displaystyle\qquad\leq(2\gamma)^{-1}d(\mathcal{R}_{2\gamma\rho_{0},2\gamma
r_{0}}\mathcal{M}^{\prime}(\mathbf{x}_{i},t_{i}-4\gamma^{2}),\operatorname{supp}\mathcal{M}^{\prime\prime}(t_{i}-4\gamma^{2})).$

######  Remark 4.13.

One could have replaced $\gamma^{-(\kappa+D)}$ by
$\gamma^{-((2-\delta)\kappa+D)}$ with any $\delta\in(0,2)$ in (c); cf.
Proposition 3.6 (b).

###### Proof of Proposition 4.12.

Suppose for contradiction that no such $\eta_{0}$ exists. Then there would
exist a sequence $\mathcal{Q}_{j}\subset\mathscr{P}(j^{-1})$, with $j$ large,
so that

  * •

there is $((\mathbf{0},0),\mathcal{M}_{j})\in\mathcal{Q}_{j}$ and

  * •

if
$(X^{\prime},\mathcal{M}^{\prime}),(X^{\prime\prime},\mathcal{M}^{\prime\prime})\in\mathcal{Q}_{j}$
then $\mathcal{M}^{\prime}$ and $\mathcal{M}^{\prime\prime}$ don’t cross
smoothly,

but that there is no $D$ and a collection of points in $\pi(\mathcal{Q}_{j})$
so that (a) & (b) & (c) hold.

Pass to a subsequence so that $\mathcal{M}_{j}\rightharpoonup\mathcal{M}$. By
Lemma 4.4, there exists a non-planar and non-generic $F$-stationary cyclic
integral $n$-varifold $V$ with $\lambda(V)\leq\Lambda-\varepsilon$ and

$\mathcal{M}=\mathcal{M}_{V}$

whenever the right hand side is nonvanishing. Set $\Pi=\operatorname{spine}V$
and $D=D(\Pi)$.

For $(X_{j}^{\prime},\mathcal{M}_{j}^{\prime})\in\mathcal{Q}_{j}$, we can pass
to a subsequence so that $X_{j}^{\prime}\to X^{\prime}$ and
$\mathcal{M}_{j}^{\prime}\rightharpoonup\mathcal{M}^{\prime}$. Then, by Lemma
4.5, $X^{\prime}\in\Pi$ and

$\mathcal{M}^{\prime}=\mathcal{M}=\mathcal{M}_{V}$

whenever the right hand side is nonvanishing. After discarding finitely many
$j$, we can assume that

$\pi(\mathcal{Q}_{j})\subset U_{\gamma}(\Pi)\cap P((\mathbf{0},0),1).$

By Lemma 2.16 we can thus find
$X_{j,1},\dots,X_{j,K_{j}}\in\pi(\mathcal{Q}_{j})$ so that

$\pi(\mathcal{Q}_{j})\subset\cup_{i=1}^{K_{j}}P(X_{j,i},\gamma)\text{ and
}K_{j}\leq C\gamma^{-D}.$

Thus, (a) holds for this collection of points.

Passing to a further subsequence (not relabeled) if necessary, we can assume
that either (b) or (c) fails for every $j$ and some $i=i(j)$. We will then
derive a contradiction. For notational simplicity, we will write
$X_{j}=(\mathbf{x}_{j},t_{j})$ instead of $X_{j,i}$.

We start with (b). Take
$(X^{\prime}_{j},\mathcal{M}^{\prime}_{j})\in\mathcal{Q}_{j}\cap\pi^{-1}(P(X_{j},\gamma))$.
We claim that, for all large $j$,

$\mathcal{R}_{2\rho_{0},2r_{0}}\mathcal{M}_{j}^{\prime}(\mathbf{0},-4)\neq\emptyset\text{
and }\mathcal{R}_{2\gamma\rho_{0},2\gamma
r_{0}}\mathcal{M}_{j}^{\prime}(\mathbf{x}_{j},t_{j}-4\gamma^{2})\neq\emptyset.$

We just prove the latter statement; the former follows similarly. Let
$\tilde{\mathcal{M}}_{j}^{\prime}$ denote the rescaled Brakke flow
corresponding to $\mathcal{M}_{j}^{\prime}$ centered at
$X_{j}=(\mathbf{x}_{j},t_{j})$ at scale $2\gamma$ (see Definition 2.6). In
other words, we set

$\tilde{\mathcal{M}}_{j}^{\prime}(\tau)=(2\gamma)^{-1}e^{\tau/2}(\mathcal{M}_{j}^{\prime}(t_{j}-4\gamma^{2}e^{-\tau})-\mathbf{x}_{j}).$

Since $\gamma$ is fixed, we find that
$\tilde{\mathcal{M}}_{j}^{\prime}\rightharpoonup\tilde{\mathcal{M}}_{V}$.
Lemma 4.6 implies that, for $j$ large,

$\mathcal{R}_{\rho_{0},r_{0}}\tilde{\mathcal{M}}_{j}^{\prime}(\mathbf{0},0)\neq
0.$

This yields the desired nonemptiness after returning to the unrescaled flow
(see Remark 2.4). This is a contradiction to our assumption that (b) failed
for all $j$.

We can thus assume (c) fails for all $j$. By our setup, there exist

$(X_{j}^{\prime},\mathcal{M}_{j}^{\prime}),(X_{j}^{\prime\prime},\mathcal{M}_{j}^{\prime\prime})\in\mathcal{Q}_{j}\cap\pi^{-1}(P(X_{j},\gamma))$

so that

(4.9) $(2\gamma)^{-1}d(\mathcal{R}_{2\gamma\rho_{0},2\gamma
r_{0}}\mathcal{M}_{j}^{\prime}(\mathbf{x}_{j},t_{j}-4\gamma^{2}),\operatorname{supp}\mathcal{M}_{j}^{\prime\prime}(t_{j}-4\gamma^{2}))\\\
<\gamma^{-(\kappa+D)}\cdot
2^{-1}d(\mathcal{R}_{2\rho_{0},2r_{0}}\mathcal{M}_{j}^{\prime}(\mathbf{0},-4),\operatorname{supp}\mathcal{M}_{j}^{\prime\prime}(-4)).$

We now derive a contradiction to (4.9) (this will complete the proof of the
proposition). It is important to shift the spatial center on the right-hand
side of (4.9). We note that

$\mathcal{R}_{2\rho_{0},2r_{0}-|\mathbf{x}_{j}|}\mathcal{M}_{j}^{\prime}(\mathbf{x}_{j},-4)\subset\mathcal{R}_{2\rho_{0},2r_{0}}\mathcal{M}_{j}^{\prime}(\mathbf{0},-4)$

by Remark 2.4. Thus, we have333Proposition 3.6 gives that
$\mathcal{R}_{3\rho_{0},r_{0}/3}V(\mathbf{0})\neq\emptyset$; this ensures that
the sets on the right-hand-side of (4.10), (4.11), (4.12) are nonempty for
large $j$.

(4.10) $(2\gamma)^{-1}d(\mathcal{R}_{2\gamma\rho_{0},2\gamma
r_{0}}\mathcal{M}_{j}^{\prime}(\mathbf{x}_{j},t_{j}-4\gamma^{2}),\operatorname{supp}\mathcal{M}_{j}^{\prime\prime}(t_{j}-4\gamma^{2}))\\\
<\gamma^{-(\kappa+D)}\cdot
2^{-1}d(\mathcal{R}_{2\rho_{0},2r_{0}-|\mathbf{x}_{j}|}\mathcal{M}_{j}^{\prime}(\mathbf{x}_{j},-4),\operatorname{supp}\mathcal{M}_{j}^{\prime\prime}(-4)).$

Let’s translate (4.10) into a statement about the rescaled flows
$\tilde{\mathcal{M}}_{j}^{\prime},\tilde{\mathcal{M}}_{j}^{\prime\prime}$
around $X_{j}$ at scale $2\gamma$. Note that under this rescaling, non-
rescaled time turns into rescaled time as:

$t=t_{j}-4\gamma^{2}\mapsto\tau=0,$
$t=-4\mapsto\tau=\tau_{j}:=2\log\gamma-\log(1+t_{j}/4).$

In particular, using $|t_{j}|\leq 1$, the dilation factor in the rescaled flow
at $\tau_{j}$ satisfies

$(2\gamma)^{-1}e^{\tau_{j}/2}=(1+t_{j}/4)^{-\frac{1}{2}}:=\beta_{j}.$

As such, (4.10) rescales (see Remark 2.4) to yield

(4.11)
$d(\mathcal{R}_{\rho_{0},r_{0}}\tilde{\mathcal{M}}_{j}^{\prime}(\mathbf{0},0),\operatorname{supp}\tilde{\mathcal{M}}_{j}^{\prime\prime}(0))\\\
<\gamma^{-(\kappa+D)}\cdot
d(\mathcal{R}_{\beta_{j}\rho_{0},\beta_{j}(r_{0}-|\mathbf{x}_{j}|/2)}\tilde{\mathcal{M}}_{j}^{\prime}(\mathbf{0},\tau_{j}),\operatorname{supp}\tilde{\mathcal{M}}_{j}^{\prime\prime}(\tau_{j})).$

Combining with Remark 2.4, (4.11),
$\beta_{j}\in(\tfrac{2}{\sqrt{5}},\tfrac{2}{\sqrt{3}})$, $r_{0}>2$,
$|\mathbf{x}_{j}|\leq 1$, we thus find that there exists a fixed
$\theta\in(2,3)$ such that

(4.12)
$\tilde{d}_{j}:=d(\mathcal{R}_{\rho_{0},r_{0}}\tilde{\mathcal{M}}_{j}^{\prime}(\mathbf{0},0),\operatorname{supp}\tilde{\mathcal{M}}_{j}^{\prime\prime}(0))\\\
<\gamma^{-(\kappa+D)}\cdot
d(\mathcal{R}_{\theta\rho_{0},r_{0}/\theta}\tilde{\mathcal{M}}_{j}^{\prime}(\mathbf{0},\tau_{j}),\operatorname{supp}\tilde{\mathcal{M}}_{j}^{\prime\prime}(\tau_{j})).$

Observe that (4.12) and the avoidance principle imply $\tilde{d}_{j}>0$.

We pass to a further subsequence so that $\tau_{j}\to\tau_{0}$ with
$\tau_{0}-2\log\gamma\in[\log\tfrac{4}{5},\log\tfrac{4}{3}]$. Note that (4.8)
guarantees that $\tau_{0}\leq-2$. Note also that

$\tilde{\mathcal{M}}_{j}^{\prime},\tilde{\mathcal{M}}_{j}^{\prime\prime}\rightharpoonup\tilde{\mathcal{M}}_{V}$

satisfy the hypotheses of Lemma 4.6. Let $\tilde{W}_{j}$,
$\tilde{w}_{j}^{\prime}$, $\tilde{w}_{j}^{\prime\prime}$ be as in that lemma
with $\tilde{w}_{j}^{\prime}\geq\tilde{w}_{j}^{\prime\prime}$. Apply Lemma 4.9
to pass

$\tilde{u}_{j}:=\tilde{d}_{j}^{-1}(\tilde{w}_{j}^{\prime}-\tilde{w}_{j}^{\prime\prime})$

to a $C^{\infty}_{\textrm{loc}}(\operatorname{reg}V\times(-\infty,0))$ limit
that satisfies

(4.13)
$\sup_{\mathcal{R}_{\rho_{0},r_{0}}V(\mathbf{0})}\tilde{u}(\cdot,-1)\leq H.$

and

(4.14)
$\sup_{\mathcal{R}_{3\rho_{0},r_{0}/3}V(\mathbf{0})}\tilde{u}(\cdot,\tau_{0})\leq
H\inf_{\mathcal{R}_{\rho_{0},r_{0}}V(\mathbf{0})}\tilde{u}(\cdot,\tau_{0}+1).$

Finally, we note that using $\theta\in(2,3)$, (4.12) passes to the limit to
yield

(4.15)
$\inf_{\mathcal{R}_{3\rho_{0},r_{0}/3}V(\mathbf{0})}\tilde{u}(\cdot,\tau_{0})\geq\gamma^{\kappa+D}.$

Combining (4.14) and (4.15) we find

(4.16)
$H^{-1}\gamma^{\kappa+D}\leq\inf_{\mathcal{R}_{\rho_{0},r_{0}}V(\mathbf{0})}\tilde{u}(\cdot,\tau_{0}+1).$

Fix a relatively open set $\Omega\subset\operatorname{reg}V$ with smooth
boundary such that

$\mathcal{R}_{2\rho_{0},r_{0}/2}V(\mathbf{0})\subset\Omega\subset\mathcal{R}_{\rho_{0},r_{0}}V(\mathbf{0}).$

By Proposition 3.6,

$2\mu(\Omega)<-2\kappa-D:=2\mu_{0}.$

We now apply Lemma 4.10 to $\Sigma=\overline{\Omega}$ on the interval
$[\tau_{0}+1,-1]$ to find

(4.17) $\int_{\Omega}\tilde{u}(\cdot,-1)\varphi
e^{-\tfrac{1}{4}|\cdot|^{2}}\geq
e^{\mu_{0}(\tau_{0}+2)}\int_{\Omega}\tilde{u}(\cdot,\tau_{0}+1)\varphi
e^{-\tfrac{1}{4}|\cdot|^{2}},$

where $\varphi>0$ is the first Dirichlet eigenfunction of $L$ on $\Omega$ and
$\int_{\Omega}\varphi e^{-\tfrac{1}{4}|\cdot|^{2}}=1$. We then claim that we
have the following chain of inequalities:

$\displaystyle H$
$\displaystyle\geq\sup_{\mathcal{R}_{\rho_{0},r_{0}}(\mathbf{0})}\tilde{u}(\cdot,-1)$
$\displaystyle\geq\int_{\Omega}\tilde{u}(\cdot,-1)\varphi
e^{-\tfrac{1}{4}|\cdot|^{2}}$ $\displaystyle\geq
e^{\mu_{0}(\tau_{0}+2)}\int_{\Omega}\tilde{u}(\cdot,\tau_{0}+1)\varphi
e^{-\tfrac{1}{4}|\cdot|^{2}}$ $\displaystyle\geq
e^{\mu_{0}(\tau_{0}+2)}\inf_{\mathcal{R}_{\rho_{0},r_{0}}V(\mathbf{0})}\tilde{u}(\cdot,\tau_{0}+1)$
$\displaystyle\geq H^{-1}\gamma^{\kappa+D}e^{\mu_{0}(\tau_{0}+2)}$
$\displaystyle\geq H^{-1}\gamma^{\kappa+D+2\mu_{0}}(4e/5)^{2\mu_{0}}.$

The first inequality follows from (4.13), the second and fourth from the
normalization of $\varphi$ and
$\Omega\subset\mathcal{R}_{\rho_{0},r_{0}}(\mathbf{0})$, the third from
(4.17), the second to last follows from (4.16) and the final inequality from
$e^{\tau_{0}}\geq(4/5)\gamma^{2}$.

Noting that $D\leq n+2$ (by definition of $D$), we have
$2\mu_{0}\geq-2\kappa-n-2$. Thus, we can rearrange the previous expression to
read

$(\tfrac{5}{4e})^{2\kappa+n+2}\leq H^{2}\gamma^{\kappa}.$

This contradicts (4.8), completing the proof. ∎

##  5\. Density drop

###  5.1. Setup

We continue to fix $n\in\\{2,3,\dots,\\}$, $\Lambda\in(0,2]$, $\varepsilon>0$
so that ($\heartsuit_{n,\Lambda}$) and ($\diamondsuit_{n,\Lambda}$) hold.
Recall the definition of the constants $C,\kappa,\rho_{0},H$ (cf. Remark
4.11). We have also fixed $\gamma$ in (4.8) and $\eta_{0}$ in Proposition
4.12. The constants $C,\kappa,\rho_{0},H,\gamma,\eta_{0}$ all depend only on
the values of $n,\Lambda,\varepsilon$.

In this section, we consider a closed interval $0\in S$ with $|S|>0$ and a
local unit-speed foliation $\\{M_{s}\\}_{s\in S}$ with
$M_{s}\subset\mathbb{R}^{n+1}$ closed connected embedded hypersurfaces having

(5.1) $\lambda(M_{s})\leq\Lambda-\varepsilon.$

Shrinking $S$, we can assume that the $M_{s}$ are uniformly smooth and
embedded, so in particular there is $t_{0}>0$ so that if
$(\mathbf{x},t)\in\operatorname{sing}\mathcal{M},\mathcal{M}\in\mathfrak{F}(M_{s}),s\in
S$ then $t\geq t_{0}$.

###  5.2. Singular points on foliation Brakke flows

For any $Q\subset S$ we will write

(5.2) $\displaystyle\operatorname{sing}(Q)$ $\displaystyle:=\cup_{s\in
Q}\cup_{\mathcal{M}\in\mathfrak{F}(M_{s})}\operatorname{sing}\mathcal{M},$
$\displaystyle\operatorname{sing}_{\textrm{gen}}(Q)$
$\displaystyle:=\cup_{s\in
Q}\cup_{\mathcal{M}\in\mathfrak{F}(M_{s})}\operatorname{sing}_{\textrm{gen}}\mathcal{M}$
$\displaystyle\operatorname{sing}_{\textrm{non-gen}}(Q)$
$\displaystyle:=\cup_{s\in
Q}\cup_{\mathcal{M}\in\mathfrak{F}(M_{s})}\operatorname{sing}_{\textrm{non-
gen}}\mathcal{M}.$

By Corollary 2.13 we see that if

$\mathcal{M},\mathcal{M}^{\prime}\in\cup_{s\in S}\mathfrak{F}(M_{s})\text{ and
}X=(\mathbf{x},t)\in\operatorname{supp}\mathcal{M}\cap\operatorname{supp}\mathcal{M}^{\prime}\text{
has }t>0,$

then:

  * •

$\mathcal{M}$, $\mathcal{M}^{\prime}\in\mathfrak{F}(M_{s_{0}})$ for $s_{0}\in
S$,

  * •

the set of tangent flows to $\mathcal{M}$, $\mathcal{M}^{\prime}$ at $X$
agree,

  * •

as do the densities $\Theta_{\mathcal{M}}(X)$,
$\Theta_{\mathcal{M}^{\prime}}(X)$.

In particular, just like how, for each $\mathcal{M}\in\cup_{s\in
S}\mathfrak{F}(M_{s})$,

$\operatorname{sing}\mathcal{M}=\operatorname{sing}_{\textrm{gen}}\mathcal{M}\cup\operatorname{sing}_{\textrm{non-
gen}}\mathcal{M}$

is a disjoint union (by [CIM15, CM15]), so is

$\operatorname{sing}(S)=\operatorname{sing}_{\textrm{gen}}(S)\cup\operatorname{sing}_{\textrm{non-
gen}}(S).$

Moreover, for any $X\in\operatorname{sing}(S)\iff
X\in\operatorname{sing}\mathcal{M}$, $\mathcal{M}\in\mathfrak{F}(M_{s})$,
$s\in S$, we can define

$\Theta(X):=\Theta_{\mathcal{M}}(X)$

unambiguously without reference to $\mathcal{M}$.

###  5.3. The improved quantity

Let $\mathfrak{N}\subset\operatorname{sing}_{\textrm{non-gen}}(S)$ be a closed
subset with respect to the parabolic topology and define,

$\mathcal{D}_{\mathfrak{N}}:S\to\\{-\infty\\}\cup[1,\infty),$
$\mathcal{D}_{\mathfrak{N}}(s):=\sup\\{\Theta(X):X\in\mathfrak{N}\cap\operatorname{sing}\mathcal{M},\;\mathcal{M}\in\mathscr{F}(M_{s})\\},$

where we use the standard convention $\sup\emptyset=-\infty$.

######  Lemma 5.1.

The function $s\mapsto\mathcal{D}_{\mathfrak{N}}(s)$ is upper semi-continuous,
i.e., for all $s_{0}\in S$,

$\limsup_{s\to
s_{0}}\mathcal{D}_{\mathfrak{N}}(s)\leq\mathcal{D}_{\mathfrak{N}}(s_{0}).$

###### Proof.

Choose $s_{i}\to s_{0}$, $s_{i}\neq s_{0}$,
$\mathcal{M}_{i}\in\mathfrak{F}(M_{s_{i}})$,
$X_{i}\in\operatorname{sing}\mathcal{M}_{i}\cap\mathfrak{N}$ with

$\lim_{i}\Theta(X_{i})=\limsup_{s\to s_{0}}\mathcal{D}_{\mathfrak{N}}(s).$

Passing to a subsequence,
$\mathcal{M}_{i}\rightharpoonup\mathcal{M}\in\mathfrak{F}(M_{s_{0}})$ and
$X_{i}\to X$. Since $\mathfrak{N}$ is closed, $X\in\mathfrak{N}$. Thus,

$\mathcal{D}_{\mathfrak{N}}(s_{0})\geq\Theta(X)\geq\lim_{i}\Theta(X_{i})=\limsup_{s\to
s_{0}}\mathcal{D}_{\mathfrak{N}}(s)$

by upper semicontinuity of density. This completes the proof. ∎

The main result of this section is as follows. Below, $\eta_{0}$ is given by
Proposition 4.12.

######  Proposition 5.2.

We have

$\liminf_{s\to
s_{0}}\mathcal{D}_{\mathfrak{N}}(s)\leq\mathcal{D}_{\mathfrak{N}}(s_{0})-\eta_{0}$

for any $s_{0}\in S$.

######  Corollary 5.3.

There is a relatively open dense subset $S^{\prime}\subset S$ with
$\operatorname{sing}(S^{\prime})\cap\mathfrak{N}=\emptyset$.

###### Proof.

Note that if $\mathcal{D}_{\mathfrak{N}}(s)<1$ then
$\mathcal{D}_{\mathfrak{N}}(s)=-\infty$ by integrality. Thus

$S^{\prime}:=\mathcal{D}_{\mathfrak{N}}^{-1}(-\infty)=\mathcal{D}_{\mathfrak{N}}^{-1}([-\infty,1))$

is open using the upper-semicontinuity from Lemma 5.1. On the other hand,
iterating Proposition 5.2 finitely many times, we see that $S^{\prime}$ is
dense. This completes the proof. ∎

The remainder of this section is devoted to the proof of Proposition 5.2. Note
that the failure of Proposition 5.2 for some $s_{0}\in S$ would mean that

(5.3) $\liminf_{s\to
s_{0}}\mathcal{D}_{\mathfrak{N}}(s)>\mathcal{D}_{\mathfrak{N}}(s_{0})-\eta_{0}.$

As such, by combining (5.3) with Lemma 5.1, we see that if Proposition 5.2 was
false, we can shrink $S$ while preserving $s_{0}\in S$ so that

(5.4)
$\mathcal{D}_{\mathfrak{N}}(s_{0})-\eta_{0}\leq\mathcal{D}_{\mathfrak{N}}(s)\leq\mathcal{D}_{\mathfrak{N}}(s_{0})+\eta_{0}\text{
for all }s\in S.$

For the sake of contradiction, we can thus assume that (5.4) holds in the
remainder of the proof.

######  Lemma 5.4.

There is $r_{0}\in(0,\tfrac{1}{2}\sqrt{t_{0}})$ so that

$\Theta_{\mathcal{M}}(X,r_{0})\leq\mathcal{D}_{\mathfrak{N}}(s)+3\eta_{0}$

for every $s\in S$, $\mathcal{M}\in\mathfrak{F}(M_{s})$, and
$X\in\mathfrak{N}\cap\operatorname{sing}\mathcal{M}$.

###### Proof.

If not, there exist $X_{i}$, $\mathcal{M}_{i}$, $s_{i}$ as above, and
$r_{i}\to 0$ so that

$\Theta_{\mathcal{M}_{i}}(X_{i},r_{i})>\mathcal{D}_{\mathfrak{N}}(s_{i})+3\eta_{0}.$

Passing to a subsequence (not relabeled), let $s_{i}\to s_{\infty}\in S$.
Using (5.4) twice we get

$\mathcal{D}_{\mathfrak{N}}(s_{i})\geq\mathcal{D}_{\mathfrak{N}}(s_{0})-\eta_{0}\geq\mathcal{D}_{\mathfrak{N}}(s_{\infty})-2\eta_{0}.$

Thus, we find that

$\Theta_{\mathcal{M}_{i}}(X_{i},r_{i})>\mathcal{D}_{\mathfrak{N}}(s_{\infty})+\eta_{0}.$

For any $\rho\in(0,\sqrt{t_{0}})$, monotonicity yields

$\Theta_{\mathcal{M}_{i}}(X_{i},\rho)>\mathcal{D}_{\mathfrak{N}}(s_{\infty})+\eta_{0}.$

for large $i$. Pass to a further subsequence (not relabeled) so that
$\mathcal{M}_{i}\rightharpoonup\mathcal{M}_{\infty}\in\mathfrak{F}(M_{s_{\infty}})$
and $X_{i}\to X_{\infty}$. Then,

$\Theta_{\mathcal{M}_{\infty}}(X_{\infty},\rho)\geq\mathcal{D}_{\mathfrak{N}}(s_{\infty})+\eta_{0}.$

and sending $\rho\to 0$, we find

(5.5) $\Theta(X_{\infty})\geq\mathcal{D}_{\mathfrak{N}}(s_{\infty})+\eta_{0}.$

However, $\mathfrak{N}$ is closed so $X_{\infty}\in\mathfrak{N}$,
contradicting (5.5). This completes the proof. ∎

Up to performing a (single) global scaling, we will assume below that
$r_{0}=\eta_{0}^{-1}$. In particular, by Lemma 5.4, we see that

(5.6)
$\operatorname{sing}(S)\cap\\{X=(\mathbf{x},t):t\in(0,4\eta_{0}^{-2}]\\}=\emptyset.$

###  5.4. Covers of high-density points in $\mathfrak{N}$

We wish to construct appropriate covers of the set of nearly top density
points in $\mathfrak{N}$:

$\mathfrak{N}_{+}:=\\{X\in\mathfrak{N}:s\in
S,\mathcal{M}\in\mathfrak{F}(M_{s}),\Theta(X)\geq\mathcal{D}_{\mathfrak{N}}(s)-\eta_{0}\\}.$

Let us introduce the notation we will use. For $\ell\in\\{0,1,\dots\\}$ we
will construct a finite set $\mathcal{T}[\ell]$ together with “geometric
realization maps”

$\iota_{\ell}:\mathcal{T}[\ell]\to\mathfrak{N}_{+}.$

We will also choose “parent” maps

$\mathfrak{p}:\mathcal{T}[\ell+1]\to\mathcal{T}[\ell].$

We will write $\mathfrak{p}^{(k)}$ for the iterated parent (grandparent) map,
with the usual convention that
$\mathfrak{p}^{(0)}=\text{id}:\mathcal{T}[\ell]\to\mathcal{T}[\ell]$.

######  Remark 5.5.

Informally, we can think of $\cup_{\ell}\iota_{\ell}(\mathcal{T}[\ell])$ as a
tree of points in $\mathfrak{N}_{+}$. However, it is technically useful to
allow multiple elements to map to the same point in $\mathfrak{N}_{+}$, but
allowing them to have different parent elements.

Now, assuming that we have inductively constructed
$\mathcal{T}[0]\xleftarrow{\mathfrak{p}}\mathcal{T}[1]\xleftarrow{\mathfrak{p}}\dots\xleftarrow{\mathfrak{p}}\mathcal{T}[\ell]$,
we proceed to define the following for each $\mathcal{X}\in\mathcal{T}[\ell]$:

  * •

the spacetime neighborhood

$P_{\ell}(\mathcal{X}):=\bigcap_{j=0}^{\ell}P(\iota_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X})),\gamma^{j});$

  * •

the restriction of $\mathfrak{N}_{+}$ to it

$\mathfrak{N}_{+,\ell}(\mathcal{X}):=\mathfrak{N}_{+}\cap
P_{\ell}(\mathcal{X});$

  * •

the set of pairs of (point, flow) so that the flow has a high density point
lying in $\mathfrak{N}_{+,\ell}(\mathcal{X})$, followed by a
translation/scaling taking $P(\iota_{\ell}(\mathcal{X}),\gamma^{\ell})\mapsto
P((\mathbf{0},0),1)$

$\begin{split}\mathcal{Q}_{\ell}(\mathcal{X})&:=\\{({\operatorname{ParDil}_{\gamma^{-\ell}}}(X^{\prime}-\iota_{\ell}(\mathcal{X})),{\operatorname{ParDil}_{\gamma^{-\ell}}}(\mathcal{M}^{\prime}-\iota_{\ell}(\mathcal{X}))):\\\
&\qquad\qquad
X^{\prime}\in\mathfrak{N}_{+,\ell}(\mathcal{X})\cap\operatorname{sing}\mathcal{M}^{\prime},\mathcal{M}^{\prime}\in\mathfrak{F}(M_{s}),s\in
S\\},\end{split}$

where $\operatorname{ParDil}_{\lambda}$ is the standard parabolic dilation of
a Brakke flow around the space-time origin $(\mathbf{0},0)$ by a factor
$\lambda>0$.

Recalling that $\mathscr{P}(\eta_{0})$ is as in Definition 4.2 and Proposition
4.12, and that

$\pi:(X,\mathcal{M})\mapsto X,$

denotes the projection onto the first coordinate in $\mathscr{P}(\eta_{0})$,
we have the following result:

######  Lemma 5.6.

If $\mathcal{X}\in\mathcal{T}[\ell]$, then:

  1. (a)

$\mathcal{Q}_{\ell}(\mathcal{X})\subset\mathscr{P}(\eta_{0})$, and

  2. (b)

the assumptions of Proposition 4.12 are satisfied with
$\mathcal{Q}_{\ell}(\mathcal{X})$ in place of $\mathcal{Q}$.

###### Proof.

We first prove (a), i.e., that $\mathcal{Q}_{\ell}(\mathcal{X})$ satisfies
(1)-(5) in Definition 4.2 (with $\eta=\eta_{0}$). We first note that since
$\mathfrak{N}_{+,\ell}\subset P_{\ell}(\mathcal{X})\subset
P(\iota_{\ell}(\mathcal{X}),\gamma^{\ell})$, we have

${\operatorname{ParDil}_{\gamma^{-\ell}}}(\mathfrak{N}_{+,\ell}(\mathcal{X})-\iota_{\ell}(\mathcal{X}))\subset{\operatorname{ParDil}_{\gamma^{-\ell}}}(P_{\ell}(\mathcal{X})-\iota_{\ell}(\mathcal{X}))\subset
P((\mathbf{0},0),1),$

so (1) holds. Condition (2) follows from $\gamma<1$ and (5.6). Condition (3)
follows from (5.1). Condition (4) follows from that
$\mathfrak{N}\subset\operatorname{sing}_{\textrm{non-gen}}(S)$. Finally, for
any $\mathcal{M}^{\prime}\in\mathfrak{F}(M_{s})$ and
$X^{\prime}\in\mathfrak{N}_{+,\ell}(\mathcal{X})\cap\operatorname{sing}\mathcal{M}^{\prime}$,
we have that

$\Theta_{\mathcal{M}^{\prime}}(X^{\prime})\geq\mathcal{D}_{\mathfrak{S}}(s)-\eta_{0}$

by definition of $\mathfrak{N}_{+}$ and

$\Theta_{\mathcal{M}^{\prime}}(X^{\prime},\eta_{0}^{-1})\leq\mathcal{D}_{\mathfrak{S}}(s)+3\eta_{0}$

by Lemma 5.4 and the fact that we scaled so that $r_{0}=\eta_{0}^{-1}$.
Putting this together, we find using monotonicity that

$\Theta_{\mathcal{M}^{\prime}}(X^{\prime},\eta_{0}^{-1}\gamma^{\ell})\leq\Theta_{\mathcal{M}^{\prime}}(X^{\prime},\eta_{0}^{-1})\leq\Theta_{\mathcal{M}^{\prime}}(X^{\prime})+4\eta_{0}.$

Translating and scaling, this yields condition (5). Thus (a) holds.

Now consider (b). The first bullet point follows from how we translated and
scaled the pairs, and the second bullet point follows from Lemma 2.11. ∎

Thus, Proposition 4.12 applies to each $\mathcal{Q}_{\ell}(\mathcal{X})$,
$\mathcal{X}\in\mathcal{T}[\ell]$ and yields the following corollary (after
undoing the various definitions of $\mathcal{Q}_{\ell}(\mathcal{X})$, etc.):

######  Corollary 5.7.

Let $\mathcal{X}\in\mathcal{T}[\ell]$. Then, there exist
$D_{\ell}(\mathcal{X})\in\mathbb{N}$ and points
$X^{\mathcal{X}}_{1},\ldots,X^{\mathcal{X}}_{K(\mathcal{X})}\in\mathfrak{N}_{+,\ell}(\mathcal{X})$
so that:

  1. (a)

$\mathfrak{N}_{+,\ell}(\mathcal{X})\subset\cup_{i=1}^{K(\mathcal{X})}P(X^{\mathcal{X}}_{i},\gamma^{\ell+1})$
with $K(\mathcal{X})\leq C\gamma^{-D_{\ell}(\mathcal{X})}$;

and denoting $\iota_{\ell}(\mathcal{X})=(\mathbf{x},t)$ and
$X^{\mathcal{X}}_{i}=(\mathbf{x}_{i},t_{i})$ we also have:

  1. (b)

if $\mathcal{M}^{\prime}\in\cup_{s\in S}\mathfrak{F}(M_{s})$ and
$\mathfrak{N}_{+}\cap P_{\ell}(\mathcal{X})\cap
P(X^{\mathcal{X}}_{i},\gamma^{\ell+1})$ intersects
$\operatorname{sing}\mathcal{M}^{\prime}$, then

$\displaystyle\mathcal{R}_{2\gamma^{\ell}\rho_{0},2\gamma^{\ell}r_{0}}\mathcal{M}^{\prime}(\mathbf{x},t-4\gamma^{2\ell})$
$\displaystyle\neq 0,$
$\displaystyle\mathcal{R}_{2\gamma^{\ell+1}\rho_{0},2\gamma^{\ell+1}r_{0}}\mathcal{M}^{\prime}(\mathbf{x}_{i},t_{i}-4\gamma^{2(\ell+1)})$
$\displaystyle\neq 0,\text{ and},$

  2. (c)

if $\mathcal{M}^{\prime},\mathcal{M}^{\prime\prime}\in\cup_{s\in
S}\mathfrak{F}(M_{s})$ and $\mathfrak{N}_{+}\cap P_{\ell}(\mathcal{X})\cap
P(X^{\mathcal{X}}_{i},\gamma^{\ell+1})$ intersects both
$\operatorname{sing}\mathcal{M}^{\prime}$ and
$\operatorname{sing}\mathcal{M}^{\prime\prime}$, then

$\gamma^{-(\kappa+D_{\ell}(X))}\cdot(2\gamma^{\ell})^{-1}d(\mathcal{R}_{2\gamma^{\ell}\rho_{0},2\gamma^{\ell}r_{0}}\mathcal{M}^{\prime}(\mathbf{x},t-4\gamma^{2\ell}),\operatorname{supp}\mathcal{M}^{\prime\prime}(t-4\gamma^{2\ell}))\\\
\leq(2\gamma^{\ell+1})^{-1}d(\mathcal{R}_{2\gamma^{\ell+1}\rho_{0},2\gamma^{\ell+1}r_{0}}\mathcal{M}^{\prime}(\mathbf{x}_{i},t_{i}-4\gamma^{2(\ell+1)}),\operatorname{supp}\mathcal{M}^{\prime\prime}(t_{i}-4\gamma^{2(\ell+1)})).$

######  Definition 5.8.

Let us now define the $\mathcal{T}[\ell]$, $\ell\in\\{0,1,2,\ldots\\}$ and
parent maps $\mathfrak{p}$:

  * •

For $\ell=0$, take $\mathcal{T}[0]\subset\mathfrak{N}_{+}$ arbitrarily so that
$\mathfrak{N}_{+}\subset\cup_{X\in\mathcal{T}[0]}P(X,1)$.

  * •

Then, inductively define

$\mathcal{T}[\ell+1]:=\bigcup_{\mathcal{X}\in\mathcal{T}[\ell]}\\{X^{\mathcal{X}}_{1},\ldots,X^{\mathcal{X}}_{K(\mathcal{X})}\\}\times\\{\mathcal{X}\\}$

and

$\iota_{\ell+1}:\mathcal{T}[\ell+1]\to\mathfrak{N}_{+},$
$\mathfrak{p}:\mathcal{T}[\ell+1]\to\mathcal{T}[\ell],$

to be projection onto the first and second factors, respectively.

######  Corollary 5.9.

For $\ell\in\\{0,1,\dots\\}$ it holds that

$\mathfrak{N}_{+}\subset\bigcup_{\mathcal{X}\in\mathcal{T}[\ell]}P_{\ell}(\mathcal{X}).$

###### Proof.

We induct on $\ell$. When $\ell=0$, since $P_{0}(X)=P(\iota_{0}(X),1)$, the
assertion follows from the definition of $\mathcal{T}[0]$. If the assertion
holds for $\ell$, then

$\mathfrak{N}_{+}=\mathfrak{N}_{+}\cap\bigcup_{\mathcal{X}\in\mathcal{T}[\ell]}P_{\ell}(\mathcal{X})=\bigcup_{\mathcal{X}\in\mathcal{T}[\ell]}\mathfrak{N}_{+,\ell}(\mathcal{X}).$

By (1) in Corollary 5.7 and our definition of $\mathfrak{p}$ and
$\iota_{\ell+1}$,

$\mathfrak{N}_{+,\ell}(\mathcal{X})\subset
P_{\ell}(\mathcal{X})\cap\bigcup_{i=1}^{K(\mathcal{X})}P(X^{\mathcal{X}}_{i},\gamma^{\ell+1})=\bigcup_{\mathcal{X}^{\prime}\in\mathfrak{p}^{-1}(\mathcal{X})}P_{\ell+1}(\mathcal{X}^{\prime}).$

Therefore, since
$\cup_{\mathcal{X}\in\mathcal{T}[\ell]}\mathfrak{p}^{-1}(\mathcal{X})=\mathcal{T}[\ell+1]$,

$\mathfrak{N}_{+}=\bigcup_{\mathcal{X}\in\mathcal{T}[\ell]}\mathfrak{N}_{+,\ell}(\mathcal{X})\subset\bigcup_{\mathcal{X}\in\mathcal{T}[\ell]}\bigcup_{\mathcal{X}^{\prime}\in\mathfrak{p}^{-1}(\mathcal{X})}P_{\ell+1}(\mathcal{X}^{\prime})=\bigcup_{\mathcal{X}^{\prime}\in\mathcal{T}[\ell]}\mathfrak{p}^{-1}(\mathcal{X}^{\prime}).$

This proves the assertion. ∎

######  Proposition 5.10.

There is $\ell_{0}$ sufficiently large so that the following holds for
$\ell\geq\ell_{0}$.

Suppose that $\mathcal{X}\in\mathcal{T}[\ell]$ and
$\iota_{\ell}(X)\in\operatorname{sing}\mathcal{M}$ for
$\mathcal{M}\in\mathfrak{F}(M_{s})$, $s\in S$, and suppose that
$\mathcal{M}^{\prime}\in\mathfrak{F}(M_{s^{\prime}})$, $s^{\prime}\in S$, is
such that $\mathfrak{N}_{+,\ell}(\mathcal{X})$ intersects
$\operatorname{sing}\mathcal{M}^{\prime}\neq\emptyset$. Then,

$|s^{\prime}-s|<16r_{0}\gamma^{\ell\kappa}\gamma^{\sum_{j=0}^{\ell-1}D_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X}))}$

###### Proof.

We begin by considering any
$\ell,\mathcal{X},\mathcal{M},\mathcal{M}^{\prime},s,s^{\prime}$ as in the
statement of the proposition. Later, we will assume a sequence of
counterexamples exists with $\ell\to\infty$ and derive a contradiction.

Write $\iota_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X}))=(\mathbf{x}_{j},t_{j})$
for $j=0,\dots,\ell$. Apply Corollary 5.7 (c) at level $j$ (in place of $\ell$
there) centered at $\mathfrak{p}^{(\ell-j)}(\mathcal{X})$ (in place of
$\mathcal{X}$ there) and $(\mathbf{x}_{j+1},t_{j+1})$ (in place of
$\mathcal{X}_{i}$ there) and $\mathcal{M}$ (in place of
$\mathcal{M}^{\prime\prime}$ there) to obtain

$\gamma^{-(\kappa+D_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X})))}(2\gamma^{j})^{-1}d(\mathcal{R}_{2\gamma^{j}\rho_{0},2\gamma^{j}r_{0}}\mathcal{M}^{\prime}(\mathbf{x}_{j},t_{j}-4\gamma^{2j}),\operatorname{supp}\mathcal{M}(t_{j}-4\gamma^{2j}))\\\
\leq(2\gamma^{j+1})^{-1}d(\mathcal{R}_{2\gamma^{j+1}\rho_{0},2\gamma^{j+1}r_{0}}\mathcal{M}^{\prime}(\mathbf{x}_{j+1},t_{j+1}-4\gamma^{2(j+1)}),\operatorname{supp}\mathcal{M}(t_{j+1}-4\gamma^{2(j+1)})).$

We iterate this from $j=0$ to $j=\ell-1$. Together the trivial estimate

$d(\mathcal{R}_{2\gamma^{\ell}\rho_{0},2\gamma^{\ell}r_{0}}\mathcal{M}^{\prime}(\mathbf{x}_{\ell},t_{\ell}-4\gamma^{2\ell}),\operatorname{supp}\mathcal{M}(t_{\ell}-4\gamma^{2\ell}))\leq
4\gamma^{\ell}r_{0},$

we obtain

$d(\mathcal{R}_{2\rho_{0},2r_{0}}\mathcal{M}^{\prime}(\mathbf{x}_{0},t_{0}-4),\operatorname{supp}\mathcal{M}(t_{0}-4))\leq
8r_{0}\gamma^{\ell\kappa}\gamma^{\sum_{j=0}^{\ell-1}D_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X}))}.$

Using that the distance between the support of co-dimension one Brakke flows
is non-decreasing [Ilm94, 10.6], we get

$d(\operatorname{supp}\mathcal{M}^{\prime}(0),\operatorname{supp}\mathcal{M}(0))\leq
8r_{0}\gamma^{\ell\kappa}\gamma^{\sum_{j=0}^{\ell-1}D_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X}))}.$

We now use this to prove the proposition.

Assume that for $\ell$ sufficiently large there are
$\mathcal{X}_{\ell},\mathcal{M}_{\ell},\mathcal{M}^{\prime}_{\ell},s_{\ell},s^{\prime}_{\ell}$
as in the setup but with

(5.7) $|s^{\prime}_{\ell}-s_{\ell}|\geq
16r_{0}\gamma^{\ell\kappa}\gamma^{\sum_{j=0}^{\ell-1}D_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X}_{\ell}))}.$

The analysis in the previous paragraph applies to the above setting to thus
yield

(5.8)
$d(\operatorname{supp}\mathcal{M}^{\prime}_{\ell}(0),\operatorname{supp}\mathcal{M}_{\ell}(0))\leq
8r_{0}\gamma^{\ell\kappa}\gamma^{\sum_{j=0}^{\ell-1}D_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X}_{\ell}))}.$

By combining (5.7) and (5.8), we find that

(5.9)
$d(\operatorname{supp}\mathcal{M}^{\prime}_{\ell}(0),\operatorname{supp}\mathcal{M}_{\ell}(0))\leq\tfrac{1}{2}|s^{\prime}_{\ell}-s_{\ell}|.$

However, (5.9) contradicts the fact that our foliation at $t=0$ is a unit
speed foliation. This completes the proof. ∎

###  5.5. The proof of Proposition 5.2

For $\ell\in\\{0,1,\dots\\}$ fixed, we form an open cover of $S$ as follows:
for $\mathcal{X}\in\mathcal{T}[\ell]$ we fix $s\in S$ with
$\iota_{\ell}(\mathcal{X})\in\operatorname{sing}\mathcal{M}$ for some
$\mathcal{M}\in\mathfrak{F}(M_{s})$. Then, let $U_{\ell}(\mathcal{X})\subset
S$ denote the relatively open interval centered at $s$ with length

$|U_{\ell}(\mathcal{X})|=32r_{0}\gamma^{\ell\kappa}\gamma^{\sum_{j=0}^{\ell-1}D_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X}))},$

or if $s$ is close to $\partial S$, then one part of $U_{\ell}(X)$ will be cut
off and this estimate will be replaced by $\leq$; this goes in the right
direction below.

Assume now that $\ell\geq\ell_{0}$, where $\ell_{0}$ is as in Proposition
5.10. Then,

$S=\bigcup_{\mathcal{X}\in\mathcal{T}[\ell]}U_{\ell}(\mathcal{X}).$

Indeed, for $s^{\prime}\in S$, (5.4) implies that there is
$\mathcal{M}^{\prime}\in\mathfrak{F}(M_{s^{\prime}})$ and
$X^{\prime}\in\mathfrak{N}_{+}\cap\operatorname{sing}\mathcal{M}^{\prime}$. By
Corollary 5.9, $X^{\prime}\in P_{\ell}(\mathcal{X})$ for some
$\mathcal{X}\in\mathcal{T}[\ell]$, so $s^{\prime}\in U_{\ell}(\mathcal{X})$ by
Proposition 5.10.

It follows that

$\displaystyle|S|$
$\displaystyle\leq\sum_{\mathcal{X}\in\mathcal{T}[\ell]}|U_{\ell}(\mathcal{X})|$
$\displaystyle\leq
32r_{0}2^{\ell}\gamma^{\ell\kappa}\sum_{\mathcal{X}\in\mathcal{T}[\ell]}\gamma^{\sum_{j=0}^{\ell-1}D_{j}(\mathfrak{p}^{(\ell-j)}(\mathcal{X}))}$
$\displaystyle=32r_{0}2^{\ell}\gamma^{\ell\kappa}\sum_{\mathcal{X}_{0}\in\mathcal{T}[0]}\sum_{\mathcal{X}_{1}\in\mathfrak{p}^{-1}(X_{0})}\cdots\sum_{\mathcal{X}_{\ell}\in\mathfrak{p}^{-1}(X_{\ell-1})}\gamma^{\sum_{j=0}^{\ell-1}D_{j}(\mathcal{X}_{j})}$

By Corollary 5.7 and the definition of $\mathfrak{p}$ we have for
$j\in\\{0,1,\ldots,\ell-1\\}$ that

$|\mathfrak{p}^{-1}(\mathcal{X}_{j})|=K(\mathcal{X}_{j})\leq
C\gamma^{-D_{j}(\mathcal{X}_{j})}.$

Therefore,

$|S|\leq
32r_{0}|\mathcal{T}[0]|\gamma^{\ell\kappa}C^{\ell}=32r_{0}|\mathcal{T}[0]|(C\gamma^{\kappa})^{\ell}.$

By (4.8), it holds that $C\gamma^{\kappa}<1$. Thus, the right hand side of
this chain of inequalities is $o(1)$ as $\ell\to\infty$. This is a
contradiction, completing the proof.

##  6\. Proof of Theorem 1.13

We fix $n\in\\{2,3,\ldots\\}$, $\Lambda\in(0,2]$ so that
($\heartsuit_{n,\Lambda}$) and ($\diamondsuit_{n,\Lambda}$) hold, any closed
embedded $M^{n}\subset\mathbb{R}^{n+1}$ with $\lambda(M)\leq\Lambda$. We need
some preparations for the proof.

###  6.1. Low-entropy local foliation

Flowing $M$ by (smooth) mean curvature flow for a short time will produce
$M^{\prime}$ an arbitrarily small $C^{\infty}$ graph over $M$ with
$\lambda(M^{\prime})<\lambda(M)\leq\Lambda$ unless $M$ is a self-shrinker. If
$M$ is a self-shrinker then either it is a round sphere (in which case Theorem
1.13 follows trivially) or, by [CM12, Theorem 0.7], there is an arbitrarily
small $C^{\infty}$-graph $M^{\prime}$ with
$\lambda(M^{\prime})<\lambda(M)\leq\Lambda$. As such, up to replacing $M$ by
$M^{\prime}$ considered above, we can assume that

$\lambda(M)\leq\Lambda-2\varepsilon.$

For $S$ a closed interval with $0\in S$ and $|S|>0$, let $\\{M_{s}\\}_{s\in
S}$ denote a unit-speed (local) foliation so that the $M_{s}$ are uniformly
smooth and embedded with

$\lambda(M_{s})\leq\Lambda-\varepsilon\text{ for all }s\in S.$

###  6.2. The generic strata

Recall the definitions of
$\operatorname{sing}(S),\operatorname{sing_{\textnormal{gen}}}(S),\operatorname{sing_{\textnormal{non-
gen}}}(S)\subset\mathbb{R}^{n+1}\times\mathbb{R}$ from (5.2) and define, for
$k=0,\ldots,n-1$, the $k$-th generic stratum by

$\mathfrak{G}^{k}:=\\{X\in\operatorname{sing}_{\textrm{gen}}(S):\Theta(X)=\lambda(\SS^{n-k}\times\mathbb{R}^{k})\\}\subset\mathbb{R}^{n+1}\times\mathbb{R}.$

Note that

$\operatorname{sing_{\textnormal{gen}}}(S)=\cup_{k=0}^{n-1}\mathfrak{G}^{k}.$

Fix $\eta_{1}=\eta_{1}(n)$ so that if
$X_{\ell}\in\operatorname{sing_{\textnormal{non-gen}}}(S)$ and $X_{\ell}\to
X\in\mathfrak{G}^{k}$, $k=0,1,\ldots,n-1$, then

$\limsup_{\ell}\Theta(X_{\ell})\leq\lambda(\SS^{n-k})-\eta_{1}.$

The existence of such $\eta_{1}$ follows from a straightforward adaptation of
[CIM15, Theorem 0.2] to the present setting.

###  6.3. Closed subsets of non-generic singular points

For $\alpha>0$ define the following subsets of the set of non-generic singular
points:

$\tilde{\mathfrak{N}}_{\alpha}:=\operatorname{sing_{\textnormal{non-
gen}}}(S)\setminus U_{\alpha}(\operatorname{sing_{\textnormal{gen}}}(S))$

and, for $k=0,\ldots,n-1$,

$\mathfrak{N}_{\alpha}^{k}:=\operatorname{sing_{\textnormal{non-
gen}}}(S)\cap\overline{\mathfrak{G}^{k}}\setminus\cup_{j=k+1}^{n-1}U_{\alpha}(\mathfrak{G}^{j}).$

Here, $\bar{\cdot}$ denotes the closure and $U_{\alpha}(\cdot)$ the open
$\alpha$-neighborhood in spacetime $\mathbb{R}^{n+1}\times\mathbb{R}$, with
respect to the parabolic metric. Finally, define

$\mathfrak{N}_{\alpha}:=\tilde{\mathfrak{N}}_{\alpha}\cup\mathfrak{N}_{\alpha}^{n-1}\cup\cdots\cup\mathfrak{N}_{\alpha}^{0}.$

######  Lemma 6.1.

$\mathfrak{N}_{\alpha}\subset\mathbb{R}^{n+1}\times\mathbb{R}$ is closed.

###### Proof.

Let’s show that each set in the union is closed. The case of
$\tilde{\mathfrak{N}}_{\alpha}$ is straightforward: if
$X_{\ell}\in\tilde{\mathfrak{N}}_{\alpha}$ are such that $X_{\ell}\to X$, then
clearly $X\not\in U_{\alpha}(\operatorname{sing_{\textnormal{gen}}}(S))$, so
$X\in\tilde{\mathfrak{N}}_{\alpha}$.

Next, consider $\mathfrak{N}_{\alpha}^{k}$. Take
$X_{\ell}\in\mathfrak{N}_{\alpha}^{k}$ with $X_{\ell}\to X$. Then,
$X\in\overline{\mathfrak{G}^{k}}\setminus\cup_{j=k+1}^{n-1}U_{\alpha}(\mathfrak{G}^{j})$
is immediate. It remains to prove that
$X\in\operatorname{sing_{\textnormal{non-gen}}}(S)$.

If not, then $X\in\operatorname{sing_{\textnormal{gen}}}(S)$, so
$X\in\cup_{m=0}^{k}\mathfrak{G}^{m}$ since
$X\not\in\cup_{j=k+1}^{n-1}U_{\alpha}(\mathfrak{G}^{j})$. Note that

(6.1)
$X_{\ell},\;X\in\overline{\mathfrak{G}^{k}}\implies\Theta(X_{\ell}),\;\Theta(X)\geq\lambda(\SS^{n-k})$

by upper semicontinuity of density. Combining (6.1) and
$\lambda(\SS^{1})>\lambda(\SS^{2})>\dots>\lambda(\SS^{n})$, we have that
$X\in\mathfrak{G}^{k}$. Thus, by the choice of $\eta_{1}$, we have

$\limsup_{\ell}\Theta(X_{\ell})\leq\lambda(\SS^{n-k})-\eta_{1}.$

This contradicts the fact that $\Theta(X_{\ell})\geq\lambda(\SS^{n-k})$ by
(6.1). ∎

###  6.4. The perturbation

It follows from Lemma 6.1 that Corollary 5.3 applies with
$\mathfrak{N}_{\alpha}$ in place of $\mathfrak{N}$ and yields a relatively
open dense subset $S^{\prime}_{\alpha}\subset S$ with
$\operatorname{sing}(S^{\prime}_{\alpha})\cap\mathfrak{N}_{\alpha}=\emptyset$.

Consider $S^{\prime}=\cap_{\alpha\in\mathbb{Q}\cap(0,1]}S^{\prime}_{\alpha}$,
which is dense by the Baire category theorem. Theorem 1.13 now follows from
Lemma 6.2 below.

######  Lemma 6.2.

$\operatorname{sing_{\textnormal{non-gen}}}(S^{\prime})=\emptyset$.

###### Proof.

Suppose, for contradiction, that there did exist some
$X\in\operatorname{sing_{\textnormal{non-gen}}}(S^{\prime})$, and take any
sequence $\alpha_{i}\in\mathbb{Q}\cap(0,1]$ with $\alpha_{i}\to 0$. Fix some
$i$. Using that

(6.2) $S^{\prime}\subset S_{\alpha_{i}}\text{ and
}\operatorname{sing}(S_{\alpha_{i}})\cap\mathfrak{N}_{\alpha_{i}}=\emptyset,$

we see that $X\not\in\mathfrak{N}_{\alpha_{i}}$. But
$\mathfrak{N}_{\alpha_{i}}\supset\tilde{\mathfrak{N}}_{\alpha_{i}}$, so
$X\not\in\tilde{\mathfrak{N}}_{\alpha_{i}}$, so in particular $X\in
U_{\alpha_{i}}(\operatorname{sing_{\textnormal{gen}}}(S))$. Since $i$ was
arbitrary,

$X\in\cap_{i}U_{\alpha_{i}}(\operatorname{sing_{\textnormal{gen}}}(S))=\overline{\operatorname{sing_{\textnormal{gen}}}(S)}=\overline{\cup_{k=0}^{n-1}\mathfrak{G}^{k}}.$

As a result, there exists some $k=0,1,\ldots,n-1$ so that

(6.3)
$X\in\overline{\mathfrak{G}^{k}}\setminus\cup_{j=k+1}^{n-1}\overline{\mathfrak{G}^{j}}.$

Fix $i$ again. Using (6.2) again, now together with
$\mathfrak{N}_{\alpha_{i}}\supset\mathfrak{N}^{k}_{\alpha_{i}}$, it follows
that $X\not\in\mathfrak{N}^{k}_{\alpha_{i}}$. Combined with
$X\in\operatorname{sing_{\textnormal{non-gen}}}(S^{\prime})$ we see that

$X\notin\overline{\mathfrak{G}^{k}}\setminus\cup_{j=k+1}^{n-1}U_{\alpha_{i}}(\mathfrak{G}^{j}).$

Since $X\in\overline{\mathfrak{G}^{k}}$ by (6.3), it must hold that

$X\in\cup_{j=k+1}^{n-1}U_{\alpha_{i}}(\mathfrak{G}^{j}).$

But since $i$ was arbitrary, this implies that

$X\in\cap_{i}\cup_{j=k+1}^{n-1}U_{\alpha_{i}}(\mathfrak{G}^{j})=\cup_{j=k+1}^{n-1}\overline{\mathfrak{G}^{j}},$

contradicting (6.3). This completes the proof. ∎

##  Appendix A The generic strong multiplicity-one property

In this appendix we consider the following condition:

######  Definition A.1.

We say that a closed embedded hypersurface $M^{n}\subset\mathbb{R}^{n+1}$
satisfies the _strong multiplicity-one property_ if the following holds.
Consider:

  * •

hypersurfaces $M_{j}\subset\mathbb{R}^{n+1}$ converging smoothly to $M$,

  * •

flows $\mathcal{M}_{j}\in\mathfrak{F}(M_{j})$,

  * •

space-time points $X_{j}\to X\in\mathbb{R}^{n+1}\times(0,\infty)$, and

  * •

scales $\lambda_{j}\to\infty$,

so that
$\operatorname{ParDil}_{\lambda_{j}}(\mathcal{M}_{j}-X_{j})\rightharpoonup\tilde{\mathcal{M}}$.
Then $\operatorname{reg}\tilde{\mathcal{M}}$ has multiplicity one and the
parabolic dimension of the singular set satisfies
$\dim_{H}\operatorname{sing}\tilde{\mathcal{M}}\leq n$.

In particular, polyhedral cones other than the flat multiplicity one
hyperplane cannot arise as limit flows of perturbations of $M$.

######  Remark A.2.

Bamler has recently proven that a condition of this type does indeed hold in
the Ricci flow setting [Bam23, Bam20].

It’s important to observe that by dimension reduction [Whi97] (and the cyclic
property of $M$ [Whi09]), if $M$ fails to have the strong multiplicity-one
property then we can adjust $X_{j},\lambda_{j}$ so as to ensure that either:

  1. (1)

$\tilde{\mathcal{M}}$ is a static/quasi-static multiplicity $\geq 2$
hyperplane, or

  2. (2)

a union of $\geq 4$ multiplicity-one half $n$-planes meeting along an
$(n-1)$-plane.

(Thus, e.g., all $M$ with $\lambda(M)<2$ automatically have the strong
multiplicity-one property; cf. Lemma 2.8.)

######  Lemma A.3 (Openness of the strong multiplicity-one property).

Suppose that $M_{k}$ are smooth closed embedded hypersurfaces in
$\mathbb{R}^{n+1}$ smoothly converging to $M$. If $M$ satisfies the strong
multiplicty-one property then so does $M_{k}$ for $k$ sufficiently large.

###### Proof.

If not, there’s $M_{k,j}$ smoothly converging to $M_{k}$, flows
$\mathcal{M}_{k,j}\in\mathfrak{F}(M_{k,j})$, space-time points $X_{k,j}\to
X_{k}\in\mathbb{R}^{n+1}\times(0,\infty)$, and scales $\lambda_{k,j}\to\infty$
so that

$\operatorname{ParDil}_{\lambda_{k,j}}(\mathcal{M}_{k,j}-X_{k,j})\rightharpoonup\tilde{\mathcal{M}}_{k}$

as $j\to\infty$, where either $\tilde{\mathcal{M}}_{k}$ satisfies (1) or (2)
above. If (1) is satisfied for infinitely many $k$ we can pass to a
subsequence so that
$\tilde{\mathcal{M}}_{k}\rightharpoonup\tilde{\mathcal{M}}$ still with
$\tilde{\mathcal{M}}$ satisfying (1). If (2) is satisfied for infinitely many
$k$, then either $\tilde{\mathcal{M}}_{k}\rightharpoonup\tilde{\mathcal{M}}$
still with $\tilde{\mathcal{M}}$ satisfying (2) as is, or (2) where or two (or
more) of the half-planes overlap.

In any case, we can (possibly after adjusting $X_{k,j},\lambda_{k,j}$ and
passing to an un-labeled subsequence) find a sequence $j_{0}(k)\to\infty$ so
that if $j(k)\geq j_{0}(k)$ then

$\operatorname{ParDil}_{\lambda_{k,j(k)}}(\mathcal{M}_{k,j(k)}-X_{k,j(k)})\rightharpoonup\tilde{\mathcal{M}}$

where $\tilde{\mathcal{M}}$ satisfies either (1) or (2). Indeed, the only case
that’s not obvious is the final possibility in which $\tilde{\mathcal{M}}_{k}$
is a union of half-planes with two or more converging. In this case we just
have to re-center the points and scales in the converging half-planes so as to
arrange that we’re in case (1).

Taking $j(k)\geq j_{0}(k)$ sufficiently large, we can ensure that $M_{k,j(k)}$
converges smoothly to $M$. In particular, the flows $\mathcal{M}_{k,j(k)}$ are
smooth for a definite interval of time, and thus the points $X_{k,j(k)}$ are
bounded away from time $0$. The data $M_{k,j(k)}$, $\mathcal{M}_{k,j(k)}$,
$X_{k,j(k)}$, $\lambda_{j(k)}$ contradict the assumed strong multiplicity-one
property of $M$. This completes the proof. ∎

We will work with initial data that can be approximated by hypersurfaces with
the strong multiplicity-one property:

######  Definition A.4.

We say that a closed embedded hypersurface $M^{n}\subset\mathbb{R}^{n+1}$
satisfies the _generic strong multiplicity one-property_ if there are $M_{k}$
smoothly converging to $M$ with $M_{k}$ having the strongly multiplicity-one
property.

The techniques in this paper used to prove Corollaries 1.18 and 1.19 can be
used to prove the following result, which drops all entropy bounds and
replaces them with the generic strong multiplicity-one assumption. (A stronger
partial result holds for $n=2$ by [CCMS20, CCS23]; see [CCS23, Theorem 9.2]
for a precise statement.)

######  Theorem A.5.

If $M^{n}\subset\mathbb{R}^{n+1}$, $n\in\\{2,3,4\\}$ has the generic strong
multiplicity one property then there exist arbitrarily small $C^{\infty}$
graphs $M^{\prime}$ over $M$ so that $\operatorname{sing_{\textnormal{non-
gen}}}\mathcal{M}^{\prime}=\emptyset$ for all
$\mathcal{M}^{\prime}\in\mathfrak{F}(M^{\prime})$.

Note that a natural generalization of Ilmanen’s multiplicity-one conjecture
would be for all _all_ initial data themselves to have the strong
multiplicity-one property directly; a weaker one is for all initial data to
satisfy the _generic_ strong multiplicity-one property.

Let us explain the necessary modifications to prove Theorem A.5. First, by
assumption we can perturb $M$ (not relabeled) so that $M$ satisfies the strong
multiplicity-one property. Now, following the setup in Section 6, we embed $M$
in a local unit speed foliation $\\{M_{s}\\}_{s\in S}$. By Lemma A.3 we can
shrink $S$ (still with $0\in S$) so that all $M_{s}$, $s\in S$, have the
strong multiplicity-one property.

We make the following modifications to the argument used in the rest of the
paper:

  1. (1)

Any $\mathcal{M}\in\mathfrak{F}(M_{s})$, $s\in S$ has
$\operatorname{reg}\mathcal{M}$ connected and multiplicity one by [CCMS20,
Corollary F.4] and the strong-multiplicity one property (here we just have to
consider tangent flows to $\mathcal{M}$ without varying the parameter $s$). In
particular, the regularity scale continues to be continuous (cf. Remark 2.5).

  2. (2)

Instead of considering arbitrary $F$-stationary varifolds we always consider
$F$-stationary varifolds that correspond to self-similar flows arising as
limits of $\mathcal{M}_{s_{j}}\in\mathfrak{F}(M_{s_{j}})$, $s_{j}\in S$. The
conclusion of Lemma 2.8 continues to hold in this setting. The $\lambda(V)<2$
assumption in subsequent results (e.g. Lemma 2.12, Corollary 2.13, Proposition
3.1, etc.) is simply used to refer to a regularity result along the lines of
Lemma 2.8, and thus these results continue to hold in this more general
setting.

  3. (3)

The class $\mathscr{P}(\eta)$ as in Definition 4.2 should be replaced with the
smaller set of such pairs that arise as rescalings of flows
$\mathcal{M}\in\mathfrak{F}(M_{s})$, $s\in S$ (and condition (3) should be
removed). A similar modification (only considering flows that arise as
rescalings of flows starting at the $M_{s}$, $s\in S$) should be made in
subsequent arguments (e.g. Lemmas 4.6 and 4.9, as well as Proposition 4.12).

The remainder of the argument is unchanged.

######  Remark A.6.

It would suffice to assume that $M$ satisfies a different form of the generic
strong multiplicity-one property in which we only considers initial data that
are leaves of the local foliation $\\{M_{s}\\}$.

######  Remark A.7.

Note that we do not know if the second main hypothesis
($\heartsuit_{n,\Lambda}$) holds when $n=5$ and $\Lambda$ is large (in other
words, we do not know if there’s a non-generic shrinker
$\Sigma^{2}\subset\mathbb{R}^{3}$ with $\mu(\Sigma)\geq-\frac{3}{2}$). Because
of this, it’s unclear if Theorem A.5 holds in $\mathbb{R}^{6}$ (even assuming
the generic strong multiplicity one property for $M$).

##  Appendix B Stable minimal cones as shrinkers

In this appendix, we prove an estimate for the first eigenvalue of the
$L$-operator on a stable minimal cone. This is not used elsewhere in the
paper. This should be related to the question of whether or not the Simons
cone can arise generically as a (static) tangent flow (cf. [Vel94, GS18,
Sto23]).

Lemma 3.1 has the following consequence.

######  Corollary B.1.

Let $V$ be a conical stationary cyclic integral $n$-varifold in
$\mathbb{R}^{n+1}$ with $\lambda(V)<2$. If $\operatorname{reg}V$ is unstable
with respect to the Euclidean area functional, then $\mu(V)=-\infty$.

If $\operatorname{reg}V$ is stable, the conclusion of Corollary B.1 need not
hold. For example it is easy to check that
$\mu(\mathbb{R}^{n})=-\tfrac{1}{2}$. We now show that non-flat stable cones
satisfy an improved inequality.

######  Proposition B.2.

Take $n\geq 2$ and $V$ any a conical stationary cyclic integral $n$-varifold
in $\mathbb{R}^{n+1}$ with444Actually, the result here would hold for a
stationary minimal cone satisfying Wickramasekera’s $\alpha$-structural
hypothesis [Wic14a] (it is easy to check that a stationary integral cyclic
varifold $V$ with $\lambda(V)<2$ satisfies the $\alpha$-structural
hypothesis). $\lambda(V)<2$. If $V$ is not a flat hyperplane, then

$\mu(V)\leq-1-\delta_{n},$

for some $\delta_{n}>0$ depending only on $n$.

###### Proof.

By [Sim68, Lemma 6.1.1] and dimension reduction we can assume that $n\geq 7$
(otherwise Corollary B.1 gives $\mu(V)=-\infty$).

Let $\Sigma\subset\SS^{n}$ denote the regular part of the link of $V$. By
[Zhu18] (cf. [Sim68]) we can find $\Sigma^{\prime}\Subset\Sigma$ so that
$\Sigma^{\prime}$ has smooth boundary and so that there is $u\in
C^{\infty}(\Sigma^{\prime})$ with $u=0$ on $\partial\Sigma^{\prime}$ (if
nonempty) and

(B.1) $\Delta_{\Sigma}u+(|A_{\Sigma}|^{2}-(n-1))u\geq 0.$

Indeed, by [Zhu18, Theorem 0.1] (since $\Sigma$ is not a totally geodesic
$\SS^{n}$ by assumption) it holds that

$\inf\left\\{\int_{\Sigma}|\nabla u|^{2}+(|A_{\Sigma}|^{2}-(n-1))u^{2}:u\in
C^{\infty}_{c}(\Sigma),\int_{\Sigma}u^{2}=1\right\\}\leq 0$

with equality if and only if $V$ is the cone over a Clifford hypersurface (and
in particular $\operatorname{sing}V=\\{\mathbf{0}\\}$). As such, if
$\operatorname{sing}V\neq\\{\mathbf{0}\\}$ this equality is strict, so we can
exhaust $\Sigma$ by smooth regions; the first Dirichlet eigenfunction on these
regions will eventually satisfy (B.1). Conversely, if
$\operatorname{sing}V=\\{\mathbf{0}\\}$ we can find $u$ satisfying (B.1) with
$\Sigma^{\prime}=\Sigma$ (in this case, this is a consequence of [Sim68, Lemma
6.1.7]).

We now consider $f(r,\omega)=w(r)u(\omega)$ for $u$ as in (B.1) and $w\in
C^{\infty}_{c}((0,\infty))$ in (1.2). We find

$\displaystyle\mu(V)\int_{0}^{\infty}w(r)^{2}e^{-\tfrac{1}{4}r^{2}}r^{n-1}dr$
$\displaystyle\leq\int_{0}^{\infty}\left(w^{\prime}(r)^{2}-((n-1)r^{-2}+\tfrac{1}{2})w(r)^{2}\right)e^{-\tfrac{1}{4}r^{2}}r^{n-1}dr.$

A standard argument shows that for any $\alpha>\frac{2-n}{2}$ we can take

$w(r)=\begin{cases}0&r\in(0,\varepsilon^{2}]\\\
\varepsilon^{\alpha}(2-\tfrac{\log
r}{\log\varepsilon})&r\in(\varepsilon^{2},\varepsilon]\\\
r^{\alpha}&r\in(\varepsilon,\varepsilon^{-1}]\\\
\varepsilon^{-\alpha}(2-\varepsilon
r)&r\in(\varepsilon^{-1},2\varepsilon^{-1}]\\\
0&r\in(\varepsilon^{-2},\infty).\end{cases}$

This yields

$\displaystyle\mu(V)\int_{0}^{\infty}e^{-\tfrac{1}{4}r^{2}}r^{n+2\alpha-1}dr$
$\displaystyle\leq\int_{0}^{\infty}\left((\alpha^{2}-(n-1))r^{-2}-\tfrac{1}{2}\right)e^{-\tfrac{1}{4}r^{2}}r^{n+2\alpha-1}dr+o(\varepsilon^{n+2\alpha-2}).$

We perform the change of variables $r=2\sqrt{s}$ in both integrals:

$\displaystyle\mu(V)\int_{0}^{\infty}e^{-s}s^{\frac{n+2\alpha}{2}-1}ds$
$\displaystyle\leq\frac{1}{2}\int_{0}^{\infty}\left(\tfrac{\alpha^{2}-(n-1)}{2}s^{-1}-1)\right)e^{-s}s^{\frac{n+2\alpha}{2}-1}ds+o(\varepsilon^{n+2\alpha-2}).$

Recalling that for $\textrm{Re}(z)>0$ the Gamma function
$\Gamma(z):=\int_{0}^{\infty}s^{z-1}e^{-s}ds$ satisfies the recurrence
relation $\Gamma(z+1)=z\Gamma(z)$, we find

$\displaystyle\mu(V)$
$\displaystyle\leq\frac{1}{2}\left(\frac{\alpha^{2}-(n-1)}{2}\frac{\Gamma(\frac{n+2\alpha}{2}-1)}{\Gamma(\frac{n+2\alpha}{2})}-1\right)+o(\varepsilon^{n+2\alpha-2})$
$\displaystyle=\frac{1}{2}\left(\frac{\alpha^{2}-(n-1)}{n+2\alpha-2}-1\right)+o(\varepsilon^{n+2\alpha-2})$
$\displaystyle=\frac{1}{2}\frac{\alpha^{2}-2\alpha-2n+3}{n+2\alpha-2}+o(\varepsilon^{n+2\alpha-2})$

Thus, we find that

$\mu(V)\leq\inf_{\alpha>\frac{2-n}{2}}\frac{1}{2}\frac{\alpha^{2}-2\alpha-2n+3}{n+2\alpha-2}.$

(Note that for $n\leq 6$ this infimum is $-\infty$; compare this with
Corollary B.1.) Furthermore,

$\displaystyle\inf_{\alpha>\frac{2-n}{2}}\frac{1}{2}\frac{\alpha^{2}-2\alpha-2n+3}{n+2\alpha-2}$
$\displaystyle=\frac{1}{4}\left(-n+\sqrt{8-8n+n^{2}}\right)$
$\displaystyle=-1-\underbrace{\frac{1}{4}\left(\sqrt{16-8n+n^{2}}-\sqrt{8-8n+n^{2}}\right)}_{:=\delta_{n}}$

(since $n\geq 7$). This completes the proof. ∎

Note that $\delta_{7}=\tfrac{1}{2}$, $\delta_{8}\approx 0.29$, and
$\delta_{n}\to 0$ as $n\to\infty$.

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