# Generic mean curvature flows with cylindrical singularities

Ao Sun, Jinxin Xue  University of Chicago, Department of Mathematics, 5734 S
University Ave, Chicago IL, 60637<EMAIL_ADDRESS>Department of
Mathematics, Jingzhai 310, Tsinghua University, Haidian District, Beijing,
100084<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the dynamics of the mean curvature flow approaching a
cylindrical singularity and show that generically such a singularity is
nondegenerate, robust, and isolated. This gives also consequences such as
generic low regularity of level set flows, generic rotational MCFs being
type-I, etc. Another discovery of the paper is the formulation of the
“firecracker” type singularity and its role playing in the study of generic
isolatedness of cylindrical singularities. The work appears as the confluence
of methods and ideas from geometric flows, dynamical systems, and semilinear
heat equations.

##  1\. Introduction

In this paper, we study generic mean curvature flows with cylindrical
singularities. Mean curvature flow (MCF) is a natural extrinsic geometric flow
with extensive studies in geometry, partial differential equations, applied
math, etc.

Let $\\{\mathbf{M}_{\tau}\\}_{\tau\in[-1,0)}$ be a MCF that is a family of
hypersurfaces in $\mathbb{R}^{n+1}$ evolving according to the equation
$\operatorname{\partial}_{\tau}x=-H(x)\mathbf{n}(x)$ where $H$ is the mean
curvature and $\mathbf{n}$ is the outer unit normal. It is known that the MCF
starting from a smooth embedded closed hypersurface always develops
singularities, thus it is central to understand singularities in order to
understand MCFs. Singularity analysis in general involves a procedure of
blowing up. Huisken [Hu1] introduced the rescaled mean curvature flow (RMCF)
to study a singularity. The RMCF is a family of hypersurfaces $M_{t}$
satisfying the equation $\operatorname{\partial}_{t}x=-\left(H-\frac{\langle
x,\mathbf{n}\rangle}{2}\right)\mathbf{n}.$ Suppose the first singularity of
the MCF $\mathbf{M}_{\tau}$ occurs at the spacetime point
$(0,0)\in\mathbb{R}^{n+1}\times\mathbb{R}$, then the MCF $\mathbf{M}_{\tau},\
\tau\in[-1,0)$ is related to the RMCF $M_{t},\ t\in[0,\infty)$ through the
relation

(1.1) $M_{t}=e^{t/2}\mathbf{M}_{-e^{-t}},$

and a (subsequential) limit of the RMCF gives the model of the singularity
called self-shrinker satisfying the equation $H-\frac{\langle
x,\mathbf{n}\rangle}{2}=0$.

The RMCF can be considered as a dynamical system, which is the viewpoint
initiated by Colding-Minicozzi in a series of papers [CM1, CM2, CM6, CM7].
Given a hypersurface $\Sigma\subset\mathbb{R}^{n+1}$, we define its
$F$-functional

$F(\Sigma):=(4\pi)^{-n/2}\int_{\Sigma}e^{-\frac{|x|^{2}}{4}}d\mathcal{H}^{n}(x),$

where $\mathcal{H}^{n}$ is the $n$-dimensional Hausdorff measure, then $M_{t}$
is the negative gradient flow of $F$ and a shrinker is a fixed point of the
flow.

The study of generic MCFs was motivated naturally by the complexity of the
dynamics of MCFs. Indeed, it is known that

  1. (1)

There are so many shrinkers that it seems not possible to give a
classification of them.

  2. (2)

When restricting our attention to generic shrinkers, classified by Colding-
Minicozzi [CM1], the singular set can be very complicated [CM6]. In
particular, there exist a MCF with a continuous family of singularities (see
the marriage ring example below).

Thus, it is natural to ask whether the shrinkers and singular sets are easier
to understand if we consider generic MCFs. The following two conjectures can
be found in the problem list of Ilmanen [I1]. Conjecture 1.2 was also
mentioned by Colding-Minicozzi-Pedersen in [CMP, Conjecture 7.1].

######  Conjecture 1.1 (Huisken conjecture).

Generic MCF in $\mathbb{R}^{3}$ has only spherical and cylindrical
singularities.

######  Conjecture 1.2 (Generic isolatedness conjecture).

Generic MCF has only isolated spacetime singularities.

Colding and Minicozzi [CM1, CM6, CM7] initiated the program of generic
dynamics of MCFs in a series of papers. In [CM1], Colding-Minicozzi studied
the variational stability of shrinkers. They introduced a quantity called
entropy, and showed that only spheres and cylinders are entropy stable
shrinkers, and they provided a scheme to avoid unstable singularities by a
perturbation close to the singular time in [CM6, CM7]. In [CCMS, SX1, SX2],
for unstable singularities modeled on compact shrinkers and conical ones, it
is known how to avoid them by a generic perturbation of initial conditions.
Note that in [Wa], Lu Wang proved that compact and asymptotically conical
shrinkers and asymptotically cylindrical shrinkers are all possible shrinkers
in $\mathbb{R}^{3}$, and Ilmanen conjectured that the asymptotically
cylindrical shrinkers must be the standard cylinder.

To approach Conjecture 1.2, if Huisken conjecture is assumed, then the
remaining thing is to prove the generic isolatedness of spherical and
cylindrical singularities. Here we say a singularity is spherical or
cylindrical if it is modeled respectively by a sphere
$\mathbb{S}^{n}(\sqrt{2n})$ or a generalized cylinder
$\Sigma^{k}:=\mathbb{S}^{n-k}(\varrho)\times\mathbb{R}^{k},\ k=1,\ldots,n$,
where $\varrho=\sqrt{2(n-k)}$ is the radius of the sphere. It is known that
spherical singularities are isolated, thus in this paper, we study generic MCF
with cylindrical singularities and make progress on Conjecture 1.2. We prove
Conjecture 1.2 in the following two settings:

  * •

Locally, each cylindrical singularity can be made isolated in a parabolic ball
by a generic perturbation of initial condition (Theorem 1.5, 1.8 and 1.9).

  * •

For MCF in $\mathbb{R}^{3}$ with first time singular set a smooth curve, we
can perturb the MCF very close to the singular time, such that the perturbed
MCF has a single first-time singularity. (Theorem 1.17).

Moreover, we also obtain the following results that may not relate to
Conjecture 1.2 and are of independent interest:

  * •

A new proof of $C^{2}$ normal form expansion of rescaled mean curvature flow
(RMCF) near a generalized cylinder (Theorem 1.3).

  * •

For rotational graphs, the MCF generically has first-time singularities being
type I (Theorem 1.15).

  * •

Globally, the first-time singular set of a generic mean convex MCF either has
finitely many isolated singularities, or contains a conjecturally existing
singularity called _firecracker_ (Definition 1.11, Theorem 1.12).

  * •

The level set flow starting from a generic mean convex hypersurface has low
regularity (Theorem 1.14).

  * •

The first-time singularity of MCF starting from a generic surface in
$\mathbb{R}^{3}$ either has finitely many isolated singularities, or some
pathological behaviour shows up (Theorem 1.18).

Throughout the paper, we only consider the MCF of closed embedded
hypersurfaces, although many of our results definitely could be applied to
other situations, e.g. noncompact MCF or immersed MCF. We shall assume

$(\star)$ $\\{\mathbf{M}_{\tau}\\}_{\tau\in[-1,0)}$ is a MCF with a
cylindrical singularity modeled on $\Sigma^{k}$ at the spacetime point $(0,0)$
and $M_{t},\ t\in[0,\infty)$ be its associated RMCF converging to $\Sigma^{k}$
in the $C^{\infty}_{loc}$-sense.

###  1.1. Generic isolatedness for RMCFs

The starting point of the work is the following result giving a normal form
for the RMCF asymptotic to the cylinder. We shall use coordinates
$z=(\theta,y)\in\mathbb{S}^{n-k}(\varrho)\times\mathbb{R}^{k}\subset\mathbb{R}^{n-k+1}\times\mathbb{R}^{k}$.

######  Theorem 1.3 ($C^{1}$ Normal form theorem).

Assume $(\star)$. Then there exists $K>0$ and $T>0$ with the following
significance: for $t>T$, the RMCF is a graph over $\Sigma^{k}\cap
B_{K\sqrt{t}}$, and the graphical function $u:\ \Sigma^{k}\cap
B_{K\sqrt{t}}\to\mathbb{R}$ of $M_{t}\cap B_{K\sqrt{t}}$ satisfies the
following in the $C^{1}$ normal form up to a rotation in $\mathbb{R}^{k}$

(1.2) $u(t,z)=\sum_{i\in\mathcal{I}}\frac{\varrho}{4t}(y_{i}^{2}-2)+o(1/t),$

as $t\to\infty$, where $\mathcal{I}\subset\\{1,2,\ldots,k\\}$.

The normal form is not a new theorem. A similar normal form was obtained in a
series of works [Zh1, Zh2] of Gang Zhou for $n=4,k=3$ and for a larger
graphical radius. Our proof uses results from the study of semi-linear heat
equations, especially the work of Velázquez [V1, V2], combined with ideas from
dynamical systems, and is much simpler.

We also mention that in the recent work on the classification of ancient
solutions of MCF [BC1, BC2, CHH, CHHW], the asymptotic expansion of the RMCF
over the cylinders at $-\infty$ time is very crucial, but the backward
expansion is very different from the forward expansion here and yields very
different geometric information.

From the normal form, the following definition follows naturally.

######  Definition 1.4.

We say a cylindrical singularity as in Theorem 3.3 _nondegenerate_ if
$\mathcal{I}=\\{1,2,\ldots,k\\}$, and _degenerate_ otherwise.

We prove several geometric and dynamic properties of the nondegenerate
singularities. The first result that we read from the normal form is the
following theorem on the isolatedness of nondegenerate cylindrical
singularities.

######  Theorem 1.5 (Isolatedness theorem).

Assume $(\star)$ and that the singularity is nondegenerate, then the
singularity is isolated. In other words, there exists a $\delta_{1}>0$ such
that $(0,0)$ is the only singularity of $\mathbf{M}_{\tau}$ in the spacetime
neighborhood $B_{\delta_{1}}\times(-\delta_{1}^{1/2},0]$.

In relation to the theorem, we mention the mean convex neighborhood
conjecture, which was proved by Choi-Haslhofer-Heshkovitz in [CHH] in
dimension $n=2$, Choi-Haslhofer-Heshkovitz-White in [CHHW] in dimension $\geq
3$ with $2$-convex assumption, and by Gang Zhou [Zh1] for singularity model
$\mathbb{S}^{1}\times\mathbb{R}^{3}$ with nondegeneracy assumption. Note that
here we also need the nondegeneracy assumption.

######  Theorem 1.6 (Mean convex neighbourhood of a nondegenerate
singularity).

Assume $(\star)$ and that the singularity is nondegenerate. Then there exist
constants $\delta_{2}>0$ and $\tau_{0}>0$, such that for any
$\tau\in(-\tau_{0},0)$, $\mathbf{M}_{\tau}\cap B_{\delta_{2}}(0)$ has positive
mean curvature and is diffeomorphic to $\Sigma^{k}$.

Another important property of nondegenerate singularity is the type-I
curvature condition. A singularity $(y,T)$ of an MCF is called _type-I_ if the
curvature blows up with speed at most $O(\frac{1}{\sqrt{T-t}})$, i.e. there
exists $r>0$ and $C>0$ such that for any $t<T$, in $M_{t}\cap B_{r}(y)$ we
have $|A|(x,t)\leq\frac{C}{\sqrt{T-t}}$, where $A$ is the second fundamental
form. Otherwise, the singularity is called _type-II_. There are examples of
type-II singularities. For instance, in [AAG], the authors constructed a
surface called “peanut”, and the MCF starting from such a peanut develops
type-II singularities that are called “degenerate neckpinching”. In [AV],
Angenent-Velázquez constructed a large family of type-II singularities.

######  Theorem 1.7 (Type-I curvature condition of a nondegenerate
singularity).

Assume $(\star)$ and that the singularity is nondegenerate. Then there exist
constants $\delta_{2}>0$ and $\tau_{0}>0$, such that for any
$\tau\in(-\tau_{0},0)$, $\mathbf{M}_{\tau}\cap B_{\delta_{2}}(0)$ has
curvature bound $|A|\leq C(-\tau)^{-1/2}$ for some constant $C$.

We next show that nondegenerate singularities are robust (Theorem 1.9) and for
degenerate ones, we can always find an arbitrarily small perturbation to make
it nondegenerate (Theorem 1.8).

######  Theorem 1.8 (Denseness of nondegenerate singularities).

Assume $(\star)$ and that the singularity is degenerate. Then for any
$\varepsilon>0$ and $\delta>0$, there exists $f:\mathbf{M}_{-1}\to\mathbb{R}$
with $\|f\|_{C^{2}}\leq\varepsilon$, the perturbed MCF
$\\{\widetilde{\mathbf{M}}_{\tau}\\}$ starting from
$\widetilde{\mathbf{M}}_{-1}:=\\{x+f(x)\mathbf{n}:x\in\mathbf{M}_{-1}\\}$ has
a singularity in a $\delta$-neighbourhood of $(0,0)$, that is modeled by
$\Sigma^{k}$, and it is nondegenerate.

The stability of nondegenerate cylindrical singularities is very subtle. In
[CM1], Colding-Minicozzi introduced a quantity called _entropy_ defined as

$\lambda(\Sigma)=\sup_{x_{0}\in\mathbb{R}^{n+1},t_{0}\in(0,\infty)}\frac{1}{(4\pi
t_{0})^{n/2}}\int_{\Sigma}e^{-\frac{|x-x_{0}|^{2}}{4t_{0}}}d\mathcal{H}^{n}(x).$

This quantity is invariant under rigid transformations. In particular,
Colding-Minicozzi proved that for a self-shrinker $\Sigma$,
$\lambda(\Sigma)=F(\Sigma)$. Moreover, Colding-Minicozzi [CM1, Theorem 0.12]
showed that generalized cylinders are all entropy stable. The entropy of
spheres and cylinders were calculated by Stone, and are ordered as follows:

$\lambda(\Sigma^{n-1})>\lambda(\Sigma^{n-2})>\cdots>\lambda(\Sigma^{1})>\lambda(\mathbb{S}^{n}(\sqrt{2n}))>1=\lambda(\mathbb{R}^{n+1}).$

A degenerate cylindrical singularity is in general not stable. For example,
the peanut MCF constructed in [AAG] has a degenerate cylindrical singularity,
and can be perturbed into a spherical or a cylindrical singularity depending
on the choice of perturbations. In general, a perturbation of an MCF with a
degenerate first time cylindrical singularity may have the first time
singularity with strictly less entropy value, no matter how small the
perturbation is. Nevertheless, our last theorem shows that perturbations can
always be constructed to make a degenerate cylindrical singularity
nondegenerate without lowering its entropy.

On the other hand, nondegenerate singularities enjoy the remarkable stability
property under perturbations.

######  Theorem 1.9 (Stability of nondegenerate singularities).

Assume $(\star)$ and the singularity is nondegenerate. Then there is an
$\varepsilon_{0}>0$ such that for all $u:\ \mathbf{M}_{-1}\to\mathbb{R}$ with
$\|u\|_{C^{2}}\leq\varepsilon_{0}$, the perturbed MCF with initial condition
$\widetilde{\mathbf{M}}_{-1}=\mathrm{Graph}\\{x+u(x)\mathbf{n}(x)\ |\
x\in\mathbf{M}_{-1}\\}$ admits a unique nondegenerate singularity in an
C$\varepsilon_{0}$-spacetime neighborhood of $(0,0)$ modeled by $\Sigma^{k}$.

Local stability of cylindrical singularities was studied by Schulze-Sesum in
[SS] for the neckpinch case, i.e. $k=1$ using geometric measure theory. Their
notion of nondegeneracy is defined using geometric measure theory, so is
different from ours, but there are still some relations. Our Theorem 1.9,
similar to Theorem 1.3 of [SS], implies that a singularity that is
nondegenerate in our sense is also nondegenerate in the sense of [SS]. We
conjecture that the definition in [SS] is equivalent to ours.

###  1.2. Generic mean convex MCFs

A hypersurface in $\mathbb{R}^{n+1}$ is called mean convex (resp. strictly
mean convex) if the mean curvature $H\geq 0$ (resp. $H>0$) at each point. MCFs
starting from mean convex initial conditions are studied intensively in
literature, see [HS, Wh1, Wh3, SW, Wh5, HK] etc. It is known that the mean
convexity is preserved under the MCF and the only possible singularities are
either spherical or cylindrical (c.f. [HS] for first-time singularities and
[Wh1, Wh3, Wh5] for all later time singularities).

However, the singularity set for mean convex MCFs can still be highly
complicated. There is an example called “marriage ring” showing that the mean
convex MCF in $\mathbb{R}^{n+1}$ can have $(n-1)$-dimensional singular set:
consider a $\mathrm{SO}(n-1)$-invariant $\mathbb{S}^{n-1}\times\mathbb{S}^{1}$
in $\mathbb{R}^{n+1}$, where $\mathbb{S}^{1}$ factor has very small radius to
ensure the hypersurface being mean convex. Then by the symmetry, the MCF
starting from such hypersurface would collapse at a singular set
$\mathbb{S}^{n-1}\times\\{*\\}$.

In [CM3], Colding-Minicozzi studied the partial regularity of the singular set
of mean convex MCF. In this paper, we obtain further information of the
singular set for generic mean convex MCF. Let $\mathcal{G}$ be the space of
strictly mean convex hypersurfaces, on which we introduce the $C^{r}$-topology
$r>2$.

######  Theorem 1.10.

There is an open dense subset of $\mathcal{G}$ such that for any MCF starting
from a hypersurface in this set, the singular set consists of at least one
isolated singularity that is either spherical or non-degenerate cylindrical.

One may wonder what is the main difficulty that prevents us to show that all
(not only one) first time singularity is isolated. Indeed, the size
$\delta_{1}$ of the isolating neighborhood is not uniform for all
singularities. In the marriage ring example, we can introduce perturbations
that preserve a discrete subgroup of $\mathbb{S}^{1}$ of arbitrarily large
cardinality $N$ such that the resulting MCF has exactly $N$ isolated
singularities, but the isolating neighborhood is less than $1/N$.

As a result, we have the following property (_isolated-or-firecracker_):

######  Definition 1.11.

A mean convex hypersurface $\mathbf{M}$ is said to have _property $($isolated-
or-firecracker$)$_, if the first-time singular set of MCF starting from
$\mathbf{M}$

  1. (1)

either consists of finitely many nondegenerate singularities,

  2. (2)

or contains a degenerate singularity $($we call it a “firecracker”$)$ which is
the limit of a sequence of nondegenerate spacetime singularities.

######  Theorem 1.12.

Generic initial data of closed mean convex hypersurfaces has property
$($isolated-or-firecracker$)$.

Here the generic initial data means a residual set of the space of strictly
mean convex hypersurfaces. The pathological firecracker singularity in the
property (isolated-or-firecracker) (2) is highly counterintuitive and we make
the following conjecture.

######  Conjecture 1.13.

The firecracker singularity exists in MCF, but does not exist generically.

Recently Simon [Si] constructed examples of minimal hypersurfaces with any
given prescribed compact singular set. The existence of a firecracker
singularity seems related to a parabolic version of Simon’s construction.

###  1.3. Generic level set flows

There is a weak formulation of MCF called level set flows (LSF). Level set
flow is the solution to a degenerate parabolic PDE

$\operatorname{\partial}_{t}f=|\nabla f|\text{div}\left(\frac{\nabla
f}{|\nabla f|}\right).$

It was first proposed by Osher-Sethian [OS] in numerical analysis. Later Chen-
Giga-Goto [CGG] and Evans-Spruck [ES] proved the existence of a viscosity
solution to level set flow PDE. Whenever $f(x,\tau)$ is a solution to the
level set flow PDE, then $\mathbf{M}_{\tau}:=\\{x:f(x,\tau)=0\\}$ is an MCF
whenever $\\{\mathbf{M}_{\tau}\\}$ is a family of codimension $1$
hypersurfaces. Therefore, one can define the weak solution to MCF as the level
sets of a level set flow function. When $\mathbf{M}_{0}:=\\{x:f(x,0)=0\\}$ is
mean convex, the MCF evolves monotonically, and one can define an _arrival
time_ function $g(x):=f(x,t)-t$, where $g$ satisfies the equation $-1=|\nabla
g|\text{div}\left(\frac{\nabla g}{|\nabla g|}\right).$

A natural question from PDE is the regularity of level set flow. Huisken [Hu2]
proved that level set flow starting from a convex hypersurface has at least
$C^{2}$-regularity, and Sesum [Se] constructed examples with at most $C^{3}$
regularity. In [SX3], we proved that starting from a generic convex
hypersurface, the level set flow does not have $C^{3}$ regularity.

When the initial hypersurface is only mean convex but not convex, the
regularity is more subtle because cylindrical singularities arise. Colding-
Minicozzi [CM4, CM5] studied how the structure of the singular set of MCF
influences the regularity of the arrival time. In particular, they showed that
if the singular time is not unique, the arrival time function is $C^{1,1}$ but
not $C^{2}$. In this paper, we have the following generic regularity result of
level set flow.

######  Theorem 1.14.

The space of strict mean convex closed hypersurfaces $($denoted by
$\mathcal{V})$ in $\mathbb{R}^{n+1}$ with $C^{r}$-topology $r\geq 2,$ has the
following structure: there is an open and dense subset
$\mathcal{U}=\mathcal{U}_{1}\cup\mathcal{U}_{2}$ of $\mathcal{V}$, such that

  * •

Any level set flow starting from a hypersurface in $\mathcal{U}_{1}$ has at
most $C^{2}$-regularity. Moreover, the level set flow has only an isolated
spherical singularity;

  * •

Any level set flow starting from a hypersurface in $\mathcal{U}_{2}$ has at
most $C^{1,1}$-regularity. Moreover, the level set flow has at least one
isolated cylindrical singularity.

The subspace $\mathcal{U}_{1}$ is studied in [SX3]. In this paper we only
focus on the space $\mathcal{U}_{2}$. $\mathcal{U}_{2}$ is defined explicitly
as follows:

$\mathcal{U}_{2}$ consists of mean convex closed hypersurfaces such that the
level set flows starting from these hypersurfaces have at least two distinct
singular times.

It is also interesting to know the relation between $\mathcal{U}_{1}$ and
$\mathcal{U}_{2}$. The peanut example suggested by Altschuler-Angenent-Giga
[AAG] (see also Angenent-Velázquez [AV]) shows that
$\overline{\mathcal{U}_{1}}\cap\overline{\mathcal{U}_{2}}\not=\emptyset$.

###  1.4. Generic MCF of rotational graphs and type-I condition

In this section, we show that if we consider only rotational graphs, we can
indeed rule out the firecracker singularities. The MCF of rotational graphs is
well-understood in [AAG], where the authors used a Liouville-Sturm theorem to
show that all the singularities of MCF of rotational graphs are isolated.

Let us use $x\in\mathbb{R},y=(y_{1},\dots,y_{n})\in\mathbb{R}^{n}$ to denote
the coordinates in $\mathbb{R}^{n+1}$,
$r=|y|=\sqrt{y_{1}^{2}+\dots+y_{n}^{2}}$. is the radius function in
$\mathbb{R}^{n}$. A hypersurface $\Gamma_{0}$ is called a _rotational graph_
if it is obtained by rotating the graph of a function $r=u(x)>0$ for $a<x<b$
around the $x$-axis. The MCF starting from such a rotational graph is denoted
by $\Gamma(t)$. Here the MCF is in the sense of level set flow.

It is a folklore conjecture that generic MCF has type-I singularities (see
[GIKW]). In [GIKW], the authors gave a numerical example to show that in the
noncompact case, the type-II blow up may not be avoidable at infinity.
However, for a closed MCF, our study of non-degenerate singularities implies
that type-I should be a generic behaviour for cylindrical singularity.

######  Theorem 1.15.

Starting from a “generic” rotational graphical closed hypersurface, the first-
time singularities of the MCF are either spherical, or nondegenerate
cylindrical. In both cases, the singularities are of type-I.

To make the genericity in Theorem 1.15 precise, we recall that the space of
rotational graphs is a Banach manifold, where the local atlas is given by the
$\mathrm{SO}(n)$-invariant $C^{r}$ functions, which form a closed linear
subspace of the Banach space $C^{r}$. Let us call the space of rotational
graphs $\mathscr{R}$. The genericity in Theorem 1.15 means an open and dense
subset in $\mathscr{R}$ for $r\geq 2$.

###  1.5. Pinch a thin ring

Recall that a marriage ring in $\mathbb{R}^{3}$ is a $S^{1}$-invariant
rotational symmetric $S^{1}\times S^{1}$ in $\mathbb{R}^{3}$, and under the
MCF it collapses to a singular set $S^{1}\times\\{*\\}$. Conjecturally, for
any closed $C^{1}$-curve $\gamma$, there exists an MCF
$\\{\mathbf{M}_{\tau}\\}_{\tau\in[0,\tau_{0})}$ such that $\mathbf{M}_{0}$ is
the boundary of a tubular neighbourhood of $\gamma$, and $M_{\tau}$ collapse
to a singular set $\gamma$. The authors heard the following conjecture from
Colding-Minicozzi.

######  Conjecture 1.16 (Colding-Minicozzi).

Any $C^{1}$-curve $\gamma\subset\mathbb{R}^{3}$ can be the singular set of a
(mean convex) MCF.

We remark that the $C^{1}$-regularity of the singular set is required by
[CM3]. We call such an MCF a _thin ring_. White [Wh2, Section 5] conjectured
that MCF in $\mathbb{R}^{3}$ can only have singular set consisting of isolated
singularities and curves.

We can show that near the singular time, we can perturb the surface to get an
isolated singularity. Intuitively, this means that we can pinch the thin ring.

######  Theorem 1.17.

Given an MCF $\\{\mathbf{M}_{\tau}\\}_{\tau\in[-1,0)}$ that is a thin ring.
Then for any $\varepsilon_{0}>0$, there exists $\tau_{0}\in[-1,0)$ such that
when $\tau>\tau_{0}$, there exists a function
$f:\mathbf{M}_{\tau}\to\mathbb{R}$ with $\|f\|_{C^{2}(\mathbf{M}_{\tau})}$,
such that the perturbed MCF starting from
$\widetilde{\mathbf{M}}_{\tau}:=\\{x+f(x)\mathbf{n}(x):x\in\mathbf{M}_{\tau}\\}$
has a single nondegenerate first-time singularity.

This theorem can be viewed as a local version of generic isolatedness
Conjecture 1.2 handling all the known examples of MCF in $\mathbb{R}^{3}$,
around the singular time.

###  1.6. Generic MCF

For generic MCFs in $\mathbb{R}^{3}$ without mean convexity assumption,
combined with [CCMS, SX1, SX2], we have the following singularity structure
theorem:

######  Theorem 1.18.

The first time singular set of an MCF starting from a generic set of surfaces
in $\mathbb{R}^{3}$ contains at least one of the following:

  1. (1)

A spherical singularity;

  2. (2)

Isolated cylindrical singularities;

  3. (3)

Firecrackers;

  4. (4)

Singularities modeled by shrinkers with cylindrical ends;

  5. (5)

Singularities modeled by higher multiplicity shrinkers.

The expectation is that only (1) and (2) types of singularities occur for
generic MCF, and then the Conjecture 1.2 holds at least for the first-time
singular set. Unfortunately, none of (3), (4), (5) can be ruled out by the
tools developed so far. The nonexistence of (4) and (5) are conjectured by
Ilmanen. It seems that (5) does not occur in generic MCF, at least in the
situation that the model shrinker is closed, see [Su]. We remark that when
$n\geq 3$, the MCF in $\mathbb{R}^{n+1}$ can have more complicated singularity
types.

###  1.7. Outline of the proof

The methodology is significantly harder than and different from [SX1, SX2,
SX3]. Let us use an example to illustrate the point. Let $M$ be an $m\times m$
matrix with distinct eigenvalues $\lambda_{1}>\lambda_{2}>\ldots>\lambda_{m}$,
the for generic vector $v$, we have $\lim\frac{M^{n}v}{\lambda_{1}^{n}}$
converges to $v_{1}$ that is the eigenvector for $\lambda_{1}$. However, to
get $v_{2}$, the eigenvector for $\lambda_{2}$, we have to consider
$\lim\frac{M^{n}v}{\lambda_{2}^{n}}$ where $v$ is generic in
$\mathbb{R}^{n}/(\mathbb{R}v_{1})$.

We consider the linearized operator
$L_{\Sigma}=\Delta-\frac{1}{2}x\cdot\nabla+(|A|^{2}+\frac{1}{2})$ restricted
to the shrinker. If the shrinker is nonspherical and noncylindrical, in [SX1,
SX2], we introduced an initial positive perturbation so that the difference
between the perturbed RMCF and the unperturbed one grows towards the direction
of the first eigenfunction, which is sufficient to decrease the entropy. This
is analogous to the procedure of finding $v_{1}$ in the last paragraph.

However, for spherical and cylindrical shrinkers, the first several
eigenfunctions correspond to rigid transformations, which does not give new
geometric evolution. New geometric information is hidden in some higher
Fourier modes, so the situation is similar to the case of finding $v_{2}$ in
the first paragraph, and we have to modulo the rigid transformations. In [SX3]
for spherical singularities, we decipher the information of regularity of the
LSF lying in the eigen direction of the first negative eigenvalue, and we
introduced a centering map to modulo rigid transformation, and make sure that
an initial perturbation is eventually growing towards the eigen direction of
the first negative eigenvalue.

Nevertheless, the situation for cylindrical singularities is much more
complicated. Lying in the kernel of $L_{\Sigma}$, there are functions
$h_{2}(y_{i})=y_{i}^{2}-2$, towards which we wish our initial perturbation to
lead, since $h_{2}$ is the essential Fourier mode determining the
isolatedness. However, there are eigenfunctions $y_{i}$ associated to positive
eigenvalue $\frac{1}{2}$ which in general cannot be eliminated by a rigid
transformation. This is the key point in the proof of $F$-instability of
cylinders in [CM1] (a perturbation of the cylinder can be constructed to lower
the value of the $F$-functional regardless rigid transformations). To address
this difficulty, we shall need an understanding of the geometric meaning of
the eigenfunctions $y_{i}$. Indeed, they correspond to translations along the
axis (i.e. $\mathbb{R}^{k}$) direction, and they can be eliminated by such a
translation if there is a nontrivial $h_{2}$-component in the normal form.
Thus, we get a situation of entanglement between eliminating $y_{i}$ and
revealing $h_{2}$. In the proof, we shall introduce a procedure of infinitely
many steps of rigid transformations dynamically along the RMCF to eliminate
all the exponentially growing modes step by step.

The second difficulty is a standing one appearing in all analysis of RMCF with
noncompact shrinkers, that is the manifold at each moment $t$ cannot be
written as a global graph over the shrinker $\Sigma^{k}$, thus a cutoff has to
be introduced. We have to get a good estimate of the solution of the
linearized equation along the RMCF in the $C^{2}$-norm, in particular close to
the boundary of the graphical scale where the cutoff is performed. Note that
neither Li-Yau nor the Feynman-Kac formula that we developed in [SX1, SX2]
apply here since the solution under study has no constant sign. To address
this difficulty, we invoke the Feynman-Kac type estimate due to Velázquez
[V1], which enables us to get good $C^{2}$ bound for the RMCF over the region
$\Sigma^{k}\cap B_{K\sqrt{t}}$ for some $K>0$ as $t\to\infty$. As we will see
in the proof that the $O(\sqrt{t})$ graphical radius is essential for yielding
geometric information for the MCF.

When the manifold is close to the limiting cylinder, we can write it within
the graphical radius as the graph of a function $u$ evolving according to the
equation $\partial_{t}u=L_{\Sigma}u+\mathcal{Q}(u)$, where $\mathcal{Q}$ is
the nonlinearity given explicitly in (A.2). The work is influenced by works on
singularities of semi-linear heat equations of the form $\partial_{t}u=\Delta
u+u^{p}$ defined on $\mathbb{R}^{k}$ by many authors such as [V1, V2, HV1, Z].
The blowup of the equation near a type-I singularity has a similar linear term
$L_{\mathbb{R}^{k}}u$. The generic isolatedness in the one dimensional case
$k=1$ was proved in [HV1], which relies crucially on the fact that the
singularity set is finite in this case. Similar results in higher dimensional
cases are missing. Instead, people describe the structure of the singularity
sets, among which we mention the results of Velázquez, Zaag, et al. The power
$p$ is important, since high power of $p$ makes the system supercritical so
that the singularity models are more complicated. In our case, the
nonlinearity $\mathcal{Q}$ is indeed more complicated, since it contains not
only high powers but also second order derivatives. Here comes the advantage
of MCF. We invoke pseudo locality property of MCF to control the nonlinearity.

The paper is organized as follows. In Section 2, we give a preliminary on the
eigenfunctions of the $L$-operator. Of particular importance for later proofs
is our interpretation of the eigenfunction $y_{i}$. In Section 3, we study
local dynamics of the RMCF near cylindrical shrinker and derive a
$H^{1}$-normal form up to scale $\sqrt{\log t}$. In Section 4, we prove the
$C^{1}$-normal form (Theorem 1.3) up to scale $O(\sqrt{t})$ and prove the
isolatedness of nondegenerate singularities (Theorem 1.5). In Section 5, we
prove the genericity and stability of nondegenerate singularities (Theorem 1.8
and 1.9). All other theorems mentioned above will be proved in Section 6 as an
application of Theorem 1.8 and 1.9. Finally, we have three appendices. In
Appendix A we derive the nonlinear term $\mathcal{Q}$ in the equation
$\partial_{t}u=Lu+\mathcal{Q}$ that is the RMCF equation for the graphical
function on $\Sigma^{k}$. In Appendix B, we present the important Feynman-Kac
estimate of Velázquez and Kammerer-Zaag adapted to our setting (Proof of
Proposition 4.8). In Appendix C, we study the centering map that is a properly
chosen rigid transformation used to modulo positive modes.

## Acknowledgment

We would like to thank Professor Minicozzi for the stimulating discussions. J.
X. is supported by grant NSFC (Significant project No.11790273) in China and
the Xiaomi endowed professorship of Tsinghua University.

##  2\. Preliminaries

In this section we summarize some previously known results on cylindrical
singularities. We will use $\\{y_{i}\\}_{i=1}^{k}$ to denote the coordinate on
$\mathbb{R}^{k}$, and $\\{\theta_{i}\\}_{i=1}^{n-k+1}$ to denote the
restriction of the coordinate functions of $\mathbb{R}^{n-k+1}$ to
$\mathbb{S}^{n-k}(\varrho)$. We will be working in the weighted Sobolev space.
Given two functions $f,g$ defined on $\Sigma$, we define an inner product
$\langle
f,g\rangle_{L^{2}}=\int_{\Sigma}f(x)g(x)e^{-\frac{|x|^{2}}{4}}d\mathcal{H}^{n}(x).$
We define the weighted $L^{2}$-norm by $\|f\|_{L^{2}}=\langle
f,f\rangle_{L^{2}}^{1/2},$ and the weighted $L^{2}(\Sigma)$ space consists of
function $f$ with $\|f\|_{L^{2}}<+\infty$. Similarly we can define weighted
higher Sobolev space $H^{k}$ with
$H^{k}(\Sigma):=\left\\{f:\sum_{i=0}^{k}\|\nabla^{i}f\|^{2}_{L^{2}}<+\infty\right\\}.$
Throughout of the paper, for simplicity of notations, we use $\|\cdot\|$ to
denote the $H^{1}$-norm if not otherwise mentioned.

Recall that the linearized operator on a self-shrinker is defined as
$L:=\Delta-\frac{1}{2}\langle x,\nabla\cdot\rangle+(|A|^{2}+1/2).$ In the
special case that the self-shrinker is $\Sigma^{k}$, we have

(2.1)
$L_{\Sigma}u=\Delta_{\mathbb{S}^{n-k}(\varrho)}u+L_{\mathbb{R}^{k}}u,\quad
L_{\mathbb{R}^{k}}u=\Delta_{\mathbb{R}^{k}}u-\sum_{i=1}^{k}\frac{1}{2}y_{i}\operatorname{\partial}_{y_{i}}u+u.$

Throughout this paper, the eigenvalue $\lambda$ of an operator $L$ is the
number such that there exists a function $f$ such that $Lf=\lambda f$. The
following fact was proved in [SWZ, Section 5.2]: suppose the eigenvalues of
$\Delta$ on $\mathbb{S}^{n-k}(\varrho)$ are given by
$\mu_{0}\geq\mu_{1}\geq\mu_{2}\geq\cdots$ with corresponding eigenfunctions
$\phi_{0},\phi_{1},\phi_{2},\cdots$, and suppose the eigenvalues of
$L_{\mathbb{R}^{k}}$ on $\mathbb{R}^{k}$ is given by
$\nu_{0}\geq\nu_{1}\geq\nu_{2}\geq\cdots$ with corresponding eigenfunctions
$\psi_{0},\psi_{1},\psi_{2},\cdots$, then the eigenvalues of $L_{\Sigma}$ are
given by $\\{\mu_{i}+\nu_{j}-1\\}_{i,j=0}^{\infty}$ with corresponding
eigenfunctions $\\{\phi_{i}\psi_{j}\\}_{i,j=0}^{\infty}$.

The spectrum of $\Delta_{\mathbb{S}^{n-k}(\varrho)}$ is
$0,-1/2,-\frac{n-k+1}{n-k},\cdots$, and the eigenfunctions are known to be the
restriction of homogeneous harmonic polynomials. The first several
eigenfunctions are given by $1$ for eigenvalue 0, and $\theta_{i}$, the
restriction of linear functions in $\mathbb{R}^{n-k+1}$ to
$\mathbb{S}^{n-k}(\varrho)$, for eigenvalue $-1/2$, and
$\theta_{i}^{2}-\theta^{2}_{j},\cdots$ for eigenvalue $-\frac{n-k+1}{n-k}$.
The spectrum of $L_{\mathbb{R}^{k}}$ on $\mathbb{R}^{k}$ is given by half
integers $1-\frac{m}{2},\ m=0,1,2,\ldots$, and the eigenfunctions for
eigenvalue $1-\frac{m}{2}$ are given by $h_{m_{1}}(y_{1})\cdots
h_{m_{k}}(y_{k})$ with $m_{1}+\ldots+m_{k}=m$, where $h_{i}$’s are Hermite
polynomials. In particular, we have $h_{0}(x)=1,h_{1}(x)=x$ and
$h_{2}(x)=x^{2}-2.$

Combining the spectra together, we obtain that the first three eigenvalues and
eigenfunctions of $L_{\Sigma^{k}}$:

eigenvalues of $L_{\Sigma^{k}}$ | corresponding eigenfunctions
---|---
$1$ | $1$
$1/2$ | $\theta_{i},y_{j},\ i=1,2,\ldots,n-k+1,\ j=1,2,\ldots,k$
$0$ | $\theta_{i}y_{j},\ h_{2}(y_{j})=y_{j}^{2}-2,\ y_{j_{1}}y_{j_{2}}$
$\max\\{-1/(n-k),-1/2\\}$ | $\ldots$
Table 1. Eigenvalues and eigenfunctions of $L_{\Sigma^{k}}$

These eigenfunctions have geometric meanings:

  * •

Constant $1$ is the mean curvature on the generalized cylinder, that is the
infinitesimal (spacetime) dilation;

  * •

$\theta_{i}$ and $y_{j}$ are both infinitesimal translations. In fact,
$\theta_{i}$’s represent the translations in the directions of the spherical
components, and $y_{i}$’s represent the translations in the axis directions.
We will see this in later discussions;

  * •

$\theta_{i}y_{j}$ represents the infinitesimal rotation;

  * •

$h_{2}(y_{i})=y_{i}^{2}-2$ is known to be the non-integrable Jacobi field. We
will see that it actually represents _non-degenerate neckpinching_ in the
related direction on the axis.

  * •

$y_{i}y_{j}$’s show up if we rotate $h_{2}(y_{i})$’s. e.g.
$\left(\frac{y_{i}+y_{j}}{\sqrt{2}}\right)^{2}-2=\frac{y_{i}^{2}-2}{2}+\frac{y_{j}^{2}-2}{2}+\sqrt{2}y_{i}y_{j}$.

Among all the eigenfunctions, we would like to call readers’ attention to
$y_{i}$ corresponding to the eigenvalue $1/2$. Perturbing the cylinder in the
$y_{i}$ direction can decrease the value of the $F$-functional strictly, which
cannot be compensated by translations or dilations. This is the key idea in
the proof of $F$-instablity of cylinders in [CM1, Section 11]. However, one of
the key observations in the present paper is that in the presence of a
nontrivial $h_{2}(y_{i})=y_{i}^{2}-2$ component, a perturbation in the
$y_{i}$-direction can be killed. Let us elaborate on this point in the
following remark.

######  Remark 2.1 (Geometric meaning of the eigenfunctions $y_{i}$).

The eigenfunctions $y_{j}$’s represent the infinitesimal translations in the
cylinder axis directions. These translations may not be seen in many cases.
For example, if the hypersurface is exactly the generalized cylinder, it is
translation invariant in the axis direction, then one can not see the
infinitesimal translation represented by $y_{i}$’s.

However, if the RMCF converges to $\Sigma^{k}$ in a “pinching” manner, then
$y_{i}$ shows up as an infinitesimal translation.More precisely, suppose
$\mathbf{M}_{\tau}$ is an MCF with a spacetime singularity at $(0,0)$, that is
modeled by a generalized cylinder $\Sigma^{1}$, and $\mathbb{R}$ has
coordinate $y$. We assume the singularity is non-degenerate and in later
section, we will prove that the dominant term in the RMCF is $h_{2}=y^{2}-2$,
namely if we write the RMCF $M_{t}$ as a graph of function $u(\cdot,t)$ over
$\Sigma^{1}$, in some large ball, then
$u(\cdot,t)=\frac{a}{t}(y^{2}-2)+o(1/t)$ for some constant $a$. As a
consequence, if we take a look at the spacetime blowing up of the MCF at a
fixed nearby spacetime point $(\bar{z},0)$ where
$\bar{z}=(0,z)\in\mathbb{R}^{n}\times\mathbb{R}$. Then the RMCF for
$(\bar{z},0)$ can be express as a graph of

(2.2)
$v(\cdot,t)=\frac{a}{t}((e^{t/2}\bar{z}+y)^{2}-2))+o(1/t)=\frac{ae^{t}}{t}\bar{z}^{2}+\frac{2a}{t}e^{t/2}\bar{z}y+\frac{a}{t}(y^{2}-2)+o(1/t).$

Because $\bar{z}$ is a constant, $\frac{ae^{t}}{t}\bar{z}^{2}$ represent a
spacetime dilation term. Then this implies that $y$ represents the translation
in the axis direction, after modulo some dilations.

##  3\. Local dynamics of cylindrical singularities: scale up to $\sqrt{\log
t}$

In this section we study the local dynamics of RMCF near a cylindrical
shrinker. We shall prove a $H^{1}$ normal form theorem that is valid in a ball
of radius $O(\sqrt{\log t})$. We shall mainly overcome the following
difficulties: (1) the first challenge here is the limit shrinker $\Sigma^{k}$
is noncompact, hence the RMCF can only be written as a graph over $\Sigma^{k}$
in some compact region; (2) the eigenspace with nonnegative eigenvalues other
than the $h_{2}$-terms grows faster than the $h_{2}$-term; (3) the analysis of
the asymptotics of $h_{2}$-terms.

Assume $(\star)$, then by Colding-Minicozzi’s Łojasiewicz inequality [CM2],
the limiting shrinker $\Sigma^{k}$ is unique. In [CM2], Colding-Minicozzi
introduced the notion of _cylindrical scale_.

######  Definition 3.1 (Graphical radius).

Let $\varepsilon_{0},\ell$ and $C_{\ell}$ be some fixed constants. Given a
hypersurface $\Sigma$, the cylindrical scale $($throughout the paper, we also
call it the _graphical radius_ $)$ $\mathbf{r}(\Sigma)$ is the largest radius
where $B_{\mathbf{r}(\Sigma)}\cap\Sigma$ is the graph over $\Sigma^{k}$ of a
function $u$ with $\|u\|_{C^{2,\alpha}}\leq\varepsilon_{0}$ and
$|\nabla^{\ell}A|\leq C_{\ell}$.

Section 6 of [CM2] gives the following estimate of the graphical radius.

######  Proposition 3.2.

Assume $(\star)$ and let $\mathbf{r}(t)$ be the graphical radius of the
manifold $M_{t}$ in the RMCF. Then we have the estimate
$\mathbf{r}(t)\geq(8(1+\alpha)\log t)^{1/2}$ for some small $\alpha>0$ and all
$t>t_{0}$ for some large $t_{0}$.

###### Proof.

In Theorem 6.1 of [CM2] we choose $1/\tau=3-\varepsilon$, then we get the
estimate $F(\Sigma_{t-1})-F(\Sigma_{t+1})\leq
t^{\frac{1}{\tau}}=t^{3-\varepsilon}$ by Lemma 6.9 of [CM2]. Defining $R(t)$
(shrinker scale in [CM2]) through
$e^{-\frac{R(t)^{2}}{4}}=F(\Sigma_{t-1})-F(\Sigma_{t+1})$, we get $R(t)\geq
2\sqrt{\frac{1}{\tau}\log t}.$ By Theorem 5.3 of [CM2] on the graphical scale
$\mathbf{r}(t)\geq(1+\mu)R(t)$, we get the estimate

$\mathbf{r}(t)\geq\frac{2}{\sqrt{\tau}}(1+\mu)\sqrt{\log
t}=2\sqrt{3-\varepsilon}(1+\mu)\sqrt{\log t}.$

∎

We remark that the graphical radius estimate and the time $t_{0}$ here is
uniform for all cylindrical singularities (see footnote 5 of [CM2]), though
the normal form time in the next theorem is not. Throughout this section, we
set $\mathbf{r}(t):=(8(1+\alpha)\log t)^{1/2}-1$. We shall prove the following
theorem in this section.

######  Theorem 3.3 ($H^{1}$-Normal form theorem).

Assume $(\star)$. There exists $T>0$ with the following significance: for any
$t>T$, the RMCF $M_{t}\cap B_{\mathbf{r}(t)}$ can be written as a graph of
function $u:\ \Sigma^{k}\cap B_{\mathbf{r}(t)}\to\mathbb{R}$, and we have the
following normal form in the $H^{1}$-norm as $t\to\infty$ up to a rotation in
$\mathbb{R}^{k}$-factor:

(3.1) $u(t,z)=\sum_{i\in\mathcal{I}}\frac{\varrho}{4t}(y_{i}^{2}-2)+o(1/t),$

where $\mathcal{I}\subset\\{1,2,\ldots,k\\}$.

As a corollary, with an interpolation argument (see [CM2, Appendix B]), we
also obtain $C^{2}$ estimate on any compact set.

######  Corollary 3.4.

Let $R$ be a fixed constant. Then for all $t$ sufficiently large, the normal
form in Theorem 3.3 holds in $C^{2}$-norm on $\Sigma^{k}\cap B_{R}$.

###  3.1. Equations of motion and invariant cones

In Proposition A.1 in Appendix A, we write the graph RMCF over a self-shrinker
is given by

$\operatorname{\partial}_{t}u=Lu+\mathcal{Q}(J^{2}u)$

where $u(\cdot,t):\ \Sigma^{k}\cap B_{\mathbf{r}(t)}\to\mathbb{R}$ whose graph
is the RMCF $M_{t}$ restricted to the ball $B_{\mathbf{r}(t)},$ and
$\mathcal{Q}$ is at least quadratic in $u$ given in (A.2) and
$J^{2}u:=(u,Du,D^{2}u)$ means the $2$-jet of $u$. We shall study the evolution
of $u$ under the differential equation.

Since $M_{t}$ cannot be written as a global graph over $\Sigma^{k}$, we
introduce a smooth cutoff function $\chi(t):\ \Sigma^{k}\to\mathbb{R}$ that is
$1$ over $B_{\mathbf{r}(t)-1}$ and vanishes outside $B_{\mathbf{r}(t)}$, so
that $|\nabla^{\ell}\chi|\leq C$ for any $\ell>0$. We define
$\mathbb{A}_{t}=\Sigma^{k}\cap(B_{\mathbf{r}(t)+1}\backslash
B_{\mathbf{r}(t)})$. Then we derive the equation for $\chi u$:

$\operatorname{\partial}_{t}(\chi u)=L(\chi
u)+\chi\mathcal{Q}(J^{2}u)+(u(\Delta\chi+\operatorname{\partial}_{t}\chi)+2\langle\nabla\chi,\nabla
u\rangle+\frac{1}{2}\langle x,\nabla\chi\rangle u),$

where the last term on the RHS is supported on the annulus
$\mathbb{A}_{\mathbf{r}(t)}$. For simplicity, we write

(3.2) $\operatorname{\partial}_{t}(\chi u)=L(\chi u)+\mathcal{B}(J^{2}u).$

The natural function space to study the equation is $H^{1}(\Sigma^{k})$, which
admits a natural direct sum decomposition into eigenspaces of the
$L$-operator. Lying in the kernel of $L$, there are functions generating
rotations, which are not important in our analysis of isolatedness of
singularities, thus we choose to modulo them. Let $g_{t}$ be a curve in the
group $\mathrm{SO}(n+1)$ of rotations, and $\overline{v}$ is the function of
the difference of the graph after the rotation by $g_{t}$. We determine
rotation so that the projection of $\overline{v}$ to the space of
infinitesimal rotations $\mathfrak{so}(n+1)$ is zero. By implicit function
theorem, one can choose a curve of matrices $G(t)\in\mathrm{SO}(n+1)$, so that
the evolution of $\overline{v}$ satisfies the equation

(3.3)
$\operatorname{\partial}_{t}\overline{v}=L\overline{v}+\overline{\mathcal{B}}(J^{2}\overline{v})+\chi\langle
G(t)x,\mathbf{n}\rangle.$

Here
$\overline{\mathcal{B}}(J^{2}\overline{v})=\mathcal{B}(J^{2}\overline{v})+O(|G(t)||v|)$,
see [BC1, Lemma 2.4, 2.5]. We will prove at the end of this section the
following lemma

######  Lemma 3.5.

Let $\mathcal{A}:=\chi\langle G(t)x,\mathbf{n}\rangle$. There is a constant
such that $\|\mathcal{A}\|_{L^{2}}\leq
C\|\overline{\mathcal{B}}(J^{2}u)\|_{L^{2}}$.

Therefore, we can absorb $\mathcal{A}:=\chi\langle G(t)x,\mathbf{n}\rangle$
into $\overline{\mathcal{B}}$, and $\overline{\mathcal{B}}$ has similar bound
as $\mathcal{B}$ by Lemma 3.5. In the following we remove all the overline
notations, for notational simplicity.

Let $E=H^{1}(\Sigma^{k})/\mathfrak{so}(n+1)$ be the function space under
consideration, and we introduce the direct sum decomposition $E=E^{+}\oplus
E^{0}\oplus E^{-}$ where $E^{+}$ (respectively $E^{0}$ and $E^{-}$) is spanned
by eigenfunctions of $L$ with positive (respectively $0$ and negative)
eigenvalues. We denote by $\Pi_{+},\ \Pi_{-},\ \Pi_{0}$ the natural projection
to the spaces $E^{+},E^{-},E^{0}$ respectively.

We introduce a double cone construction that is used to suppress the
$E_{+}$-components and manifest the $E_{0}$-component. Let $\kappa\in(0,1)$ be
a small number. We introduce two cones $\mathcal{K}_{\geq 0}$ and
$\mathcal{K}_{0}$ as follows

$\mathcal{K}_{\geq 0}(\kappa):=\left\\{u=(u_{+},u_{0},u_{-})\in E^{+}\oplus
E^{0}\oplus E^{-}\ |\ \|u_{+}+u_{0}\|\geq\kappa\|u\|\right\\}$

is a $\kappa$-cone around $E^{+}\oplus E^{0}$ and

$\mathcal{K}_{0}(\kappa):=\left\\{u=(u_{+},u_{0},u_{-})\in E^{+}\oplus
E^{0}\oplus E^{-}\ |\ \|u_{0}\|\geq\kappa\|u\|\right\\}$

is a $\kappa$-cone around $E^{0}$. Both are narrower when $\kappa$ is closer
to 1.

######  Theorem 3.6 (Cone theorem).

Let $M_{t}$ be the RMCF as in Theorem 3.3 and $u:\ \Sigma^{k}\cap
B_{\mathbf{r}(t)}\to\mathbb{R}$ be the graphical function of $M_{t}\cap
B_{\mathbf{r}(t)}$. Then for all $\kappa\in(0,1)$, there exists $n$ such that

  1. (1)

if $u(n)$ does not enter $\mathcal{K}_{\geq 0}(\kappa)$ for all $n$, then
$\|u(n)\|\leq\frac{C}{n^{1+\alpha}}$.

  2. (2)

if $u(n)\in\mathcal{K}_{\geq 0}(\kappa)$, then $u(n+1)\in\mathcal{K}_{\geq
0}(\kappa)$;

  3. (3)

suppose $u_{+}(t)\to 0$ as $t\to\infty$, and if
$u(n)\in\mathcal{K}_{0}(\kappa)$, then
$u(n^{\prime})\in\mathcal{K}_{0}(\kappa)$ for all $n^{\prime}\geq n.$

Theorem 3.6 will be proved in Section 3.2. Theorem 3.3 follows immediately
from the cone theorem.

###### Proof of Theorem 3.3.

By assumption, we have $u\to 0$ in the $C^{\infty}_{loc}$-sense, thus
$u_{+}\to 0$. By item (2) of the cone Theorem 3.6, we see that

  1. (1)

either $u(t)$ remains in the cone $\mathcal{K}_{0}(\kappa)$ for all time
sufficiently large,

  2. (2)

or $u(t)$ does not enter the cone $\mathcal{K}_{0}(\kappa)$ for all time
sufficiently large.

In the former case, since $\kappa$ can be chosen to be arbitrarily small, we
see that the $u_{0}$ part dominates. Since we have modulo rotations, the
$E^{0}$ part is spanned by $\\{h_{2}(y_{i})=y_{i}^{2}-2\\}$. The following
Lemma 3.7 shows that the coefficients of $h_{2}(y_{i})$ are either 0 or
$\frac{\varrho}{4}\frac{1}{t}$ as given in the statement. By Lemma 3.5, the
rotations are of order $O(1/t^{2})$, thus is absorbed in the $o(1/t)$ term in
the normal form.

In the latter case occurs, then by item (1) of the cone theorem, we get that
$\|u(n)\|\leq\frac{C}{n^{1+\alpha}}$. This corresponds to the case
$\mathcal{I}=\emptyset$ in the normal form. ∎

######  Lemma 3.7.

The coefficients of the $h_{2}(y_{i})=y_{i}^{2}-2$ in Theorem 3.3 are
explicitly given by $a_{i}(t)=\frac{b}{t}+o(\frac{1}{t}),$ where $b$ is either
$0$ or $\frac{\varrho}{4}$.

###  3.2. Dynamics of the RMCF near cylinder

In this subsection, we shall prove the cone theorem 3.6. For later use, we
shall prove something stronger that is the evolution of the difference between
two RMCFs. Let $u_{1},u_{2}$ be two graphical RMCF over $\Sigma^{k}$, and
$\|u_{i}(\cdot,t)\|_{C^{2,\alpha}}\leq\varepsilon_{0}$ inside $\Sigma^{k}\cap
B_{\mathbf{r}(t)}$. Then we write $v=\chi(u_{1}-u_{2})$, and we can calculate

(3.4) $\operatorname{\partial}_{t}v=Lv+\delta\mathcal{B}.$

where
$\delta\mathcal{B}=\delta\mathcal{B}(J^{2}u_{1},J^{2}u_{2})=\mathcal{B}(J^{2}u_{1})-\mathcal{B}(J^{2}u_{2})$.
In $\delta\mathcal{B}$, we replace $u_{i}$ by $\chi u_{i}$ creating an error
supported on $\mathbb{A}_{\mathbf{r}(t)}$ denoted by
$\mathcal{E}(J^{2}u_{1},J^{2}u_{2}).$ Thus we can write

(3.5) $\delta\mathcal{B}=Pv+\mathcal{E}(J^{2}u_{1},J^{2}u_{2}),\quad
P=\int_{0}^{1}D_{u}\mathcal{B}(sJ^{2}(\chi u_{1})+(1-s)J^{2}(\chi u_{2}))ds.$

The following result describes the dynamics of the RMCF close to the shrinker.

######  Proposition 3.8.

Let $v=\chi(t)(u_{1}(t)-u_{2}(t)):\ \Sigma^{k}\cap
B_{\mathbf{r}(t)}\to\mathbb{R}$ be as above a solution to (3.4) satisfying
$\|v(n)\|\geq\|v(t)\|_{C^{2}(\mathbb{A}_{\mathbf{r}(t)})}\frac{1}{t^{1+\alpha}}$
and $\|v(t)\|_{C^{2}}\leq\varepsilon_{0}$ for all $t\in[n,{n+1}]$ for some $n$
sufficiently large. then we have

$\|v({n+1})-e^{L}v(n)\|\leq\varepsilon_{0}\|v(n)\|.$

The cone Theorem 3.6 follows immediately from this proposition and the
following lemma.

######  Lemma 3.9.

Suppose there is a sequence $t_{n}\to\infty$ such that
$\|v(t_{n})\|\geq\frac{1}{t_{n}^{1+\alpha}}$. Then we have for all $t>t_{n}$
that $\|v(t)\|\geq\frac{1}{t^{1+\alpha}}$ for some $n$ sufficiently large.

Proposition 3.8 will be proved later in this subsection, and Lemma 3.9 will be
proved in Section 3.3. Let us first use Proposition 3.8 and Lemma 3.9 to prove
the Cone Theorem 3.6.

###### Proof of Cone Theorem 3.6.

To prove the Cone Theorem 3.6 as stated, we shall apply Proposition 3.8 with
$u_{2}=0$ and $u_{1}=u$ the graphical function of the RMCF $M_{t}$. By the
definition of the graphical radius, we get
$\|u(t)\|_{C^{2}(B_{\mathbf{r}(t)})}\leq\varepsilon_{0}$. Next, we either have
$\|v(t)\|\leq\frac{1}{t^{1+\alpha}}$ for all $t$ sufficiently large, or there
is a sequence $t_{n}\to\infty$ such that
$\|v(t_{n})\|\geq\frac{1}{t_{n}^{1+\alpha}}$. In the latter case, Lemma 3.9
verifies the assumption of Proposition 3.8. Thus we can apply Proposition 3.8,
which means that the solution to the nonlinear equation (3.4) is well
approximated by that of the linear equation $\partial_{t}u=Lu$.

Item (2) of the statement follows immediately from Proposition 3.8 (details
can be found in the proof of Theorem 3.1 of [SX3]). Introducing
$\mathcal{K}_{>0}(\kappa)$ similar to $\mathcal{K}_{\geq 0}(\kappa)$, we also
have that if $u(n)\in\mathcal{K}_{>0}(\kappa)$ implies
$u(n+1)\in\mathcal{K}_{>0}(\kappa)$, which implies $\|u_{+}(n)\|$ grows
exponentially due to the presence of positive eigenvalue of $L$. If $u_{+}\to
0$ as in item (3), then we get $u(n)\notin\mathcal{K}_{>0}(\kappa)$ for all
$\kappa>0$ and all $n$ large. From item (2) and the fact that
$\mathcal{K}_{0}(\kappa)\subset\mathcal{K}_{\geq 0}(\kappa)$, we get that
$u(n)\in\mathcal{K}_{0}(\kappa)$ implies $u(n+1)\in\mathcal{K}_{0}(\kappa)$,
which proves item (3).

∎

In the following, we work on the proof of Proposition 3.8. The argument is
similar to the proof of Proposition 4.6 in [SX2] with the main difference
being that the graphical radius here is of order $\sqrt{\log t}$ which is much
smaller than $e^{t/2}$ used in [SX2].

###### Proof of Proposition 3.8.

Let $w$ be a solution to the linearized RMCF equation, i.e.
$\operatorname{\partial}_{t}w=Lw$ with the initial condition
$w(n)=\chi(n)v(n)$ for some $n$ large. We next estimate the evolution of
$v-w$. We first compute the time derivative of $\|v-w\|^{2}_{L^{2}}$, which is

$\begin{split}\operatorname{\partial}_{t}\int(v-w)^{2}e^{-\frac{|x|^{2}}{4}}=&2\int(v-w)(L(v-w)+\delta\mathcal{B})e^{-\frac{|x|^{2}}{4}}\\\
=&-2\int|\nabla(v-w)|^{2}e^{-\frac{|x|^{2}}{4}}+2\int(v-w)^{2}e^{-\frac{|x|^{2}}{4}}+2\int(v-w)\delta\mathcal{B}e^{-\frac{|x|^{2}}{4}}.\end{split}$

Similarly, we compute the time derivative of $\|\nabla(v-w)\|^{2}_{L^{2}}$:

$\begin{split}&\operatorname{\partial}_{t}\int|\nabla(v-w)|^{2}e^{-\frac{|x|^{2}}{4}}=2\int\nabla(v-w)\cdot\nabla(L(v-w)+\delta\mathcal{B})e^{-\frac{|x|^{2}}{4}}\\\
=&-2\int|\mathcal{L}(v-w)|^{2}e^{-\frac{|x|^{2}}{4}}+2\int|\nabla(v-w)|^{2}e^{-\frac{|x|^{2}}{4}}+2\int(\mathcal{L}(v-w))\delta\mathcal{B}e^{-\frac{|x|^{2}}{4}}.\end{split}$

We bound $2\int(\mathcal{L}(v-w))\delta\mathcal{B}e^{-\frac{|x|^{2}}{4}}$ by
$\|\mathcal{L}(v-w)\|_{L^{2}}^{2}+\|\delta\mathcal{B}\|^{2}_{L^{2}}$. Since we
have $\delta\mathcal{B}=Pv+\mathcal{E}$, we get
$\|\delta\mathcal{B}\|^{2}_{L^{2}}\leq
2\|Pv\|^{2}_{L^{2}}+2\|\mathcal{E}\|^{2}_{L^{2}}$. Now we estimate
$\|\delta\mathcal{B}\|_{L^{2}}^{2}$. Notice that $\delta\mathcal{B}$ is
defined in equation (3.4) independent of $w$, and
$\delta\mathcal{B}=Pv+\mathcal{E}$, where $\mathcal{E}$ is only supported on
$\mathbb{A}_{\mathbf{r}(t)}$. If
$\|v\|_{C^{2}(B_{\mathbf{r}(t)})}\leq\varepsilon_{0}$ by the definition of
graphical radius, it is straightforward that

(3.6)
$\|\delta\mathcal{B}\|_{L^{2}}^{2}\leq\varepsilon_{0}^{2}\|v\|_{L^{2}}^{2}+e^{-\frac{\mathbf{r}(t)^{2}}{4}}\|v\|^{2}_{C^{2}(\mathbb{A}_{\mathbf{r}(t)})}\leq\varepsilon_{0}^{2}\|J^{2}v\|_{L^{2}}^{2}+\frac{\|v\|^{2}_{C^{2}(\mathbb{A}_{\mathbf{r}(t)})}}{t^{2(1+\alpha)}},$

where we used Proposition 3.2.

The term $\|J^{2}v\|_{L^{2}}^{2}$ is bounded by a multiple of
$\|Lv\|_{L^{2}}^{2}$ using Bochner formula (Details can be found in [CM3,
Section 5] and [SX3, Proposition 4.6].) In conclusion, we have

(3.7) $\operatorname{\partial}_{t}\|v-w\|^{2}\leq
C\|v-w\|^{2}+\|v\|^{2}_{C^{2}(\mathbb{A}_{\mathbf{r}(t)})}/t^{2(1+\alpha)}$

with initial condition $\|v(n)-w(n)\|^{2}=0$. By assumption, we have
$\|v\|^{2}_{C^{2}(\mathbb{A}_{\mathbf{r}(t)})}/{n^{2(1+\alpha)}}\leq\|v(n)\|^{2}$.
Note that $v(n+1)-w(n+1)=v(n+1)-e^{L}v(n)$. Thus, integrating the last
differential inequality over time 1, we get the estimate in the statement.

∎

###  3.3. ODE system of neutral modes

In this section we prove Lemma 3.7. Throughout this subsection, we set
$v(t)=\chi(t)u(t)$. The idea is that after modulo the rotations, the remaining
neutral modes are $h_{2}(y_{i})$’s and $y_{i}y_{j}$’s. Let us consider $M(t)$
is a $n$ by $n$ matrix, where the entries $m_{ii}(t)$ are the coefficients of
$h_{2}(y_{i})$, and $m_{ij}(t)=m_{ij}(t)$ are the coefficients of
$\frac{\sqrt{2}}{2}y_{i}y_{j}$. Here we impose the coefficient
$\frac{\sqrt{2}}{2}$ before $y_{i}y_{j}$ because the $L^{2}$ norm of
$y_{i}^{2}$ is $1/4$ of the $L^{2}$ norm of $(y_{2}^{2}-2)^{2}$. $M(t)$ is a
symmetric matrix, and $\Pi_{0}u(t)\chi(t)=\operatorname{tr}(M(t)B)$, where $B$
is the matrix that $b_{ii}=h_{2}(y_{i})$ and $b_{ij}=y_{i}y_{j}$. We want to
understand the longtime behaviour of $M(t)$. The idea is essentially identical
to [V2]. The key is to show that $M$ must be positive definite.

We remark that the because of the nature of our nonlinear term, in the theory
semilinear heat equation in [V2] etc, the solution is up to a minus sign.

######  Lemma 3.10.

Suppose $\|\Pi_{\neq 0}v(t)\|\leq\varepsilon(t)\|\Pi_{0}v(t)\|$, where
$\varepsilon(t)\to 0$ as $t\to\infty$. Then $M(t)$ satisfies the following
ODE:

(3.8) $M^{\prime}(t)=\gamma
M^{2}(t)+O(\varepsilon_{1}(t)|M|^{2})+O(t^{-2(1+\alpha)}),$

where $\varepsilon_{1}(t)\to 0$ as $t\to\infty$, and $\gamma$ is a fixed
constant.

###### Proof.

First notice that any term involving in $\nabla\chi$ or $\chi(1-\chi)$ would
be only nonzero near the boundary, hence can be estimated by
$O\left(\frac{1}{t^{1+\alpha}}\right)$. Recall that $m_{ii}(t):=\int\chi
vh_{2}(y_{i})$ and $m_{ij}(t):=\int\chi v\frac{\sqrt{2}}{2}y_{i}y_{j}$. From
Proposition A.1, we write the reminder $\mathcal{Q}(J^{2}u)$ as

$\mathcal{Q}(J^{2}u)=-\frac{1}{2\varrho}(u^{2}+4u\Delta_{\theta}u+2|\nabla_{\theta}u|^{2})+O(\|u\|_{C^{2,\alpha}}(|u|^{2}+|\nabla
u|^{2}+|\nabla^{2}u|)).$

Then we take the time derivative of $m_{ii}$ and $m_{ij}$ to get

(3.9)
$\begin{split}m_{ii}^{\prime}(t)=&\int(Lv+\chi\mathcal{Q}(u))h_{2}(y_{i})e^{-\frac{|x|^{2}}{4}}+O\left(\frac{1}{t^{2(1+\alpha)}}\right)\\\
=&\int-\left(\frac{1}{2\varrho}(v^{2}+4v\Delta_{\theta}v+2|\nabla_{\theta}v|^{2})\right)h_{2}(y_{i})e^{-\frac{|x|^{2}}{4}}+O\left(\frac{1}{t^{2(1+\alpha)}}\right),\\\
m_{ij}^{\prime}(t)=&\int-\left(\frac{1}{2\varrho}(v^{2}+4v\Delta_{\theta}v+2|\nabla_{\theta}v|^{2})\right)\frac{\sqrt{2}}{2}y_{i}y_{j}e^{-\frac{|x|^{2}}{4}}+O\left(\frac{1}{t^{2(1+\alpha)}}\right).\end{split}$

Suppose $g$ is a $0$-eigenfunction of $L$. In the following, $L^{p}_{w}$ is
the Gaussian weighted $L^{p}$-norm. Then

$\int(\Pi_{\neq 0}v)^{2}ge^{-\frac{|x|^{2}}{4}}\leq\|\Pi_{\neq
0}v\|^{2}_{L^{4}}\|g\|_{L^{2}},\quad\int\Pi_{\neq
0}vg^{2}e^{-\frac{|x|^{2}}{4}}\leq\|\Pi_{\neq
0}v\|_{L^{4}}\|g^{2}\|_{L^{4/3}}.$

let us estimate $\|\Pi_{\neq 0}v\|_{L^{4}}$. The idea is similar to the proof
of Lemma 2.4 in [V2], so we only sketch the proof. Let us write $h=\Pi_{\neq
0}v=v-\Pi_{0}v$. Then $h$ satisfies the equation
$\operatorname{\partial}_{t}h=Lh+\mathcal{Q}(J^{2}u)-\Pi_{0}v$. Let us write
$D=\mathcal{Q}(J^{2}u)-\Pi_{0}v$. Compared with [V2, Lemma 2.4], the only
difference is the nonlinear term $D$. Nevertheless, we can compute that
$\|\mathcal{Q}(J^{2}u)\|_{L^{5N}}\leq C(\|u\|^{2}_{L^{5N}}+\|\nabla
u\|^{2}_{L^{5N}})$ by used the $C^{2,\alpha}$-bound of $u$, which is the
ingredient using in [V2, (2.13)]. Thus, we can proceed exactly the same as the
proof of [V2, Lemma 2.4], so see that $\|\Pi_{\neq
0}v\|_{L^{4}}\leq\varepsilon_{2}(t)\|\Pi_{0}v\|$ for some
$\varepsilon_{2}(t)\to 0$ as $t\to\infty$.

Once we obtain the smallness of $\|\Pi_{\neq 0}v\|_{L^{4}}$, we can see that
if we replace $v$ in (3.9) by $\Pi_{0}v$, we will only get some extra small
term that is the scale of $C\varepsilon_{2}(t)|M|^{2}$. For $\Pi_{0}v$, its
Fourier expansion has only finitely many terms, then we can write it into the
Fourier expansion and do calculations to calculate $a^{\prime}_{ii}$ and
$a_{ij}^{\prime}$ explicitly. By orthogonality, we only need to calculate
$\int(h_{2}(y_{i}))^{3}e^{-|y|^{2}/4}$ and $\int
y_{i}^{2}y_{j}^{2}y_{k}^{2}e^{-|y|^{2}/4}$. If we write
$h_{2}(y_{i})=a(y_{i}^{2}-2)$, we can calculate that
$\gamma=-\frac{4a}{\varrho}$. ∎

###### Proof of Lemma 3.7.

The proof is the same as [V2, Section 3]. The only difference here is that
here (3.8) consists of a term $O(t^{-2(1+\alpha)})$. However, this term is
integrable in $t$, and it does not influence the argument in [V2]. Therefore
we obtain the desired formula as in [V2, Lemma 3.4].

For the readers’ convenience, let us sketch the proof here. Let
$\lambda_{1}(t),\dots,\lambda_{k}(t)$ be the eigenvalues of the matrix $M(t)$.
Then by classical perturbation theory for linear operator (see proof of [V2,
Lemma 3.1])
$\lambda_{i}^{\prime}=\gamma\lambda_{i}^{2}+O(\varepsilon_{1}(t)\sum_{i=1}^{k}\lambda^{2}_{i})+O(t^{-2(1+\alpha)})$.
Thus, the eigenvalues satisfy a Riccati equation system. Note that the
majorant part of the ODE is $\lambda^{\prime}=\gamma\lambda^{2}$, and the
solution converging to $0$ as $t\to\infty$ is either $-\frac{1}{\gamma t}$, or
constant $0$. Then one can show that each $\lambda_{i}$ has either of the
asymptotics.

Once we obtain the asymptotic expansion for the eigenvalues of the matrix
$M(t)$, let $W(t)$ denote the eigenspace associated to the eigenvalues
$\lambda_{i}$’s with $\lambda_{i}\approx-\frac{1}{\gamma t}$, and let
$\Pi_{W(t)}$ denote the orthogonal projection to $W(t)$. Then the classical
eigen theory (see [V2, Lemma 3.6]) shows that $E:=\lim_{t\to\infty}\Pi_{W(t)}$
exists, and $E$ is the orthogonal projection on an $l$-dimensional space. Then
up to a rotation, $E$ is a diagonal matrix with first $l$ entries $1$. Thus we
have the desired expansion. ∎

###### Proof of Lemma 3.9.

We follow the proof of Proposition 3.8 until equation (3.7), which does not
need the assumption of
$\|v(t)\|_{C^{2}(\mathbb{A}_{\mathbf{r}(t)})}/t^{(1+\alpha)}\leq\|v(t_{n})\|$.
Letting $x=\|v_{-}(t)\|^{2},\ y=\|v_{0}(t)\|^{2},z=\|v_{+}(t)\|^{2}$, then we
get $\begin{cases}\dot{x}&\leq-cx+O(t^{-2(1+\alpha)})\\\
|\dot{y}|&\leq\varepsilon(x+y+z)+O(t^{-2(1+\alpha)})\\\
\dot{z}&\geq\frac{1}{2}z+O(t^{-2(1+\alpha)})\\\ \end{cases}$, where
$c=\max\\{-1/(n-k),-1/2\\}<0$. Such an ODE system was firstly studied by
Fillipas-Liu in [FL], and later in [ChM, Appendix B]. If we have (1) for all
time $|x(t)|\geq\kappa(|y(t)|+|z(t)|)$ for some constant $\kappa$, and (2)
$|x(t)|+|y(t)|+|z(t)|\to 0$, then we get
$|x|+|y|+|z|\leq\frac{C}{t^{1+2\alpha}}$ by integrating the differential
inequalities.

Next, suppose there is a sequence of time $t_{n}\to\infty$ such that
$t_{n}^{-(1+\alpha)}\leq\|v(t_{n})\|$. By the above reasoning,
$|x(t)|\geq\kappa(|y(t)|+|z(t)|)$ cannot hold for all time, i.e.
$v(s_{n})\in\mathcal{K}_{\geq 0}(\kappa)$ along a sequence $s_{n}\to\infty$.
Without loss of generality, we choose some $s_{n}=t_{n}$ such that we have
$(\star)$ both $t_{n}^{-(1+\alpha)}\leq\|v(t_{n})\|$ and
$v(t_{n})\in\mathcal{K}_{\geq 0}(\kappa)$ where $\kappa$ is chosen so that the
cone $\mathcal{K}_{\geq 0}(\kappa)$ is very narrow. This can be achieved,
since otherwise we have $v(t_{n})\notin\mathcal{K}_{\geq 0}(\kappa)$, and we
can repeat the ODE argument in the last paragraph to decrease $\|v(t)\|$ and
derive a contradiction to the existence of the sequence $t_{n}$.

With $(\star)$, we repeat the proof of Proposition 3.8 for a very short time
interval $[t_{n},t_{n}+\Delta]$, where $\Delta$ is chosen such that
$\|v(t)\|_{C^{2}(\mathbb{A}_{\mathbf{r}(t)})}/t^{(1+\alpha)}\leq\|v(t_{n})\|$
for $t\in[t_{n},t_{n}+\Delta]$ by continuity. Equation (3.7) remains to be
true on the $\Delta$-interval. Thus we get
$\|v({t_{n}+\Delta})-e^{L+\Delta}v(t_{n})\|\leq\varepsilon_{0}\|v(t_{n})\|$ as
Proposition 3.8. Then Item (2) of the cone Theorem 3.6 holds, and the cone
condition $v(t)\in\mathcal{K}_{\geq 0}(\kappa)$ holds for all
$t\in[t_{n},t_{n}+\Delta]$. Then either $\|v(t)\|$ grows if
$v(t)\in\mathcal{K}_{>0}(\kappa^{\prime})$ for some $\kappa^{\prime}$, in
which case, we continue to have $t^{-(1+\alpha)}\leq\|v(t)\|$ since
$t^{-(1+\alpha)}$ decays; or we have $v(t)\in\mathcal{K}_{0}(\kappa)$, in
which case $v_{0}$ term dominates. Reasoning as in the proof of Lemma 3.7, the
term $y=\|v_{0}\|$ obeys the ODE $\dot{y}=-y^{2}$ to the leading order, which
has solution $y(t)=\frac{1}{t+C}$ where $C=y(t_{n})^{-1}$. By assumption
$(\star)$, we have $y(t_{n})\geq\frac{1}{t_{n}^{1+\alpha}}$. It is easy to
verify that this implies
$\frac{1}{t+x(t_{n})^{-1}}\geq\frac{1}{(t_{n}+t)^{1+\alpha}}$ for all $t>0$.
Thus, we have proved that $\|v(t)\|\geq\frac{1}{t^{1+\alpha}}$ holds on the
$\Delta$-interval and we can iterate Proposition 3.8.

This completes the proof. ∎

Finally, we estimate the rotations $G(t)$. The argument is similar to [BC1,
Page 39]. The reason why we choose modulo rotations is to single out the
$h_{2}$ and $y_{i}y_{j}$-terms in the kernel of $L$.

###### Proof of Lemma 3.5.

From the definition of $\mathcal{A}$, we have
$\int(\mathcal{B}(J^{2}u)-\mathcal{A}(t))\mathcal{A}(t)e^{-\frac{|x|^{2}}{4}}\,dx=0$
and by Plancheral and Cauchy-Schwarz inequality,

(3.10) $\|\mathcal{A}\|_{L^{2}}\leq C\|\mathcal{B}(J^{2}u)\|_{L^{2}}.$

First, by (3.6), we get that
$\|\mathcal{A}\|_{L^{2}}\leq\varepsilon_{0}\|u\|+Ct^{-(1+\alpha)}$. Since
$\mathfrak{so}(n+1)$ has finite dimensions, the estimate is also valid if we
use $C^{k}$-norm on the LHS. This implies that the $\mathcal{A}$ term can be
absorbed into the nonlinear term $\mathcal{B}$. Once we obtain this, we can
proceed with the analysis in the previous sections as we discussed. ∎

##  4\. Isolatedness of nondegenerate singularities

In this section, we show that cylindrical singularities with a nondegenerate
normal form (see Definition 1.4) are isolated, hence prove Theorem 1.5.

Let us first give an overview of the proof. From the relation between MCF and
RMCF, we note that a $d$-neighborhood of $0\in\mathbb{R}^{n+1}$ is rescaled to
a ball of radius $de^{t/2}$ for the RMCF. Note that the normal forms that we
obtained in the last section are only valid in a ball of radius $O(\sqrt{\log
t})$ for the RMCF, which is much smaller than $de^{t/2}$. Thus, the proof of
Theorem 1.5 consists of two steps:

  1. (1)

Extending the graphical scale from $O(\sqrt{\log t})$ to $O(\sqrt{t})$
(Theorem 1.3).

  2. (2)

Extending the graphical scale from $O(\sqrt{t})$ to $O(e^{t/2})$ if the
singularity is nondegenerate.

For the first step, we first use a regularization property of the drift
Laplacian of [KZ, V1, V2] to study the linear equation $\partial_{t}u=Lu$.
Next for the nonlinearity of RMCF, we also need to use the pseudolocality
property of MCF to control the RMCF. For the second step, to further extend
the graphical scale, we need to use the pseudolocality property of MCF again.
The normal form in the graphical radius $O(\sqrt{t})$ allows us to analyze the
shape of the RMCF near the boundary of the graphical radius, and allows us to
apply pseudolocality argument.

###  4.1. Proof of the isolatedness theorem

We first present a pseudolocality lemma that we will use in both steps above.

######  Lemma 4.1 (Pseudolocality of MCF graph over cylinders).

For any $\varepsilon_{0}>0$, and a vector $V=(v_{1},\dots,v_{k})$ with
$0<|V|<\varepsilon_{0}$, there exists $\delta>0$, $C>0$, $T_{0}>0$, $R_{1}>0$
and $\varepsilon_{1}>0$ with the following significance. Suppose $T>T_{0}$,
$\mathbf{M}_{\tau}$ is a MCF, at time $\tau=0$ is a graph of some function $u$
over the cylinder $\Sigma^{k}$ inside a ball of radius
$\varepsilon_{0}\sqrt{T}$, with
$\|u(\theta,y)-C|T^{-1/2}y-V|^{2}\|_{C^{1}}<\varepsilon_{1}$. Then for
$\tau\in[0,1]$, $\mathbf{M}_{\tau}\cap B_{R_{1}}$ is a smooth MCF.

Roughly speaking, this lemma shows that if an MCF is almost a cylinder in a
large range, but the radius is strictly larger than the radius of the self-
shrinking cylinder, then it does not shrink to a singularity before the time
that the self-shrinking cylinder shrinks to singularities. With the
pseudolocality lemma we can prove Theorem 1.5 assuming Theorem 1.3.

###### Proof of Theorem 1.5 assuming Theorem 1.3.

Theorem 1.3 shows that the RMCF is a graph over $\Sigma^{k}$ in a ball of
radius $\varepsilon_{0}\sqrt{t}$, when $t$ is sufficiently large. Let us cover
the RMCF $M_{t}\cap(B_{\varepsilon_{0}\sqrt{t}}\backslash
B_{2^{-1}\varepsilon_{0}\sqrt{t}})$ with several balls of radius
$R_{1}:=2^{-1}\varepsilon_{0}\sqrt{t}$, and we can apply Lemma 4.1 to get that
the MCF starting from $M_{t}$ is smooth in $B_{R_{1}}(p)$ for time $1$, where
$|p|\in(2^{-1}\varepsilon_{0}\sqrt{t},\varepsilon_{0}\sqrt{t})$. Rescale back
we can see that $\mathbf{M}_{\tau}$ is smooth in
$B_{e^{-t/2}R_{1}}(e^{-t/2}p)$ for $\tau\in[-e^{-t},0]$. Because $t$ can be
chosen arbitrarily large,
$\mathbf{M}_{\tau}\cap(B_{\delta}(0)\backslash\\{0\\})$ is smooth for
$\tau\in[-1,0]$ for some $\delta>0$ (for example, we can choose it to be
$e^{-T/2}\varepsilon_{0}\sqrt{T}$ for some fixed sufficiently large $T$).

Moreover, the region
$\frac{\varepsilon_{0}}{2}\sqrt{t}\leq|y|\leq\varepsilon_{0}\sqrt{t},\
t\in[T,\infty)$ for the RMCF scale, corresponds to the region
$\frac{\varepsilon_{0}}{2}\sqrt{-\log(-\tau)}\leq|y|\leq\varepsilon_{0}\sqrt{-\log(-\tau)},\
\tau\in[-e^{-T},0)$, which shrinks to the point $0$ as $\tau\to 0$. Thus we
get that the singularity is isolated in the $\delta$-spacetime neighborhood. ∎

Now we will be focused on the proof of Theorem 1.3 and related properties. The
remaining part of the section is organized as follows. In Section 4.2, we give
the proof of the $C^{1}$ normal form Theorem 1.3 up to scale $\sqrt{t}$ as
well as the mean convex neighborhood Theorem 1.6. In Section 4.3, we study the
pseudo locality of MCF, which gives the proof of Lemma 4.1 as well as a
technical result in the proof of Theorem 1.3 on extending the graphical scale
to $O(\sqrt{t})$.

###  4.2. $C^{1}$ normal form up to scale $\sqrt{t}$, Proof of Theorem 1.3

In this section, we give the proof of Theorem 1.3. We first state a result
extending the graphical scale from $O(\sqrt{\log t})$ to $O(\sqrt{t})$.

######  Proposition 4.2.

Suppose $M_{t}$ is an RMCF converging to $\Sigma^{k}$ in the
$C^{\infty}_{loc}$-sense. Then for any $\varepsilon_{0}>0$, there exist
$T_{0}>0$ and $K>0$, such that when $t>T_{0}$, $M_{t}$ can be written as a
graph of function $u(\cdot,t)$ over $\Sigma^{k}\cap B_{K\sqrt{t}}$, with
$\|u(\cdot,t)\|_{C^{2}(B_{K\sqrt{t}})}\leq\varepsilon_{0}$.

Proposition 4.2 will be proved in Section 4.3. Let us first use Proposition
4.2 to prove Theorem 1.3.

####  4.2.1. Proof of Theorem 1.3 assuming Proposition 4.2

The main difficulty is as follows. In the equation of motion for $u$, we have
$\operatorname{\partial}_{t}u=Lu+\mathcal{Q}(J^{2}u)$ (see Lemma A.1).
However, in the radius $O(\sqrt{t})$, $h_{2}$ is not small (when
$|y|\approx\varepsilon_{0}\sqrt{t}$, $h_{2}\approx\varepsilon_{0}$).
Therefore, in the $C^{0}$ norm, the nonlinear term $\mathcal{Q}(J^{2}u)$ is
$O(\varepsilon_{0}^{2})$, which leads to cumulative errors to the estimate of
$u$ as $t\to\infty$. The way to address the difficulty is to invoke a
formalism of [AV], also used in [Zh1, Zh2]. Then Velázquez’s regularization
argument for semilinear PDE in [V1] could be applied.

###### Proof of Theorem 1.3.

Using Proposition 4.2, we have the equation of $u$ in (A.5). Now we rewrite
(A.5) as $\operatorname{\partial}_{t}u=\mathscr{L}u+\tilde{m},$ where we
introduced the nonlinear operator

$\mathscr{L}f=\frac{\varrho^{2}}{(\varrho+f)^{2}}\Delta_{\theta}f+\Delta_{\mathbb{R}^{k}}f-\frac{1}{2}y_{i}\operatorname{\partial}_{i}f+\frac{\varrho+f}{2}-\frac{\varrho^{2}}{2(\varrho+f)},$

and $\tilde{m}=O(|\nabla u|^{4}+|\nabla
u|^{2}|\nabla^{2}u|+|\nabla_{\theta}u|^{2})$.

We next study the solution to the equation $\partial_{t}f=\mathscr{L}f.$ The
proof of the next lemma is postponed.

######  Lemma 4.3.

Suppose $1\leq l\leq k$. Let

$f(\theta,y,t)=\sqrt{\varrho^{2}+\frac{\varrho^{2}(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t}}-\varrho=\frac{\varrho(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t\left[\sqrt{1+\frac{(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t}}+1\right]}.$

Then we have

(4.1)
$\operatorname{\partial}_{t}f-\mathscr{L}f=-\frac{(n-k)^{2}(\sum_{i=1}^{l}y_{i}^{2})}{t^{2}(\varrho^{2}+\frac{\varrho^{2}(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t})^{3/2}}.$

Moreover in the $H^{1}$-norm, we have

(4.2) $f-\frac{\varrho}{4t}(\sum_{i=1}^{l}y_{i}^{2}-2l)=O(t^{-2}).$

We can check that $w:=u-f$ satisfies the following equation

$\operatorname{\partial}_{t}w=\mathscr{L}u-\mathscr{L}f+\tilde{m}-\frac{\varrho^{4}(\sum_{i=1}^{l}y_{i}^{2})}{(2t)^{2}(\varrho^{2}+\frac{\varrho^{2}(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t})^{3/2}}.$

Using the mean value theorem we obtain the equation to be

(4.3)
$\operatorname{\partial}_{t}w=Lw+\mathcal{M}(w)-\frac{\varrho^{4}(\sum_{i=1}^{l}y_{i}^{2})}{(2t)^{2}(\varrho^{2}+\frac{\varrho^{2}(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t})^{3/2}},$

where $\mathcal{M}(w)$ is the nonlinear term consists of the terms of
$w\Delta_{\theta}w$, $w^{2}$, $|\nabla w|^{2}$, and higher power terms of $w$
and $|\nabla w|$, as well as $t^{-1}|y|w$, $t^{-1}|y||\nabla w|$,
$t^{-1}|y||\nabla w|$; it also consists of terms like $t^{-2}|y|^{2}$ and
higher powers, coming from the derivatives of $f$.

Using Kato’s inequality $\Delta g\cdot\text{sgn}(g)\leq\Delta|g|$, we get
$\operatorname{\partial}_{t}|w|-L_{\mathbb{R}^{k}}|w|\leq C(|w|^{2}+|\nabla
w|^{2}+\frac{|y|^{2}}{t^{2}}+\frac{1}{t^{2}})$. Similarly, we can obtain the
equations for $|\nabla_{\theta}w|$ and $|\nabla_{y}w|$, and letting
$Z=(|w|+|\nabla w|)\chi_{\varepsilon_{0}\sqrt{t}}$, we get

(4.4)
$\begin{split}\operatorname{\partial}_{t}Z-L_{\mathbb{R}^{k}}Z-\frac{C}{t}Z\leq
C(Z^{2}+\frac{|y|^{2}}{t^{2}}+\frac{1}{t^{2}}+\chi_{1}(y,t)),\end{split}$

where $\chi_{1}(y,t)$ is $1$ outside $\varepsilon_{0}\sqrt{t}$, and $0$
elsewhere. Now we are already in the setting of Velázquez [V1] (2.14),

######  Lemma 4.4 (Proposition 2.3 in [V1]).

Let $Z\geq 0$ satisfy equation (4.4) with $\|Z\|_{L^{2}}=o(t^{-1})$. Then we
have $\max_{|y|\leq\varepsilon_{0}\sqrt{t}}Z(y,t)=o(1).$ as $t\to\infty$.

For the proof of Lemma 4.4, we refer readers to [V1, equation (2.14)] (proof
of Proposition 2.3 in [V1]) and thereafter. The idea of the proof is also
similar to the proof of Proposition 4.8, which is given in Appendix B.

In our case, we have $\|Z\|_{L^{2}}\leq C\|w\|=o(t^{-1})$ because of (4.2).
Thus, by the last lemma, we get
$\|w(\sqrt{t}\xi,t)\|_{L^{\infty}(|\xi|\leq\varepsilon_{0})}=o(1)$ as
$t\to\infty$. This shows that $u(\sqrt{t}\xi,t)$ converges to
$f(\theta,\sqrt{t}\xi,t)=\sqrt{\varrho^{2}+\frac{\varrho^{2}(t\sum_{i=1}^{l}\xi_{i}^{2}-2l)}{2t}}-\varrho$
in $C^{1}$-norm for $\xi$ in the ball of radius $\varepsilon_{0}$. This
completes the proof of Theorem 1.3 for $C^{1}$-normal form. ∎

###### Proof of Lemma 4.3.

To find an approximation solution of
$\operatorname{\partial}_{t}f=\mathscr{L}f$, we use the idea of [AV] to apply
the change of variable $x=\sqrt{t}\xi$, then if
$\operatorname{\partial}_{t}f=\mathscr{L}f$ and
$g(\theta,\xi,t)=f(\theta,\sqrt{t}\xi,t)+\varrho$, we get

$\operatorname{\partial}_{t}g=\frac{\Delta_{\xi}g}{t}-\frac{1}{2}\xi\cdot\nabla_{\xi}g+\frac{g^{2}-\varrho^{2}}{2g}+\frac{1}{2\sqrt{t}}\xi\cdot\nabla_{\xi}g,$

and when $t$ is sufficiently large, the dominate part of the right hand side
is $-\frac{1}{2}\xi\cdot\nabla_{\xi}g+\frac{g^{2}-\varrho^{2}}{2g}$. So an
approximation solution is
$g(\theta,\xi,t)=\sqrt{\varrho^{2}+\ell(\sum_{i=1}^{l}\xi_{i}^{2})}$, which
yields an approximation solution
$\tilde{g}(\theta,y,t)=\sqrt{\varrho^{2}+\frac{\ell(\sum_{i=1}^{l}y_{i}^{2})}{t}}-\varrho.$
Notice that we expect the dominant mode to be
$\frac{\varrho}{4t}((\sum_{i=1}^{l}\xi_{i}^{2})-2l)$. So we slightly modify
the function to get an approximated solution $f$ as in the statement.

We can compute
$\operatorname{\partial}_{y_{j}}f=\frac{\varrho^{2}y_{j}\delta_{jl}}{2t\sqrt{\varrho^{2}+\frac{\varrho^{2}(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t}}}$
and

$\operatorname{\partial}^{2}_{y_{j}y_{m}}f=\frac{\varrho^{4}\delta_{jm}\delta_{jl}\delta_{ml}}{4t\sqrt{\varrho^{2}+\frac{\varrho^{2}(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t}}}-\frac{\varrho^{4}y_{j}y_{m}\delta_{jl}\delta_{ml}}{(2t)^{2}(\varrho^{2}+\frac{\varrho^{2}(\sum_{i=1}^{l}y_{i}^{2}-2l)}{2t})^{3/2}}.$

This gives (4.1) and (4.2).

∎

####  4.2.2. Mean convex neighborhood and Type-I

In this section, we give the proof of the mean convex neighborhood Theorem 1.6
and the Type-I curvature condition Theorem 1.7. In fact, Lemma 4.1 shows that
when $T$ is sufficiently large and $\varepsilon_{2}$ is sufficiently small,
the MCF is still a graph over the cylinder. Then the RMCF is roughly the graph
of function $\sim\varepsilon_{0}|V|^{2}e^{t/2}$. This gives a description of
the neighbourhood of the non-degenerate singularity.

###### Proof of Theorem 1.6.

We only need to prove that for the corresponding RMCF $\\{M_{t}\\}$, when $t$
is sufficiently large, $M_{t}\cap B_{\delta_{2}e^{t/2}}$ is mean convex.

First, we show that when $t$ is sufficiently large, $M_{t}\cap
B_{\delta_{2}\sqrt{t}}$ is mean convex, where $\delta_{2}$ is chosen slightly
smaller than $K$ in Proposition 4.2. In previous section, we have proved that
$M_{t}\cap B_{\delta_{2}\sqrt{t}}$ can be written as a graph of the function
$u(\cdot,t)$ over $\Sigma^{k}$, with $u(\sqrt{t}\xi,t)$ converges to
$\sqrt{\varrho^{2}+\frac{\varrho^{2}}{2t}{(t|\xi|^{2}-2k)}}-\varrho$ in
$C^{1}$-norm for $\xi$ in the ball of radius $\varepsilon_{0}$. In particular,
this shows that $u(\sqrt{t}\xi,t)$ converges to
$\frac{\varrho|\xi|^{2}}{1+\sqrt{1+|\xi|^{2}}}$ in $C^{1}$-norm in the ball of
radius $\varepsilon_{0}$.

The explicit calculation of mean curvature is given in Appendix A. The mean
curvature of the graph of $u(\theta,y,t)=\frac{\varrho
t^{-1}|y|^{2}}{1+\sqrt{1+t^{-1}|y|^{2}}}+w(\theta,y,t)$ is explicitly given by
(A.3), and it is bounded from below by $\varrho/2-C\varepsilon_{0}$, where
$\varrho/2$ is the mean curvature of $\Sigma^{k}$, and $\varepsilon_{0}$ is
the $C^{2}$-bound of $u$. Thus, if initially we fixed $\varepsilon_{0}$ to be
a sufficiently small number, we see the mean curvature is positive.

Finally, from the scale $\delta_{2}\sqrt{t}$ to $\delta_{2}e^{t/2}$, we only
need to be aware that previously we have proved that when $t$ is sufficiently
large, we can break $M_{t}\cap B_{\delta_{2}e^{t/2}}$ into the union of
several pieces, each piece is very close to a cylinder (with variant radius).
In particular, when $t$ is sufficiently large, each piece is mean convex. This
concludes the proof.

∎

###### Proof of Theorem 1.7.

We only need to prove that for the corresponding RMCF $\\{M_{t}\\}$, when $t$
is sufficiently large, $M_{t}\cap B_{\delta_{2}e^{t/2}}$ has bounded $|A|$.
This is true just as the proof of Theorem 1.6. ∎

###  4.3. Pseudolocality extension

In this sectoin, we give the proof of Proposition 4.2. The following
pseudolocality argument was first studied by Ecker-Huisken in [EH]. Later
Ilmanen-Neves-Schulze [INS, Section 9] gave a simple proof using White’s
regularity theorem [Wh4].

######  Theorem 4.5.

There exist $\varepsilon_{0}>0$ such that for any
$\varepsilon<\varepsilon_{0}$, there exist $R_{0}>0$ and $\tau_{0}>0$ with the
following significance: suppose $M_{t}$ is an RMCF and $M_{t_{0}}$ is a graph
of a function $u(\cdot,t_{0})$ over $\Sigma^{k}$ in the ball of radius
$R>R_{0}$ and $\|u(\cdot,t)\|_{C^{1}(B_{R})}\leq\varepsilon$. Then when
$t\in[t_{0},t_{0}+\tau_{0}]$, $M_{t}$ is a graph of function $u(\cdot,t)$ over
$\Sigma^{k}$ in the ball of radius $e^{(t-t_{0})/2}R$ with
$\|\nabla^{k}u(\cdot,t)\|_{L^{\infty}}\leq Ce^{-(k-1)(t-t_{0})/2}\varepsilon$,
$k=0,1$; and when $t\in[t_{0}+\tau_{0}/2,t_{0}+\tau_{0}]$, we have
$\|\nabla^{k}u(\cdot,t)\|_{L^{\infty}}\leq C_{k}\varepsilon$,
$k\in\mathbb{N}$.

In the theorem, the factor $e^{-(k-1)(t-t_{0})/2}$ is from the exponential
expansion property of the RMCF. In particular, when $k=0$, the $L^{\infty}$
norm of $u$ increases very fast. We need to use the equation of motions
$\partial_{t}u=Lu+\mathcal{Q}$ to show that the growth rate is actually not
that fast.

The last theorem follows from the following pseudolocality theorem for MCF
considering the scaling between MCF and RMCF.

######  Theorem 4.6.

For any $\tau_{0}\in(-1,0)$ and $\varepsilon_{0}>0$, there exists $\eta_{0}>0$
with the following significance: for any $\eta\in(0,\eta_{0})$, if
$\mathbf{M}_{-1}\cap B_{\eta^{-1}}$ is a graph of function $v$ over
$\Sigma^{k}$, with $\|v\|_{C^{1}}\leq\eta$, then

  1. (1)

$\mathbf{M}_{\tau}\cap B_{\frac{\eta^{-1}}{2}}$ is smooth MCF for
$\tau\in(-1,\tau_{0})$.

  2. (2)

$\mathbf{M}_{\tau}\cap B_{\frac{\eta^{-1}}{2}}$ has a singularity before time
$-\tau_{0}$.

  3. (3)

$\mathbf{M}_{\tau}\cap B_{\frac{\eta^{-1}}{2}}$ can be written as a graph of
function $u(\cdot,\tau)$ over $\sqrt{-\tau}\Sigma^{k}$ for
$\tau\in(0,\tau_{0})$, with $\|u(\cdot,\tau)\|_{C^{1}}\leq\varepsilon_{0}$.

###### Proof.

We first prove item (1). We prove it by contradiction. Suppose not, then we
can find a sequence of hypersurfaces $\mathbf{M}_{-1}^{k}$, that each one of
them is a graph of function $v_{k}$ inside the ball of radius $k$, with
$\|v_{k}\|_{C^{1}}\leq k^{-1}$, but each one of them has singularity before
time $\tau_{0}$. Then we can use the compactness of Brakke flow (see [I2,
Section 7]) to passing to a limit weak MCF $\mathbf{M}_{\tau}^{\infty}$. By
Ecker-Huisken’s curvature estimate (see [EH]. This is actually the key
ingredient to pseudolocality of MCF, and this is also why we need to shrink
the radius from $\eta^{-1}$ to $\frac{\eta^{-1}}{2}$),
$\mathbf{M}_{\tau}^{\infty}$ is a smooth MCF when $\tau$ is close to $-1$.
Also, $\mathbf{M}_{-1}^{k}$ has varifold limit the standard cylinder
$\Sigma^{k}$. Thus, the limit flow $\mathbf{M}_{\tau}^{\infty}$ is exactly the
shrinking soliton, with only one singularity at time $0$. Then by the lower
semi-continuity of Gaussian density, we know that the singularity of
$\mathbf{M}_{\tau}^{k}$ must have singular time converging to $0$, which is a
contradiction. The proof of item (2) and (3) is basically verbatim and we omit
it here. ∎

####  4.3.1. Proof of Lemma 4.1

The proof of Lemma 4.1 is a straightforward application of the idea of
pseudolocality.

###### Proof of Lemma 4.1.

We use the pseudolocality property of MCF. We observe that the graph of
function $C|y-V|^{2}$ when $|y|<\varepsilon_{0}\sqrt{T}$ is a paraboloid over
the cylinder. Moreover, for any fixed $R>0$, inside the ball of radius $R$,
$C|T^{-1/2}y-V|^{2}\to C|V|^{2}$ as $T\to\infty$. So the graph of function
$C|T^{-1/2}y-V|^{2}$ converges to the cylinder
$\mathbb{S}^{n-k}(\varrho+C|V|^{2})\times\mathbb{R}^{k}$ smoothly compactly as
$T\to\infty$. By the pseudolocality of MCF, as $T\to\infty$, the MCF
$\mathbf{M}_{\tau}^{T}$ starting from the graph of function
$C|T^{-1/2}y-V|^{2}$ converges to the shrinking cylinder MCF with initial
radius $\varrho+C|V|^{2}$, which is smooth for time $\tau\in[0,1]$. This shows
that when $T$ is sufficiently large, $\mathbf{M}_{\tau}^{T}$ is sufficiently
close to the shrinking cylinder MCF with initial radius $\varrho+C|V|^{2}$
with $\tau\in[0,1]$, inside the ball of radius $2R_{1}$. Finally, when
$\varepsilon_{1}$ is sufficiently small, a similar argument shows the lemma. ∎

####  4.3.2. Proof of Proposition 4.2

In this section, we give the proof of Proposition 4.2 assuming the following
lemma whose proof is postponed to the end of this section.

######  Lemma 4.7.

There exist $t_{0}>0$, $\tau>0$ and $R_{0}>0$ and $K>0$ with the following
significance: suppose $M_{T}$ is a graph over $\Sigma^{k}\cap B_{R}$ as a
function of $u(\cdot,T)$ with
$\|u(\cdot,T)\|_{C^{2}(B_{R})}\leq\varepsilon_{0}$, where $T>t_{0}$ and
$R>R_{0}$. Then when $t\in[T+\tau/2,T+\tau]$, we have
$\|u(\cdot,t)\|_{C^{2}}\leq\varepsilon_{0}$ in $\Sigma^{k}\cap B_{r(t)}$,
where $r(t)=K\sqrt{(t-T)+(R/K)^{2}}$.

Now we are ready to prove Propsition 4.2.

###### Proof of Propsition 4.2.

Let us fix $t_{0},\tau$ and $R_{0}$ as in Lemma 4.7. We suppose $T$ is
sufficiently large so that
$\|u(\cdot,T)\|_{C^{2}(B_{R_{0}})}\leq\varepsilon_{0}$. Now we apply Lemma
4.7, which shows that for $t\in[T+\tau/2,T+\tau]$,
$\|u(\cdot,t)\|_{C^{2}(B_{r(t)})}\leq\varepsilon_{0}$.

Then we can apply Lemma 4.7 by replacing $T$ by $T+\tau/2$, and replacing
$R_{0}$ by $r(T+\tau/2)$. Then we obtain that for $t\in[T+\tau,T+(3/2)\tau]$,
$\|u(\cdot,t)\|_{C^{2}(B_{\tilde{r}(t)})}\leq\varepsilon_{0}$, where
$\tilde{r}(t)=K\sqrt{(t-(T+\tau/2))+(r(T+\tau/2)/K)^{2}}=r(t)$. Therefore, we
have extended the estimate
$\|u(\cdot,t)\|_{C^{2}(B_{r(t)})}\leq\varepsilon_{0}$ to the interval
$t\in[T+\tau/2,T+(3/2)\tau]$.

Iterating the above process shows that
$\|u(\cdot,t)\|_{C^{2}(B_{\tilde{r}(t)})}\leq\varepsilon_{0}$ for
$t>T+\tau/2$. Finally, $r(t)\sim\sqrt{t}$ when $t$ is sufficiently large. Thus
when $t$ is sufficiently large, we obtain
$\|u(\cdot,t)\|_{C^{2}(B_{K\sqrt{t}})}\leq\varepsilon_{0}.$

∎

####  4.3.3. Proof of Lemma 4.7

The goal of this section is to prove Lemma 4.7.

From now on we choose sufficiently small $\varepsilon_{0}$ and $\tau_{0}$ so
that Theorem 4.5 holds and $\varepsilon_{0}e^{\tau_{0}/2}\leq 1/2$, so that
Proposition A.1 applies for the graph function with $C^{2}$-norm less than
$3\varepsilon_{0}$.

Recall that we have the equation $\operatorname{\partial}_{t}(\chi u)=L(\chi
u)+\mathcal{B}(J^{2}u)$ as in Section 3.1. To get $C^{2}$ bound of $u$, we
note that $\mathcal{B}$ is quadratically small in the $C^{2}$ norm of $u$,
thus it remains to give a good $C^{2}$ control on the linear part. For the
linear equation, we have the following estimates.

######  Proposition 4.8 (Lemma 2.13 of [KZ]).

Let $v:\ \Sigma^{k}\to\mathbb{R}$ be a solution to the equation

$\operatorname{\partial}_{t}v=Lv+P(\cdot,t)v$

where $|P|$ is uniformly bounded by $C_{0}/t$ as $t\to\infty$. Then there
exist $K_{0}>0$ and $\delta_{0}>0$ such that we have the estimates

$\displaystyle\|v(\cdot,s^{\prime})\|_{C^{2}(B_{K_{0}\sqrt{s^{\prime}}})}$
$\displaystyle\leq
C(K_{0},\delta_{0})(s^{\prime})^{2}t_{0}^{-2+C_{0}}\|v(\cdot,t_{0})\|_{L^{2}},$

whenever $s^{\prime}=K_{0}+s$, $e^{(s-t_{0})/2}\leq K_{0}\sqrt{s}$, $s\geq
t_{0}+\delta_{0}$, and $s\geq\frac{1}{15}K_{0}$.

We postpone the proof to Appendix B. Roughly speaking, this proposition shows
that the solution to the variational equation has nice $C^{2}$ control on the
scale up to $O(\sqrt{t})$ provided we have $L^{2}$ control of the initial
value.

######  Lemma 4.9.

Let $\tau,R,r(t)$ be as in Lemma 4.7, the constant $K_{0}$ be as in
Proposition 4.8, and $v$ be the solution to the Cauchy problem
$\begin{cases}\operatorname{\partial}_{t}v=Lv,\\\ v(\cdot,T)=\chi
u(\cdot,T).\end{cases}$ Then when $T$ is sufficiently large, for
$t\in[T+\tau/2,T+\tau]$ and $K_{0}^{2}T\geq R$, we have
$\|v(\cdot,t)\|_{C^{2}(B_{r(t)})}\leq C/t.$

###### Proof.

We only need to carefully choose parameters in Proposition 4.8. In Proposition
4.8, let $K_{0}\leq\tau/4$, $t_{0}=T$, $s^{\prime}=t$, and
$\delta_{0}=\tau/4$. With these choice of parameters, one can check that the
conditions “$s^{\prime}=K_{0}+s$, $e^{(s-t_{0})/2}\leq K_{0}\sqrt{s}$, $s\geq
t_{0}+\delta_{0}$, and $s\geq\frac{1}{15}K_{0}$” hold whenever $T$ is
sufficiently large.

Then Proposition 4.8 shows that $\|v(\cdot,t)\|_{C^{2}(B_{r(t)})}\leq
C\|v(\cdot,T)\|.$ Here $C$ is a fixed constant once we fix $K_{0}$ and
$\delta_{0}$.

Notice that we have proved that $\|u\|_{H^{1}(B_{(8(1+\alpha)\log
t)^{1/2}})}\leq Ct^{-1}$ (Theorem 3.3). Next, since we have $r(t)\leq
e^{(t-t_{0})/2}R$ whenever $K_{0}$ is fixed and $R$ is sufficiently large,
Theorem 4.5 shows that when $\|\chi u\|_{C^{2}(B_{r(t)})}\leq
3\varepsilon_{0}$. Thus we have $\|(\chi
u)(\cdot,t)\|_{H^{1}(B_{r(t)})}\leq\frac{C}{t}.$ This completes the proof. ∎

We are now ready to give the proof of Lemma 4.7.

###### Proof of Lemma 4.7.

From Theorem 4.5, $\|\nabla^{2}u\|_{L^{\infty}}$ is nicely bounded in the ball
of radius $r(t):=K_{0}\sqrt{(t-T)+(R_{0}/K_{0})^{2}}$. Therefore we only need
to bound the $C^{1}$-norm of $u$ in $r(t)$.

Let $v$ be as in Lemma 4.9, and let $w=v-\chi u$. Then $w$ satisfies the
equation $\operatorname{\partial}_{t}w=Lw+\mathcal{B}(J^{2}u)$ with the
initial condition $w(\cdot,T)=0$. By DuHamel principle, Theorem 4.5, and the
quadratic decay of $\mathcal{B}$, we have $\|w\|_{C^{0}(B_{r(t)})}\leq
C\tau\varepsilon_{0}^{2}$ when $t\in[T,T+\tau].$ Here we point out that in
$\mathcal{B}(J^{2}u)$, most of the terms related to $J^{2}u$ are uniformly
bounded, except for a term $\frac{1}{2}\langle x,\nabla\chi\rangle u$.
However, this term is supported outside $B_{r(t)}$ because $\nabla\chi$ is
supported outside $B_{r(t)}$, so it does not influence the estimate.

Use Lemma 4.9, when $t\in[T+\tau/2,T+\tau]$, we have

$\|u\|_{C^{0}(B_{r(t)})}\leq C\tau\varepsilon_{0}^{2}+Ct^{-1}\leq
C\tau\varepsilon_{0}^{2}+CT^{-1}.$

If $\tau$ and $\varepsilon_{0}$ is chosen sufficiently small, and $T$ is very
large, we have $\|u\|_{C^{0}(B_{r(t)})}\leq\varepsilon^{2}_{0}$. Finally,
using the interpolation inequality (see Lemma B.1 in [CM2]) with
$\|\nabla^{(n+4)}u\|_{L^{\infty}(B_{r(t)})}\leq C_{n+4}$ and
$\|\nabla^{(n+6)}u\|_{L^{\infty}(B_{r(t)})}\leq C_{n+6}$ (Theorem 4.5), we
have

$\displaystyle\|\nabla u\|_{L^{\infty}(B_{K\sqrt{(t-T)+(R/K)^{2}}})}$
$\displaystyle\leq
C\|u\|_{L^{\infty}(B_{r(t)})}^{\frac{1}{2}+\frac{1}{2n+4}}\|\nabla^{(n+2)}u\|_{L^{\infty}(B_{r(t)})}^{\frac{1}{2}-\frac{1}{2n+4}}\leq
C\varepsilon_{0}^{1+\frac{1}{n+2}},\quad\mathrm{and}$
$\displaystyle\|\nabla^{2}u\|_{L^{\infty}(B_{K\sqrt{(t-T)+(R/K)^{2}}})}$
$\displaystyle\leq
C\|u\|_{L^{\infty}(B_{r(t)})}^{\frac{1}{2}+\frac{1}{2n+6}}\|\nabla^{(n+6)}u\|_{L^{\infty}(B_{r(t)})}^{\frac{1}{2}-\frac{1}{2n+6}}\leq
C\varepsilon_{0}^{1+\frac{1}{n+3}}$

on a slightly smaller ball as well. When $\varepsilon_{0}$ is sufficiently
small, we have $\|u\|_{C^{2}(B_{r(t)})}\leq\varepsilon_{0}$. Thus we obtain
Lemma 4.7. ∎

Let us conclude this section with the following remark.

######  Remark 4.10.

Once we obtain the $O(\sqrt{t})$ graphical scale, we can repeat all the
analysis in Section 3 with this larger graphical scale. All the results hold
with $O(t^{-(1+\alpha)})$ replaced by $O(e^{-K^{2}t})$. Nevertheless, because
the nature of the Ricatti ODE (3.8), the $H_{1}$-normal form still have
remainder $o(1/t)$ rather than exponential.

##  5\. Denseness and stability of nondegenerate singularities

In this section, we study the perturbed RMCF and give the proof of Theorem 1.8
and 1.9.

###  5.1. The setup of the proof

Let us consider a perturbation of $M_{0}$ given by
$\widetilde{M}^{\eta}_{0}:=\\{x+\eta v_{0}(x)\mathbf{n}(x)\ |\ x\in M_{0}\\}$.
Taking $\eta$ sufficiently small, the RMCF $\widetilde{M}^{\eta}_{t}$ starting
from $\widetilde{M}^{\eta}_{0}$ remains $o(\eta)$-close to the unperturbed
RMCF $M_{t}$ for $t\in[0,T]$ (Proposition 3.3 of [SX1]).

Evolving the RMCF $\widetilde{M}^{\eta}_{t}$ and writing it as the graph of a
function over $\Sigma^{k}\cap B_{\mathbf{r}(T)}$, it is most likely that the
Fourier modes with positive eigenvalues of $L$ grows exponentially, which may
quickly dominate the $h_{2}$-component that we added by the perturbation at
time $T$ by applying the last lemma. This is related to an essential drawback
of RMCF is that it can only detect a single spacetime. If the RMCF zoom in at
a point that is not a singularity, the RMCF will not converge to a self-
shrinker. In our case, a perturbation can shift the spacetime position of the
singularity. Thus we need to keep track of the spacetime position of the
singularity, to make sure that we are zooming in at the correct point.

In terms of the local dynamics of the RMCF near the shrinker, positive Fourier
modes correspond to rigid transformations and have exponential growth, and we
can choose proper rigid transformations to modulo them. In particular, by
Remark 2.1, the presence of $h_{2}$ term can be used to kill the $y_{i}$-modes
by a translation in the axis direction of the cylinder.

We use $\mathcal{T}$ to denote the group of transformations generated by
translations, rotations and dilations. It is natural to equip $\mathcal{T}$
with vector norm because $\mathcal{T}$ is a subgroup of the direct sum of the
affine group and the dilation group of the Euclidean space. We will only
consider the connected component with the identity element id. We use
$\mathfrak{t}$ to denote the Lie algebra of $\mathcal{T}$. Recall the function
space $E=H^{1}/\mathfrak{so}(n+1)$ and the cones on $E$ in Section 3.1.

We have the following proposition whose proof is postponed to Section 5.4.

######  Proposition 5.1.

For any $K>0$, there exists $R_{0}$ sufficiently large, such that any
$R>R_{0}$ has the following significance. Let $u$ be a function on
$\Sigma^{k}\cap B_{R+10}$ satisfying $\|u\|_{C^{2}}\leq\varepsilon_{0}$ for
some sufficiently small $\varepsilon_{0}$ and
$\|\Pi_{h_{2}(y_{i})}\chi_{R}u\|\geq\kappa\|\chi_{R}u\|$ for some $\kappa>0$
and some $1\leq i\leq k$, and $\|\chi_{R}u\|\geq Ce^{-R^{2}/4}\|u\|_{C^{2}}$.
Suppose $M$ is the graph of function $u$. Then there exists an element
$\mathbf{T}\in\mathcal{T}$ such that $\|\mathbf{T}\|\leq C\|\chi_{R}u\|$ and
$\mathbf{T}(M)$ is a graph of function $v$ over $\Sigma^{k}\cap B_{R+9}$
satisfying

(5.1)
$\Pi_{y_{i}}(\chi_{R}v)=\Pi_{\theta_{j}}(\chi_{R}v)=\Pi_{\theta_{j}y_{\ell}}(\chi_{R}v)=\Pi_{1}(\chi_{R}v)=0,\quad\forall\
j,\ell\in\\{1,\ldots,k\\}$

and $\|\Pi_{h_{2}(y_{i})}\chi_{R}v\|\geq\kappa\|\chi_{R}v\|$.

Here (5.1) means that all Fourier modes $1,\theta_{j},\theta_{j}y_{\ell}$
corresponding to rigid transformations can be eliminated. Moreover, the mode
$y_{i}$ can also be eliminated in the presence of a nontrivial
$h_{2}(y_{i})$-component (see Remark 2.1).

To prove the denseness of nondegenerate singularities, we need to understand
how to perturb a degenerate singularity to obtain a nondegenerate singularity.
From the above discussion, we need to find perturbations so that after long
time, if we write the perturbed RMCF over the original RMCF as a graph,
$h_{2}(y_{i})$ becomes dominant. Let us first recall Lemma 5.2 of [HV1] and
its $H^{1}$-version proved in [SX3].

######  Lemma 5.2.

Let $\mathcal{S}(t,s)$ be the fundamental solution to the variational equation
$\partial_{\tau}v=L_{M_{\tau}}v$ such that the function
$\mathcal{S}(\tau,s)v(s)$ solve the equation with initial condition $v(s):\
M_{s}\to\mathbb{R}$. Then the operator $\mathcal{S}(t,0):\ L^{2}(M_{0})\to
H^{p}(M_{t})$ has dense image.

We refer the readers to [SX3, Appendix B] for a proof of Lemma 5.2 for $p=1$,
and the $p>1$ case is similar. We shall first apply this lemma to the
variational equation $\partial_{t}v=L_{M_{t}}v$ to find a initial condition
$v_{0}:\ M_{0}\to\mathbb{R}$ such that $\mathcal{S}(T,0)v_{0}$ lies in the
cone $\mathcal{K}_{0}(\kappa)$ for $\kappa$ close to 1 and for some $T$
sufficiently large such that $M_{T}$ is written over $\Sigma^{k}\cap
B_{\mathbf{r}(T)}$ as a function of $u$ with
$\|u\|_{C^{2}}\leq\varepsilon_{0}$. We will show that the perturbed RMCF graph
function will keep staying in the cone, after appropriate translation,
dilation and rotation.

###  5.2. Denseness of nondegenerate singularities

In this section we prove Theorem 1.8 assuming Proposition 5.1. As we have
explained in the introduction, the proof differs quite a lot from the
perturbation argument as in [SX1, SX2, SX3], due to the presence of the
$y_{i}$-eigenmodes. The difficulty is overcome by combining the observation in
Remark 2.1 as well as an argument applying a centering map for infinitely many
steps along the flow. Moreover, the neutral eigenspace is in general not a one
dimensional space, but essentially $k$ dimensions, spanned by $h_{2}(y_{i})$
for $i=1,2,\dots,k$. Thus, it is possible that the original RMCF is degenerate
in some directions of $y_{i}$, and nondegenerate in some other directions
$y_{j}$. Thus we need to understand how the perturbation could make both
nondegenerate directions and degenerate directions become nondegenerate.

###### Proof of Theorem 1.8.

Let us assume that $M_{t}$ is the RMCF zooming in at $(0,0)$, corresponds to a
MCF $\mathbf{M}_{\tau},\ \tau\in[-1,0)$. Moreover, we assume the leading term
of $M_{t}$ is $\sum_{i\in\mathcal{I}}\frac{\varrho}{4t}(y_{i}^{2}-2)$, where
$\mathcal{I}$ is a proper subset of $\\{1,2,\dots,k\\}$. To proceed, without
loss of generality and for simplicity, we assume that $k=2$ and the normal
form of $M_{t}$ has leading term $\frac{\varrho}{4t}(y_{2}^{2}-2)$. In other
words, the $y_{1}$-direction is degenerate and the $y_{2}$-direction is
nondegenerate.

Before we begin the proof, let us make two remarks: First, the case of all the
directions being degenerated is simpler than this mixed type problem, so we
only discuss the more difficult mixed case, i.e. some directions are
degenerate, others are nondegenerate here. Indeed, the totally degenerate case
can be treated in the same way as the proof of Theorem 1.9 in the next
subsection. Second, we assume that the neutral mode contains no non-diagonal
elements, i.e. those terms $y_{i}y_{j}$. The general case can be treated as
the proof of Lemma 3.7 in Section 3.3.

Step 1, construct the initial perturbation.

Throughout proof, $\chi=\chi(t)$ is a smooth cut-off function that is constant
$1$ on $\Sigma^{k}\cap B_{K\sqrt{t}-1}$ and $0$ outside $\Sigma^{k}\cap
B_{K\sqrt{t}}$. When a hypersurface $M_{t}$ is close to $\Sigma^{k}\cap
B_{K\sqrt{t}}$, we can just pushforward $\chi$ to be a function over $M_{t}$.

Let $\varepsilon_{0}$ be a small number as in Proposition 5.1 and suppose $T$
is a large time such that $M_{T}\cap B_{K\sqrt{T}}$ can be written as a graph
of $u:\ \Sigma^{k}\cap B_{K\sqrt{T}}\to\mathbb{R}$ with
$\|u\|_{C^{2}}\leq\varepsilon_{0}$. We first apply Lemma 5.2 to find some
initial condition $v_{0}$ of the variational equation
$\partial_{t}v=L_{M_{t}}v$, for some fixed small $\xi>0$ such that $v(T)$
satisfies $\|\Pi_{h_{2}(y_{1})}\chi v(T)\|\geq(1-\xi)\|\chi v(T)\|$, and
because $C^{2}\hookrightarrow H^{p}$, we may choose $v(T)$ such that
$\|v\|_{C^{2}}\leq(K^{2}T)\|\chi v\|\ll C^{-1}e^{K^{2}T}\|\chi v\|$. Here
$\bar{v}$ is the pullback of $v$ from $M_{T}\cap B_{K\sqrt{T}}$ to
$\Sigma^{k}\cap B_{K\sqrt{T}}$. Now we use the variational equation to
approximate the RMCF. Suppose $\widetilde{M}^{\eta}_{t}$ is the RMCF starting
from $\\{x+\eta v_{0}\mathbf{n}(x)\ |\ x\in M_{0}\\}$. Then
$\widetilde{M}^{\eta}_{T}$ can be written as a graph of function
$\eta\tilde{v}(T)$ over $M_{T}$, and $\|\eta\tilde{v}(T)-\eta
v(T)\|_{C^{2}}\leq C\eta^{1+\sigma_{0}}$ for some $\sigma_{0}>0$ (Proposition
3.3 of [SX1]). Next, we know that the function $\eta\tilde{v}$ satisfies (3.4)
and (3.5) when pulled back to $\Sigma^{k}\cap B_{K\sqrt{t}}$, whenever $t>T$
and $\|\eta\tilde{v}\|_{C^{2}}$ is sufficiently small.

Step 2, use the rigid transformation to eliminate all Fourier modes with
positive eigenvalues.

From now on we fix a sufficiently small $\eta$ and omit the superscript
$\eta$. Because both $M_{T}$ and $\widetilde{M}_{t}$ are close to
$\Sigma^{k}\cap B_{K\sqrt{T}}$, from our construction of the perturbation,
$\widetilde{M}_{T}$ can be written as a graph of function $\tilde{u}$ over
$\Sigma^{k}\cap B_{K\sqrt{T}}$ with $\|\Pi_{h_{2}(y_{1})}\chi
w\|\geq(1-3\xi)\|\chi w\|$ for $w=\tilde{u}-u$ (details for the difference of
two graphs that are close to each other can be found in [SX3, Appendix C] and
[SX1, Appendix C]). In the following we take $1-3\xi=\kappa$ as in Proposition
5.1. We shall run the two equations (c.f. (3.4) and (3.2) respectively)

(5.2) $\partial(\chi w)=L(\chi w)+\delta\mathcal{B}(\chi
w)\quad\mathrm{and}\quad\partial(\chi u)=L(\chi u)+\mathcal{B}(\chi u)$

simultaneously. We shall verify the two cone conditions

(5.3) $\|\Pi_{h_{2}(y_{1})}\chi w\|\geq\kappa\|\chi
w\|\quad\mathrm{and}\quad\|\Pi_{h_{2}(y_{2})}\chi u\|\geq\kappa\|\chi u\|$

at each integer moment $n$ sufficiently large.

When the first cone condition in (5.3) is violated. We first apply Proposition
5.1 to perform a rigid transformation to eliminate all the Fourier modes of
$\chi w$ with positive eigenvalues except possibly $y_{2}$ (the eigenfunction
$y_{1}$ can be eliminated due to the presence of $h_{2}(y_{1})$ by the cone
condition at time $n-1$). However, it may happen that the $y_{2}$ component
cannot be killed by the rigid transformation since the first cone condition in
(5.3) does not guarantee a nondegenerate $h_{2}(y_{2})$-component in $\chi w$.
In this situation, we introduce

$\bar{w}(n)=w(n)-\Pi_{y_{2}}w(n)\quad\mathrm{and}\quad\bar{u}(n)=u(n)+\Pi_{y_{2}}w(n).$

We continue to run the two equations (5.2) with initial conditions
$\chi\bar{w}(n)$ and $\chi\bar{u}(n)$ respectively. Note that we have
$\tilde{u}(n)=w(n)+u(n)=\bar{w}(n)+\bar{u}(n)$ remains the graphical function
of the perturbed RMCF $\widetilde{M}_{t}$ satisfying the RMCF equation.

For $\bar{u}(n)$, we verify the second cone condition in (5.3). If it is
violated, we apply Proposition 5.1 to perform a rigid transformation to
eliminate all the Fourier modes of $\chi\bar{u}(n)$ with positive eigenvalues
including $y_{2}$.

In this way, we guarantee both cone conditions (5.3) by composing two rigid
transformations at each time $n$.

Step 3, study the dynamics projected to $E^{0}$.

With the cone conditions (5.3), we study the $\chi w$-equation in (5.2)
projected to the $E^{0}$ subspace. Suppose $\Pi_{0}\chi
w=a_{1}(t)h_{2}(y_{1})+a_{2}(t)h_{2}(y_{2})$, then we have (3.4)

$\partial_{t}(\chi w)=L(\chi w)+P(\chi w)+\mathcal{E},\quad
P=-\int_{0}^{1}D_{u}\mathcal{B}(su+(1-s)\bar{u})ds=-\int_{0}^{1}D_{u}\mathcal{B}(u-sw)\,ds.$

Since $w$ is the perturbation, in the following we shall also assume $\|\chi
w\|_{C^{2}}\leq\|\chi u\|_{C^{2}}$, otherwise, $\tilde{u}$ has already grew
large enough such that $h_{2}(y_{i})$ is very large for $i=1,2$. Note from
Proposition A.1, we have that
$\mathcal{B}=-\frac{1}{\varrho}\left(\frac{u^{2}}{2}+2u\Delta_{\theta}u+|\nabla_{\theta}u|^{2}\right)+O(J^{2}u)^{3}$.
Thus we get

$\partial_{t}(\chi w)=L(\chi w)+(u-\frac{1}{2}(\chi w))(\chi
w)+O(J^{2}u)^{2}(J^{2}(\chi w))+\mathcal{E}.$

Using Proposition 4.8 we bound the $L^{2}$ norm of the boundary term
$\mathcal{E}$ by $\|\chi w\|t^{C}e^{-K^{2}t}$. When projected to the
$h_{2}(y_{1})$\- and $h_{2}(y_{2})$-components, we get the following system of
ODEs (see proof of Lemma 3.7 for the quadratic term in $\mathcal{B}$ and Item
(3) of Proposition A.1 for higher order terms)

$\begin{cases}\dot{a}_{1}&=(O(\frac{1}{t^{2}})-\frac{1}{2}a_{1})a_{1}+O((|a_{1}|+|a_{2}|)/t^{2})\\\
\dot{a}_{2}&=\frac{1}{t}a_{2}-\frac{1}{2}a_{2}^{2}+O((|a_{1}|+|a_{2}|)/t^{2}).\end{cases}$

Choosing $\kappa$ sufficiently close to 1, we have $|a_{1}|\gg|a_{2}|$. Note
that in the first equation, if $a_{1}$ is very small, negligible, then the
equation is approximated by $\dot{a}_{1}=\frac{1}{t^{2}}a_{1}$, whose solution
has the form $a_{1}(t)=Ce^{-\frac{1}{t}}$, i.e. the solution stabilizes, thus
as $t$ is large, the behavior of the solution is dominated by
$\dot{a}_{1}=-a_{1}^{2}$ and the solution eventually stabilizes to
$\frac{1}{\varrho t}$. For the second equation, the solution to the equation
$\dot{a}_{2}=\frac{1}{t}a_{2}$ is $a_{2}(t)=-\frac{a_{2}(0)}{t}$ where
$|a_{2}(0)|\ll a_{1}(0)$. Thus eventually, the solution $a_{2}$ would be
negligible compared to $a_{1}$. Thus we have added a $h_{2}(y_{1})$-component
eventually comparable with the $h_{2}(y_{2})$-component in $u(t)$.

Finally, for each rigid transformation $\mathbf{T}_{n}$ at time $n$ we have

$\|\mathbf{T}_{n}-\mathrm{id}\|\leq C(\|\chi w(n)\|+\|\chi u(n)\|)\leq
C\varepsilon_{0}$

by Proposition 5.1. We denote by $\widetilde{\mathbf{T}}_{n}$ the rigid
transformation of the initial condition of the MCF corresponding to
$\mathbf{T}_{n}$. Using the correspondence between MCF and RMCF, we have
$\|\widetilde{\mathbf{T}}_{n}-\mathrm{id}\|\leq
e^{-\frac{n}{2}}\|\mathbf{T}_{n}-\mathrm{id}\|$. This means the composition
$\cdots\circ\widetilde{\mathbf{T}}_{T+2}\circ\widetilde{\mathbf{T}}_{T+1}\circ\widetilde{\mathbf{T}}_{T}$
converges to some $\mathbf{T}_{\infty}$, and
$\|\mathbf{T}_{\infty}-\mathrm{id}\|$ is of order $O(\varepsilon_{0})$.

∎

###  5.3. Stability of nondegenerate singularities

In this section we prove Theorem 1.9. The proof is similar to and simpler than
the proof of Theorem 1.8, since the behaviour of $h_{2}(y_{i})$’s are uniform.

###### Proof of Theorem 1.9.

Suppose $\\{M_{t}\\}_{t\in[0,\infty)}$ is the RMCF corresponding to
$\mathbf{M}_{\tau},\ \tau\in[-1,0)$, zooming in at $(0,0)$. Let $\varepsilon
u_{0}:\ \mathbf{M}_{-1}\to\mathbb{R}$ be an initial perturbation for any $u$
with $\|u\|_{C^{2}}\leq 1$ and $\varepsilon$ sufficiently small. We consider
the RMCF $\widetilde{M}_{t}$ starting from
$\widetilde{M}_{0}:=\\{x+\varepsilon u_{0}(x)\mathbf{n}(x):x\in M_{0}\\}$.

The proof consists of the following two steps.

  1. (1)

The cone preservation property and the centering map.

  2. (2)

The slow dynamics on $E^{0}$.

By the normal form theorem and the cone theorem, for any $\varepsilon>0$
small, there exists $T_{0}>0$ such that for all $t>T_{0}$, the unperturbed
RMCF $M_{t}$ can be written as a graph over $\Sigma^{k}\cap B_{K\sqrt{t}}$ of
a function $u:\ \Sigma^{k}\cap B_{K\sqrt{t}}\to\mathbb{R}$ satisfying $\chi
u\in\mathcal{K}_{0}(1-\xi)$. For sufficiently small initial perturbation, the
perturbed RMCF $\widetilde{M}_{t}$ is written as the graph of a function
$w(t):\ \Sigma^{k}\cap B_{K\sqrt{t}}\to\mathbb{R}$ that is sufficiently close
to $u$, thus we have $\chi w\in\mathcal{K}_{0}(1-2\xi)$ for some fixed
$T>T_{0}$. Thus we get both $u(T)$ and $w(T)$ are in the cone
$\mathcal{K}_{0}(\kappa)$ for $1-2\xi>\kappa$.

To apply the cone theorem, we have to make sure that $w_{+}\to 0$ in the
$C^{\infty}_{loc}$ sense as $t\to\infty$. However, this is in general not
true. To fix this issue, we apply the centering map along with the flow
whenever the orbit $w(n)$ leaves the cone $\mathcal{K}_{0}(\kappa)$. For each
$n>T_{0}$, we introduce a $\widetilde{\mathbf{T}}_{n}$ in the group of rigid
transformation as follows.

  1. (1)

If $w(n)\in\mathcal{K}_{0}(\kappa)$, then $\widetilde{\mathbf{T}}_{n}=0$.

  2. (2)

If $w(n)\in\mathcal{K}_{0}(\kappa)$ but $w(n+1)\notin\mathcal{K}_{0}(\kappa)$,
then we apply Proposition 5.1 to choose a rigid transformation
$\widetilde{\mathbf{T}}_{n}$ such that centered MCF
$\widetilde{\mathbf{T}}_{n}\widetilde{\mathbf{M}}_{t}$ has corresponding RMCF
$\overline{M}_{n+1}$ at time $n+1$ written as a graph $\bar{w}(n+1):\
\Sigma^{k}\to\mathbb{R}$ with $\bar{w}(n+1)\in\mathcal{K}_{0}(\kappa)$.

Since $|w_{0}(n)|$ decays to zero like $1/t$, using the correspondence between
MCF and RMCF, we get the estimate
$|\widetilde{\mathbf{T}}_{n}|\leq\frac{e^{-t/2}}{t}$. This implies that the
composition of the sequence of centering maps converges. When $\varepsilon\to
0$, the centering map converges to identity.

We shall perform a step of centering map at each time $n$. Thus we always have
$\bar{w}(n)\in\mathcal{K}_{0}(\kappa)$. Thus we get for large $t$, the neutral
modes in $E^{0}$ dominate.

We next analyze the neutral dynamics in the finite dimensional space $E^{0}$.
We give a basis $(h_{2}(y_{1}),\ldots,h_{2}(y_{k}))$ of $E^{0}$. For the
unperturbed flow $M_{t}$, by the normal form, we see that the vector
$\Pi_{0}u(t)=\sum_{i=1}^{k}\frac{1}{t}h_{2}(y_{i})+O(1/t^{2})$, thus we get
that $\Pi_{0}u(t)$ almost points to the diagonal direction $(1,\ldots,1)$ in
$E^{0}$. To show that the perturbed flow $\bar{M}_{t}$ has also a
nondegenerate singularity, it is enough to show that $\Pi_{0}w(t)$ also
approaches the diagonal direction $(1,\ldots,1)$ in $E^{0}$. In other words,
it is enough to show that the solutions to the equations
$\dot{a}_{i}=-a_{i}^{2}+o(1/t^{2})$ is robust under small perturbation of
initial conditions. This is similar to the proof of Lemma 3.7 and we omit it
here.

Iterating the procedure, we get that $\overline{\mathbf{M}}_{t}:=\prod_{i\geq
n}\widetilde{\mathbf{T}}_{i}\widetilde{\mathbf{M}}_{t},t\in[-1,0)$ gives an
MCF with corresponding RMCF $\overline{M}_{t},t\in[0,\infty)$ converges to
$\Sigma^{k}$ in the $C^{\infty}_{loc}$-sense with leading terms in the normal
form $\sum_{i=1}^{k}\frac{1}{t}h_{2}(y_{i})$. Thus the perturbed MCF
$\overline{\mathbf{M}}_{t}$ admits a nondegenerate singularity at the
spacetime point $(0,0)$ modeled on $\Sigma^{k}$.

∎

###  5.4. Centering map and eliminating positive spectrums

Finally we prove Proposition 5.1. The main idea is as follows. The
infinitesimally small transformations are generated by eigenfunctions with
eigenvalues $\geq 0$ except $h_{2}(y_{i})$. Thus, on the linear level, by
performing a small rigid transformation, we can eliminate those corresponding
Fourier modes in the function $u$. In particular, by Remark 2.1, the Fourier
modes $y_{i}$ can be eliminated in the presence of $h_{2}(y_{i})$. Some
nonlinearity is created by the introduction of the cutoff $\chi$. We shall
apply the implicit function theorem to kill all the positive Fourier modes.

###### Proof of Proposition 5.1.

We use $v_{\mathbf{T}}$ to denote the graph of $\mathbf{T}(M)$ over
$\Sigma^{k}$. If we write $\\{\phi_{i}\\}_{i=1}^{\infty}$ to be the
eigenfunctions of $L$, then we define the map

$\Gamma:\mathcal{T}\to E^{+}\oplus
E^{0}\oplus\mathfrak{so}(n+1),\quad\mathrm{via}\
\mathbf{T}\mapsto\\{\langle\chi_{R}v_{\mathbf{T}},\phi_{i}\rangle\\}_{\\{\lambda_{i}\geq
0\\}}.$

Here we remark that

(5.4) $\displaystyle\langle f,\phi_{i}\rangle_{H^{1}}$ $\displaystyle=\int
f\phi_{i}e^{-\frac{|x|^{2}}{4}}+\int\langle\nabla f,\nabla\phi_{i}\rangle
e^{-\frac{|x|^{2}}{4}}=\int f(1-\mathcal{L})\phi_{i}e^{-\frac{|x|^{2}}{4}}$
$\displaystyle=\int
f(1-(L-1))\phi_{i}e^{-\frac{|x|^{2}}{4}}=(2-\lambda_{i})\int
f\phi_{i}e^{-\frac{|x|^{2}}{4}}=(2-\lambda_{i})\langle
f,\phi_{i}\rangle_{L^{2}}.$

So here we choose $\langle\cdot,\cdot\rangle$ to be the $L^{2}$ inner product
that can also be used to understand $H^{1}$ inner product.

Then we consider the linearization of $\Gamma$. For any
$V\in\\{0\\}\times\mathbb{R}^{k}$, $U\in\mathbb{R}^{n-k+1}\times\\{0\\}$,
$\gamma\in\mathbb{R}_{+}$ and $\mathcal{R}\in\mathrm{SO}(n+1)$, Notice that
$\mathbf{T}_{V}$, $\mathbf{T}_{U}$, $\mathbf{T}_{\gamma}$ and
$\mathbf{T}_{\mathcal{R}}$ are naturally associated with the eigenfunctions of
$L$ with nonnegative eigenvalues. For our purpose, we only need to consider
$\mathcal{R}$ that is generated by $y_{i}\theta_{\alpha}$ on the sphere, i.e.
those rotations which contribute nothing to the rotations of $\mathbb{R}^{k}$
or $\mathbb{R}^{n-k+1}$ themselves.

For simplicity, we use $r$ to denote $\varrho$. From Appendix, we have the
$\phi_{i}$ component of $D\Gamma$:

$(D_{V}\Gamma)_{\phi_{i}}=-\int\langle\chi\nabla
u,V\rangle\phi_{i}e^{-\frac{|x|^{2}}{4}},\quad(D_{U}\Gamma)_{\phi_{i}}=\int\chi\left(\langle
U,\theta\rangle-\frac{\varrho\langle\nabla
u,U^{\top}\rangle}{\varrho+u}\right)\phi_{i}e^{-\frac{|x|^{2}}{4}},$
$(D_{\gamma}\Gamma)_{\phi_{i}}=\int\chi(\varrho+u-\langle\nabla
u,y\rangle)\phi_{i}e^{-\frac{|x|^{2}}{4}},$
$\begin{split}(D_{y_{i}\theta_{\alpha}}\Gamma)_{\phi_{i}}=&\int\chi\left(y_{i}\theta_{\alpha}-\left\langle\nabla
u,(\varrho+u)\theta_{\alpha}y_{i}\operatorname{\partial}_{y_{i}}+\frac{\varrho}{\varrho+u}\theta_{\alpha}y_{i}(\Pi_{\theta_{\alpha}}\theta)\right\rangle\right)\phi_{i}e^{-\frac{|x|^{2}}{4}}.\end{split}$

We can see that $D\Gamma=A+O(e^{-R^{2}/4})\|u\|_{C^{1}}$, where
$O(e^{-R^{2}/4})$ comes from the cutoff function $\chi_{R}$ and $A$ is a
matrix satisfying

  1. (1)

the diagonal of $A$ are some constants independent of $R$, except for the
$D\Gamma(\mathbf{T}_{V})$ term;

  2. (2)

$D\Gamma(\mathbf{T}_{\operatorname{\partial}_{y_{i}}})$ projects to the
$\phi_{i}=y_{i}$ direction is given by
$\int\chi(u\frac{y_{i}^{2}-2}{2})e^{-\frac{|x|^{2}}{4}}$, which is exactly the
coefficient of the $h_{2}(y_{i})$ term of $\chi u$.

  3. (3)

The off-diagonal elements of $A$ are either zero, or with the scale
$\|\Pi_{\neq h_{2}}u\|_{L^{2}}$,

Indeed, for item (1), we notice that
$\nabla(\phi_{i}e^{-\frac{|x|^{2}}{4}})=(\nabla\phi_{i}-\phi_{i}\frac{y}{2})e^{-\frac{|x|^{2}}{4}}$,
and plugging in $\phi_{i}=1,\theta_{\alpha},y_{i},y_{i}\theta_{\alpha}$ we get
$e^{\frac{|x|^{2}}{4}}\nabla(\phi_{i}e^{-\frac{|x|^{2}}{4}})=-\frac{y}{2},$
$-\theta_{\alpha}\frac{y}{2},$
$(\operatorname{\partial}_{y_{i}}-y_{i}\frac{y}{2}),$
$(\theta_{\alpha}\operatorname{\partial}_{y_{i}}-\theta_{\alpha}y_{i}\frac{y}{2})$
respectively. Item (3) can be checked directly, e.g.

$\langle
D_{\operatorname{\partial}_{y_{i}}}\Gamma,1\rangle_{L^{2}}=\frac{1}{2}\Pi_{{y_{i}}}\chi
u,\ \langle
D_{\operatorname{\partial}_{y_{i}}}\Gamma,\theta_{\alpha}\rangle_{L^{2}}=\frac{1}{2}\Pi_{\theta_{\alpha}y_{i}}\chi
u,\ \langle
D_{\operatorname{\partial}_{\theta_{\alpha}y_{i}}}\Gamma,\theta_{\alpha}y_{i}\rangle_{L^{2}}=\frac{1}{2}\Pi_{\theta_{\alpha}h_{2}(y_{i})}\chi
u,$ $\langle D_{\gamma}\Gamma,y_{i}\rangle_{L^{2}}=(k+2)\Pi_{y_{i}}\chi
u+O(\|\Pi_{-}\chi u\|_{L^{2}}),\ \langle
D_{\gamma}\Gamma,\theta_{\alpha}\rangle_{L^{2}}=(k+1)\Pi_{\theta_{\alpha}}\chi
u+O(\|\Pi_{-}\chi u\|_{L^{2}}),$ $\langle
D_{\gamma}\Gamma,y_{i}\theta_{\alpha}\rangle_{L^{2}}=(k+1)\Pi_{y_{i}\theta_{\alpha}}\chi
u+O(\|\Pi_{-}\chi u\|_{L^{2}}).$

In conclusion, the linear operator $D\Gamma$ is invertible if $\chi
u\in\mathcal{K}_{0}(\kappa)$ and $\|\chi u\|\geq e^{-R^{2}}\|u\|_{C^{2}}$.
Then by implicit function theorem, there exists $\mathbf{T}\in\mathcal{T}$
such that $\mathbf{T}(M)$ is a graph of function $v$ over $\Sigma^{k}\cap
B_{R+9}$ satisfying (5.1). Also, we have the estimate

(5.5) $\|\mathbf{T}-\text{Id}\|\leq C(\|\Pi_{>0}\chi
u\|+e^{-R^{2}/4}\|u\|_{C^{2}}).$

We next estimate how the projection to $h_{2}(y_{i})$ would change under
$\mathbf{T}$. Recall that $\nabla
h_{2}(y_{i})=2ay_{i}\operatorname{\partial}_{y_{i}}$ for some constant $a$.
Then
$\nabla(h_{2}(y_{i})e^{-\frac{|x|^{2}}{4}})=(2ay_{i}\operatorname{\partial}_{y_{i}}-h_{2}(y_{i})\frac{1}{2}y)e^{-\frac{|x|^{2}}{4}}$.
Then we can calculate

$\langle
D_{\operatorname{\partial}_{y_{j}}}\Gamma,h_{2}(y_{i})\rangle=-(2a\delta_{ij}-\frac{1}{2})\Pi_{y_{i}}\chi
u+O(e^{-R^{2}/4})\|u\|_{C^{2}},\quad\langle
D_{\theta_{\alpha}}\gamma,h_{2}(y_{i})\rangle=O(e^{-R^{2}/4})\|u\|_{C^{2}},$
$\langle D_{\gamma}\Gamma,h_{2}(y_{i})\rangle=(k+1)\Pi_{h_{2}(y_{i})}\chi
u+O((1-\kappa)\|\chi u\|)+O(e^{-R^{2}/4})\|u\|_{C^{2}},$ $\langle
D_{y_{j}\theta_{\alpha}}\Gamma,h_{2}(y_{i})\rangle=O((1-\kappa)\|\chi
u\|)+O(e^{-R^{2}/4})\|u\|_{C^{2}}.$

Whenever $\phi_{i}$ is none of $1,\theta_{\alpha},y_{i},y_{i}\theta_{\alpha}$,
we can use the orthogonality and integration by parts to get

$\langle D_{V}\Gamma,\phi_{i}\rangle=-\int\langle V,\nabla(\chi
u)\rangle\phi_{i}e^{-\frac{|x|^{2}}{4}}+\int
u\phi_{i}\langle\nabla\chi,V\rangle e^{-\frac{|x|^{2}}{4}},$

$\langle D_{U}\Gamma,\phi_{i}\rangle=\int(1-\chi)\langle
U,\theta\rangle\phi_{i}e^{-\frac{|x|^{2}}{4}}-\int\frac{\varrho}{\varrho+u}\langle\nabla(\chi
u),U^{\top}\rangle\phi_{i}e^{-\frac{|x|^{2}}{4}}-\int\frac{\varrho
u}{\varrho+u}\langle\nabla\chi,U^{\top}\rangle\phi_{i}e^{-\frac{|x|^{2}}{4}},$

$\langle
D_{\gamma}\Gamma,\phi_{i}\rangle=\int(1-\chi)\varrho\phi_{i}e^{-\frac{|x|^{2}}{4}}+\int\chi
u((k+1)\phi_{i}+\langle y,\nabla\phi_{i}\rangle)e^{-\frac{|x|^{2}}{4}}+\int
u\phi_{i}\langle\nabla\chi,y\rangle e^{-\frac{|x|^{2}}{4}}$

$\begin{split}\langle
D_{y_{i}\theta_{\alpha}}\Gamma,\phi_{i}\rangle=&\int(1-\chi)y_{i}\theta_{\alpha}\phi_{i}e^{-\frac{|x|^{2}}{4}}-\int\phi_{i}\left\langle\nabla(\chi
u),(\varrho+u)\theta_{\alpha}y_{i}\operatorname{\partial}_{y_{i}}+\frac{\varrho}{\varrho+u}\theta_{\alpha}y_{i}(\Pi_{\theta_{\alpha}}\theta)\right\rangle
e^{-\frac{|x|^{2}}{4}}\\\
&-\int\phi_{i}u\left\langle\nabla\chi,(\varrho+u)\theta_{\alpha}y_{i}\operatorname{\partial}_{y_{i}}+\frac{\varrho}{\varrho+u}\theta_{\alpha}y_{i}(\Pi_{\theta_{\alpha}}\theta)\right\rangle
e^{-\frac{|x|^{2}}{4}}\end{split}$

Clearly, there are several terms showing up:

  * •

terms with $\nabla(\chi u)$, which can be bounded by the $H^{1}$-norm of the
projection of $\chi u$ to $\phi_{i}$ or $\phi_{i-1}$. Here we recall that
taking $y_{j}$-derivative of an eigenfunction may either eliminate the
eigenfunction, or raise the eigenvalue by $1/2$.

  * •

terms with $(1-\chi)$. These terms can be bounded by the $L^{2}$-norm of the
projection of $(1-\chi)f$, where $f$ is some fixed function appearing above.
Because $\chi$ is the cutoff function, $(1-\chi)f$ has $L^{2}$-norm bounded by
$e^{-R^{2}/4}\|u\|_{C^{2}}$.

  * •

terms with $\nabla\chi$. These terms can be bounded by the $L^{2}$-norm of the
projection of a function supported on $B_{R+1}\backslash B_{R}$. Such function
has $L^{2}$-norm bounded by $e^{-R^{2}/4}\|u\|_{C^{2}}$.

In the above calculations we are using $L^{2}$-inner product. By (5.4), we get
the $H^{1}$ inner product $\langle D\Gamma,\phi_{i}\rangle_{H^{1}}$ is of the
order $C(\|\chi u\|+e^{-R^{2}/4}\|u\|_{C^{2}})\leq C\|\chi u\|$.

As a consequence, if we write $\mathbf{T}(M)$ as a graph of $v$ over $B_{R}$,
then we have

(5.6) $\displaystyle\|\Pi_{h_{2}(y_{i})}\chi v\|$
$\displaystyle\geq\|\Pi_{h_{2}(y_{i})}\chi u\|-C(\|\Pi_{>0}\chi
u\|+e^{-R^{2}/4}\|u\|_{C^{2}})\|\chi u\|$
$\displaystyle\geq(\kappa-C\varepsilon_{0})\|\chi u\|\quad\text{and}$
$\displaystyle\|\Pi_{<0}\chi v\|$ $\displaystyle\leq\|\Pi_{<0}\chi
u\|+C(\|\Pi_{>0}\chi u\|+e^{-R^{2}/4}\|u\|_{C^{2}})\|\chi u\|$
$\displaystyle\leq\|\Pi_{\neq u}\chi u\|+\varepsilon_{0}\|\chi
u\|\leq(1-\kappa+\varepsilon_{0})\|\chi u\|.$

Thus, when $\varepsilon_{0}$ is sufficiently small, we get
$\|\Pi_{h_{2}(y_{i})}\chi v\|\geq\kappa\|\chi v\|$. ∎

##  6\. Applications

In this section, we give some applications of our main result to level set
flows, rotational graphs, etc.

###  6.1. Regularity of level set flow

In this section we prove Theorem 1.14. We first prove the following lemma:

######  Lemma 6.1.

If the $\\{\mathbf{M}_{\tau}\\}$ is a MCF with a nondegenerate singularity
that is modeled by $\Sigma^{k}$ with $k\geq 1$, then $\\{\mathbf{M}_{\tau}\\}$
has at least two distinct singular time.

###### Proof.

Without loss of generality, assume $(0,0)$ is the nondegenerate singularity of
$\\{\mathbf{M}_{\tau}\\}$. We show that $\mathbf{M}_{\tau}$ is nonvanishing
after time $0$. Recall that in Proof of Theorem 1.5 assuming Theorem 1.3, we
prove that thers exists $\delta>0$ such that
$\mathbf{M}_{\tau}\cap(B_{\delta}(0)\backslash\\{0\\})$ is smooth for
$\tau\in[-1,0]$ for some $\delta>0$. This shows that $\mathbf{M}_{\tau}$ is
nonvanishing after time $0$, and hence $\mathbf{M}_{\tau}$ has at least two
distinct singular times. ∎

###### Proof of Theorem 1.14.

Let us define $\mathcal{U}_{1}$ to be the set of mean convex hypersurfaces
that has a single spherical singularity, and the spherical singularity is the
most generic one in the sense of [SX3]. Recall we define $\mathcal{U}_{2}$ to
be the set consisting of mean convex closed hypersurfaces such that the level
set flows starting from these hypersurfaces have a nondegenerate cylindrical
singularity.

The openness of $\mathcal{U}_{1}$ is proved in [SX3], and the openness of
$\mathcal{U}_{2}$ is proved by the stability of nondegenerate cylindrical
singularity Theorem 1.9. Hence
$\mathcal{U}=\mathcal{U}_{1}\cup\mathcal{U}_{2}$ is open. Next we prove
denseness. Given a mean convex hypersurface $\Sigma$, the first time
singularity is either spherical or cylindrical. If the first-time singularity
is spherical, from [SX3], $\Sigma\in\overline{\mathcal{U}_{1}}$. If the first-
time singularity is spherical, the genericity Theorem 1.8 shows that
$\Sigma\in\overline{\mathcal{U}_{2}}$. Then we proved the denseness of
$\mathcal{U}$.

Finally, for hypersurfaces in $\mathcal{U}_{1}$, the level set flow starting
from them has $C^{2}$ but not $C^{3}$ regularity argued as [SX3]; for
hypersurfaces in $\mathcal{U}_{2}$, the level set flow starting from them has
at least two distinct singular time by Lemma 6.1, hence by [CM5] the level set
flow has $C^{1,1}$ but not $C^{2}$ regularity. Then the proof is concluded. ∎

###  6.2. Generic isolated first time singularities

###### Proof of Theorem 1.12.

For any closed mean convex hypersurface $\mathbf{M}$ and any $\varepsilon>0$,
we show that in a $\varepsilon$-$C^{2}$ neighbourhood of $\mathbf{M}$, there
is a hypersurface $\widetilde{\mathbf{M}}$ has property (isolated-or-
firecracker). We apply the perturbations to $\mathbf{M}$ in Theorem 1.8 with
initial perturbation $C^{2}$-norm less than $\varepsilon_{1}$,
$\varepsilon_{2}$, $\varepsilon_{3}$, $\cdots$, to perturb a degenerate
singularity to a nondegenerate one. If any one of the perturbations such that
the first-time singularities of the perturbed flow are all nondegenerate, then
we are done.

Otherwise, we choose $\varepsilon_{i}<2^{-i}\varepsilon$, and we also choose
the perturbation which does not change the nondegeneracy of any existing
nondegenerate singularities. Then we will get a sequence of perturbations,
which generate a sequence of nondegenerate singularities, and each one of them
is isolated. By the semi-continuity of Gaussian density, a subsequence of
these nondegenerate singularities converges to a limit singularity, which is
by definition a firecracker. ∎

###  6.3. Generic rotational symmetric MCF has first-time type I singularity

Combining with the analysis of MCF starting from a rotational graph by [AAG],
in this section, we prove a generic type I singularity theorem for rotational
graphs. Altschuler-Angenent-Giga [AAG] studied the evolution of $\Gamma(t)$
with $\Gamma_{0}$ being a smooth compact hypersurface that is a rotational
graph.

######  Theorem 6.2 (Theorem 1.1, Theorem 1.2 in [AAG]).

The evolution of $\Gamma(t)$ has the following properties:

  1. (1)

Outside $x$-axis, $\Gamma(t)$ is always smooth.

  2. (2)

There is a finite sequence of times $0=t_{0}<t_{1}<\dots<t_{l}$ such that the
hypersurfaces $\Gamma(t)$ are smooth and compact, including the $x$-axis, when
$t_{j-1}<t<t_{j}$, $j=1,2\dots,l$. The solution is empty for $t>t_{l}$.

  3. (3)

Let $\Gamma_{*}(t)$ be a connected component of $\Gamma(t)$ that becomes
singular at time $t_{j}$. Then either $\Gamma_{*}(t)$ shrink to a point on the
$x$-axis, or one or more necks of $\Gamma_{*}(t)$ have been pinched.

  4. (4)

The number of singularities is no more than the number of positive local
maxima and minima of the function $r=u(x)$ which defines the initial
hypersurfaces.

Here item (4) is crucial for our application. In the previous section we have
seen that generically the first singular set either consists of nondegenerate
singularities, or there exists a firecracker: a singularity that is the limit
of nondegenerate singularities. Item (4) here rules out the second
possibility. To apply item (4), we need the initial rotational graph function
to have finitely many local maxima and minima. The following lemma shows that
it is easy to find such rotational graphs. Recall that $\mathscr{R}$ is the
space of rotational graphs.

######  Lemma 6.3.

There is an open and dense subset of $\mathscr{R}$ such that the function
generating a smooth closed rotational graph has finitely many critical points.

###### Proof.

Openness is a consequence of the continuity of derivatives for $C^{\infty}$
perturbations. Now we prove denseness. Suppose $\Gamma$ is a smooth closed
rotational graph with function $u(x)$, that is defined on $[a,b]$. We need to
show that there exists an arbitrarily close rotational graph that has finitely
many critical points. If $u$ itself has finitely many critical points, then we
are done. Now we consider that $u$ has infinitely many critical points.
Because $\Gamma$ is smooth and closed, we have $\lim_{x\to a^{+}}u(x)=\infty$,
$\lim_{x\to b^{-}}u(x)=-\infty$. Thus we may assume the critical points of $u$
are all lying between $[a+\delta,b-\delta]$ for some $\delta>0$.

Now we consider a smooth cut-off function $\varphi$, that is supported on
$[a+\delta/2,b-\delta/2]$, strictly monotone increasing on
$[a+\delta/2,(a+b)/2]$ and strictly monotone decreasing on
$[(a+b)/2,b-\delta/2]$. Then by the standard argument as the existence of
Morse function, $u+s\varphi$ has isolated (hence finitely many) critical
points for a.e. $s\in\mathbb{R}$. This implies the denseness. ∎

Now we give the proof of Theorem 1.15.

###### Proof of Theorem 1.15.

Let us start with openness, which is the easier part. Suppose $\Gamma_{0}$ is
a smooth closed rotational graphical hypersurface, with MCF $\Gamma(t)$. If
the first singularity of $\Gamma_{0}$ is spherical, then Huisken’s theorem
shows that for any nearby smooth closed rotational graphical hypersurface
$\widetilde{\Gamma}_{0}$, the MCF $\widetilde{\Gamma}(t)$ also has the first
singularity to be spherical.

If the first-time singularities of $\Gamma_{0}$ are nondegenerate cylindrical,
our Theorem 1.9 shows that after a small perturbation, they are still
nondegenerate cylindrical.

Next we prove the closedness. Suppose $\Gamma_{0}$ is a smooth closed
rotational graphical hypersurface, with MCF $\Gamma(t)$. If the first
singularity of $\Gamma_{0}$ is spherical, then we are done. So we consider the
first singularities of $\Gamma_{0}$ are cylindrical. By Lemma 6.3, we may
assume that the graph generating $\Gamma_{0}$ has finitely many critical
points.

Now we apply the perturbation argument (denseness part) in Section 5, but only
restricted to the subspace of $L^{2}_{2}$ and $H^{1}_{2}$ that are
$SO(n)$-invariant. Here we use the fact that the rotational symmetry is
preserved by MCF, and the linearized equation is also rotational symmetric.
Moreover, the eigenfunction $h_{2}$ is also rotational symmetric. So the
arguments in Section 5 still apply here.

We perturb $\Gamma_{0}$ with respect to each one of its degenerate
singularity. Because the number of local maxima and local minima of a function
does not change under small perturbation of the function, and each time when
we apply the perturbation, we have at least one more nondegenerate cylindrical
singularity, so after finitely many times of perturbations, we get a
rotational graph $\widetilde{\Gamma}_{0}$, such that the first-time
singularities of the MCF $\widetilde{\Gamma}(t)$ are nondegenerate
cylindrical.

By Huisken’s theorem [Hu1], a spherical singularity must be type-I. By Theorem
1.6, a nondegenerate cylindrical singularity must be type-I. Then we conclude
the proof. ∎

###  6.4. Pinch a thin ring

In this section we show how to pinch a thin ring near the singular time.

###### Proof of Theorem 1.17.

In fact, we are able to choose the location to pinch the ring. Pick any point
$p$ on the singular set $\gamma$. Up to a translation, we may assume $p$ is
the origin.

Let $\\{M_{t}\\}_{t\in[0,\infty)}$ be the RMCF zooming in at $(0,0)$. By
Corollary 3.2 in [CM3], given $R>0,\varepsilon>0$, there exists $T>0$ that is
sufficiently large, such that for any $t>T$, $q\in e^{t/2}\gamma$, $M_{t}\cap
B_{R}(q)$ is a graph over a cylinder (possibly after a rotation), with graph
function has $C^{1}$-norm less than $\varepsilon$. Because $\gamma$ is a
continuous curve, the Fourier mode of the RMCF can not be dominated by
$h_{2}(y)$.

Now let us choose $f$ that is close to $\frac{1}{t}(y^{2}-2)$ after pullback
to the cylinder, in $B_{K\sqrt{t}}(0)$, where $K$ is chosen such that
$K\sqrt{t}$ is the graphical radius of the RMCF. Outside $B_{K\sqrt{t}+1}(0)$,
we choose $f=K^{2}$, and we smoothify $f$ in $B_{K\sqrt{t}+1}(0)\backslash
B_{K\sqrt{t}}(0)$.

Then the local dynamics show that the MCF would generate a nondegenerate
singularity in a neighbourhood of $(0,0)$. Moreover, this singularity is
isolated. Finally, by pseudolocality of MCF over the cylinder, when $T$ is
chosen sufficiently large, the singularity is generated before the MCF of the
graph over cylinder with radius $K+\sqrt{2}$ inside a ball of radius $R$ for
some sufficiently large $R$. This shows that the perturbation makes a single
singularity at the first singular time. ∎

##  Appendix A Nonlinear estimate

In this appendix, we derive the equation of motion for the graphical function
$u$ of a manifold evolving under RMCF approaching a cylinder
$\Sigma^{k}=\mathbb{S}^{n-k}(\varrho)\times\mathbb{R}^{k}$ where
$\varrho=\sqrt{2(n-k)}$ is the radius of the sphere. We will use coordinates
$x=(\theta,y)\in\Sigma^{k}$ where $\theta\in\mathbb{R}^{n-k+1}$ denotes point
on $\mathbb{S}^{n-k}(\varrho)$ and $y$ denotes point on $\mathbb{R}^{k}$. Note
$|\theta|=\varrho$. The following computations are locally around a point
$x_{0}=(\theta_{0},y_{0})$, and we choose local orthonormal frame
$\\{\theta_{\alpha}\\}$ and $\\{y_{i}\\}$. Greek letters correspond to the
spherical part and $i,j,k$’s correspond to the $\mathbb{R}^{k}$ part. We will