# On the rate of convergence of the rescaled mean curvature flow

Rory Martin-Hagemayer  and  Natasa Sesum

###### Abstract.

We estimate from above the rate at which a solution to the rescaled mean
curvature flow on a closed hypersurface may converge to a limit self-similar
solution, i.e. a shrinker. Our main result implies that any solution which
converges to a shrinker faster than any fixed exponential rate must itself be
shrinker itself.

###### Contents

  1. 1 Introduction
  2. 2 Geometric Quantities and Evolution Equations
    1. 2.1 Preliminary Calculations
    2. 2.2 The Differential Operator $\mathcal{L}$
    3. 2.3 Evolution of Geometric Quantities
      1. 2.3.1 The Normalizer Lemma
      2. 2.3.2 Evolution Equations
    4. 2.4 Second Derivatives
  3. 3 Energy Functional
    1. 3.1 Evolution of the Energy Functional
  4. 4 Dirichlet-Einstein Quotient
    1. 4.1 Evolution of the Dirichlet-Einstein Quotient
    2. 4.2 Order of Terms in $\dot{N}$
  5. 5 Controlling $\nabla^{l}S$
  6. 6 Rate of convergence

##  1\. Introduction

We say that a map $\bar{\varphi}:M\times[0,T)\to\mathbb{R}^{n+1}$ is a _mean
curvature flow_ starting from an embedding $\varphi_{0}:M\to\mathbb{R}^{n+1}$
provided

$\begin{cases}\frac{\partial\bar{\varphi}}{\partial\tau}(p,\tau)=H(p,\tau)\nu(p,\tau)\\\
\bar{\varphi}(p,0)=\bar{\varphi}_{0}(p)\par\end{cases}$ (1.1)

where $H(p,\tau)$ is the mean curvature at $\bar{\varphi}(p,\tau)$ and
$\nu(p,\tau)$ is the normal vector at $\bar{\varphi}(p,\tau)$. The mean
curvature flow can be extended in time so long as the norm of second
fundamental form $|A|^{2}$ remains bounded. In other words, the mean curvature
flow develops a singularity precisely if $|A|^{2}$ approaches infinity,
developing a singularity at some time $T$. There are several ways to rescale
mean curvature flow around a singularity. Given a mean curvature flow
$\tilde{\varphi}:M\times[-1,0)\to\mathbb{R}^{n+1}$, one can rescale by the
equations

$\tilde{\varphi}(\cdot,t)=\frac{1}{\sqrt{-\tau}}\bar{\varphi}(\cdot,\tau)$

where $t=-ln(-\tau)$. One can verify that this rescaled mean curvature flow
moves by the equation in the following definition.

######  Definition 1.1.

We say that $\tilde{\varphi}:M\times[0,\infty)\to\mathbb{R}^{n+1}$ is a
_rescaled mean curvature flow_ provided it moves by the equation

$\frac{\partial}{\partial
t}\tilde{\varphi}(p,t)=H(p,t)\nu(p,t)+\frac{\tilde{\varphi}(p,t)}{2}$ (1.2)

where $H(\cdot,t)$ is the mean curvature at $\tilde{\varphi}(\cdot,t)$
measured after rescaling.

If $M_{t}$ is a family of hypersurfaces moving by (1.2), and if as
$t\to\infty$ we have that the $\lim_{t\to\infty}M_{t}=\Sigma$, for some
compact nonempty surface $\Sigma$, then Huisken’s monotinicity formula implies
that $\Sigma$ is a shrinker, satisfying

$H+\frac{x\cdot\nu}{2}\equiv 0.$

In this paper we prove the following theorem.

######  Theorem 1.2.

Let $\widetilde{\varphi}:M\times[0,\infty)\rightarrow\mathbb{R}^{n+1}$ be a
solution to the rescaled mean curvature flow,

$\frac{\partial\widetilde{\varphi}}{\partial t}=\vec{H}+\frac{x}{2},$ (1.3)

such that we can find some $\tau_{1},\tau_{2}\in[-1,0)$ ($\tau_{1}<\tau_{2}$)
such that the corresponding (non-rescaled) mean curvature flow
$\varphi:M\times[\tau_{1},\tau_{2})\to\mathbb{R}^{n+k}$ is sufficiently close
in measure to the homothetically shrinking flow of a (closed) shrinker
$\Sigma$. Then there exists $T>0$ such that on $[T,\infty)$ we may express
$M_{t}$ as a normal graph of a function $v(p,t)$ over $\Sigma$. Furthermore
one of the following must hold.

  1. (1)

There exist $C,m>0$ such that $\|v(\cdot,t_{i})\|_{C^{2,\alpha}(\Sigma)}\geq
Ce^{-mt_{i}}$.

  2. (2)

$M_{t}$ moves by diffeomorphisms and is identically $\Sigma$.

This theorem is analogous to results achieved by Sun and Wang for the Kähler-
Ricci flow [14] and Kotschwar for the Ricci flow [10]. In the former paper Sun
and Wang proved that a Kähler-Ricci flow starting in a sufficiently small
$C^{k,\gamma}$ neighborhood of a Kähler-Einstein manifold of Einstein constant
1, then it must converge at a polynomial rate under their normalized Kähler-
Ricci flow to another Kähler-Einstein metric. In [10] Kotschwar showed that if
$(M,\bar{g})$ is a closed Ricci soliton and $g(t)$ is a smooth solution to the
normalized Ricci flow such that up to a sequence of diffeomorphisms
$\psi_{i}\in\text{Diff}(M)$ we have $\psi_{i}^{*}g(t_{i})\to\bar{g}$ in
$C^{k}_{\bar{g}}$ for all k and for some sequence of times $t_{i}\to\infty$,
then either $\|\psi_{i}g(t_{i})-\bar{g}\|_{C^{2}_{\bar{g}}}\geq Ce^{-mt_{i}}$
for some $m,C>0$ or there is a smooth family of diffeomorphisms
$\Phi_{t}\in\text{Diff}(M)$ such that $\Phi_{t}^{*}g(t)=\bar{g}$ for all
$t>0$.

The approach we take here is quite similar to that of Kotschwar, to whom our
result is more visibly similar. We use the work of Kotschwar, specifically his
energy functional approach, to draw insights into how to set up our proof of
this result.

We follow the general outline of the proof by contradiction used by Kotschwar
in [10] in the setting of Ricci flow. First we prove that we may express the
flow $M_{t}$ as a normal graph over the shrinker $\Sigma$ past some time
$T_{0}$. We use a theorem of Felix Schultze in [13] to establish the needed
graphicality. It allows us to use the Lojasiewicz inequality for mean
curvature flow found in [13], which comes from a graphical representation of
the flow. We use this to establish needed bounds on energy functional $E(t)$
that we introduce later in the text, and that are used in the proof of Theorem
1.2.

More precisely, assuming that Theorem 1.2 were false, we would have that for
every $C,m>0$, we eventually have

$\|{v(\cdot,t_{i})}\|_{C^{2,\alpha}(\Sigma)}<Ce^{-mt_{i}}$

for $i$ large enough, for our family of hypersurfaces $M_{t}$ given by the
graphs of $v(t)$ over $\Sigma$, with $M_{t}$ not being a shrinker. We then
show that this implies we may also find constants $C_{m}^{\prime}$ such that
for Huisken’s functional $\Omega(M)=\int_{M}e^{-\frac{|x|^{2}}{4}}d\mu$ we
have

$\Omega(M_{t_{i}})-\Omega(\Sigma)\leq C^{{}^{\prime}}_{m}e^{-mt_{i}}.$ (1.4)

We actually choose $E(t)$ so that recalling Huisken’s monotonicity formula we
can write

$\Omega(M_{t})-\Omega(M_{s})=\int_{t}^{s}E(\xi)d\xi.$

If we had some sort of exponential control of $E(t)$, like

$E(t)\geq E(T_{0})e^{-K^{\prime}(t-T_{0})},$ (1.5)

this would allow us to write (pushing $s\to\infty$ in Huisken’s monotonicity
formula) that

$\Omega(M_{t})-\Omega(\Sigma)\geq\frac{E(T_{0})}{K^{\prime}}e^{-K^{\prime}(t-T_{0})},$

which we could combine with (1.4) to write that

$E(T_{0})\leq Ce^{(K^{\prime}-m)(t_{i}-T_{0})}.$

If we had such an inequality holding for all $i$ and all $m$ large, we could
conclude that $E(T_{0})=0$, and thus that $M_{T_{0}}$ were a shrinker. In [8],
Hong Huang proved that the mean curvature flow satisfies a backwards
uniqueness property, so if $M_{T_{0}}$ were a shrinker, our flow would have to
be a shrinker for all times, which would contradict our original assumption.

Above discussion implies that we devote our efforts precisely towards proving
(1.5). Inspired by the proof in [10] our key tool for doing so will be
analyzing the Dirichlet-Einstein quotient, which we define below.

######  Definition 1.3.

Given a family of hypersurfaces moving by the normal rescaled mean curvatire
flow (2.1) or the recaled mean curvature flow (2) (which is not a shrinker),
we define the _Dirichlet-Einstein quotient_ $N(t)$ by the following formula

$N(t)=\frac{\dot{E}(t)}{E(t)}=\frac{d}{dt}\text{ln}(E(t)).$ (1.6)

The usefulness of the Dirichlet-Einstein quotient comes from the fact that if
we are able to bound the Dirichlet-Einstein quotient from below, that is, we
are able to prove that we have some $K^{\prime}>0$ such that

$N(t)\geq-K^{\prime}.$

This then implies

$\text{ln}(E(t))=\text{ln}(E(T_{0})+\int_{T_{0}}^{t}N(s)ds\geq\text{ln}(E(T_{0})-K^{\prime}(t-T_{0})$

and from this we are able to establish (1.5).

The outline of the paper is as follows. In section 2 we derive evolution
equation of quantities that we need to define our energy functionals and their
evolution equations in later sections. In section 3 we define the energy
functional $E(t)$ which is strongly linked to Huisken’s monotonicity formula
and compute its first and second derivative in time. In section 4 we introduce
Dirichlet-Einstein quotient $N(t)$, compute its rate of change in time and
prove the lower bound on $\dot{N}(t)$ in terms of $S$ and its derivatives.

##  2\. Geometric Quantities and Evolution Equations

To derive certain estimates, we will sometime consider the so called _normal
rescaled mean curvature flow_ (2.1)

$\frac{\partial{\varphi}}{\partial t}=\vec{H}+\frac{x^{\perp}}{2},$ (2.1)

which we show is equivalent to (1.1) up to tangential diffeomorphisms. In
order to make our computations simpler we will first understand how quantities
evolve on a hypersurface moving by (2.1). For convenience we define the
quantity $S=H+\frac{x\cdot\nu}{2}$ and rewite the flow (2.1) in the form

$\frac{\partial\varphi}{\partial t}=S\nu.$

Later on, we will define our energy functional by the equation

$E(t)=\int_{M_{t}}S^{2}e^{-\frac{|x|^{2}}{4}}d\mu$

and analyze its evolution in time to determine its behavior as $t\to\infty$.
In this section we compute the evolution equations of key geometric quantities
which will be necessary later on to perform the calculations on our energy
functional which will be necessary to prove theorem 1.3. To calculate the
evolution equation for $E(t)$, we would write

$\frac{\partial}{\partial t}E(t)=2\int_{M_{t}}S\frac{\partial}{\partial
t}Se^{-\frac{|x|^{2}}{4}}d\mu+\int_{M_{t}}S^{2}\frac{\partial}{\partial
t}[e^{-\frac{|x|^{2}}{4}}]d\mu+\int_{M_{t}}S^{2}e^{-\frac{|x|^{2}}{4}}\frac{\partial}{\partial
t}d\mu$

so at the very least we will have to systematically work our way up to the
calculation of $\partial_{t}S$, $\partial_{t}e^{-\frac{|x|^{2}}{4}}$, and
$\partial_{t}d\mu$. We do so, calculating some other important evolution
equations along the way, below.

###  2.1. Preliminary Calculations

In this subsection we prove the following key lemma about the evolution of
geometric quantities which are necessary for our computations on the energy
functional later. The calculations are done for quantities while evolving by
(2.1). The computations are standard, so we will omit the details.

######  Lemma 2.1.

The following calculations hold true on any family of hypersurfaces
$M_{t}\subset\mathbb{R}^{n+1}$ moving by (2.1)

$\frac{\partial}{\partial t}g_{ij}=-2A_{ij}S$ (2.2)
$\partial_{t}g^{jk}=2A^{jk}S$ (2.3) $\frac{\partial}{\partial
t}\Gamma^{k}_{ij}=-g^{kl}\Big{[}\frac{\partial}{\partial
x_{j}}\big{(}A_{il}S\big{)}+\frac{\partial}{\partial
x_{i}}\big{(}A_{jl}S\big{)}-\frac{\partial}{\partial
x_{l}}\big{(}A_{ij}S\big{)}\Big{]}+SA^{kl}\Big{[}\frac{\partial}{\partial
x_{j}}g_{il}+\frac{\partial}{\partial x_{i}}g_{jl}-\frac{\partial}{\partial
x_{l}}g_{ij}\Big{]}$ (2.4) $\frac{\partial\nu}{\partial t}=-\nabla S.$ (2.5)
$\frac{\partial}{\partial t}A_{ij}=\nabla_{i}\nabla_{j}S-SA_{jl}A_{il}.$ (2.6)

_Proof:_ In this proof we use Mantegazza’s computations of the evolution of
basic geometric quantities in sections 1.2 and 2.3 of [11], but do so keeping
track of the fact that for us $\frac{\partial\varphi}{\partial t}=S\nu$,
rather than $H\nu$. One may use the Gauss-Weingarten relations (see e.g. [11]
pg 4)

$\begin{cases}\frac{\partial^{2}\varphi}{\partial x_{i}\partial
x_{j}}=\Gamma^{k}_{ij}\frac{\partial\varphi}{\partial x_{k}}+A_{ij}\nu\\\
\frac{\partial}{\partial
x_{j}}\nu=-A_{jl}g^{ls}\frac{\partial\varphi}{\partial x_{s}}\end{cases}$

and exponential coordinates at a point to compute the rate of change of the
metric $g_{ij}$ on $M_{t}$ by using the formula of the first fundamental form
as follows

$\begin{split}\frac{\partial}{\partial t}g_{ij}=&\frac{\partial}{\partial
t}\langle\frac{\partial\varphi}{\partial
x_{i}},\frac{\partial\varphi}{\partial x_{j}}\rangle\\\
=&\frac{\partial}{\partial x_{i}}\langle\frac{\partial\varphi}{\partial
t}^{T},\frac{\partial\varphi}{\partial x_{j}}\rangle+\frac{\partial}{\partial
x_{j}}\langle\frac{\partial\varphi}{\partial
t}^{T},\frac{\partial\varphi}{\partial
x_{i}}\rangle-2\Gamma^{k}_{ij}\langle\frac{\partial\varphi}{\partial
t}^{T},\frac{\partial\varphi}{\partial
x_{k}}\rangle-2A_{ij}\langle\frac{\partial\varphi}{\partial t},\nu\rangle\\\
=&-2A_{ij}S,\end{split}$

which gives (2.2).

One may use the fact that $g_{ij}g^{jk}=\delta_{i}^{k}$ and (2.2) to calculate
that

$\partial_{t}g^{jk}=2A^{jk}S,$

which is (2.3).

Combining (2.2), (2.3) and (2.1) easily yields (2.4).

Straightforward computation shows that $\langle\frac{\partial\nu}{\partial
t},\frac{\partial\varphi}{\partial
x_{i}}\rangle=-\langle\nu,\frac{\partial^{2}\varphi}{\partial t\partial
x_{i}}\rangle=-\langle\nu,\frac{\partial(S\nu)}{\partial
x_{i}}\rangle=-\frac{\partial S}{\partial x_{i}}$ which can be expressed more
succinctly as

$\frac{\partial\nu}{\partial t}=-\nabla S,$

which gives (2.5).

Using the formula for the second fundamental form, the Gauss-Weingarten
relations, we perform the following calculation to determine the rate of
change of entries of the second fundamental form under (2.1).

$\begin{split}\frac{\partial}{\partial t}A_{ij}=&\frac{\partial}{\partial
t}\langle\nu,\frac{\partial^{2}\varphi}{\partial x_{i}\partial
x_{j}}\rangle\\\ =&\langle\nu,\frac{\partial^{2}(S\nu)}{\partial x_{i}\partial
x_{j}}\rangle-\langle\nabla S,\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\rangle\\\ =&\frac{\partial^{2}S}{\partial x_{i}\partial
x_{j}}-S\langle\nu,\frac{\partial}{\partial
x_{i}}\big{(}A_{jl}\frac{\partial\varphi}{\partial
x_{l}}\big{)}\rangle-\langle\frac{\partial S}{\partial
x_{l}}\frac{\partial\varphi}{\partial
x_{s}}g^{ls},\Gamma^{k}_{ij}\frac{\partial\varphi}{\partial
x_{k}}+A_{ij}\nu\rangle\\\ =&\frac{\partial^{2}S}{\partial x_{i}\partial
x_{j}}-SA_{jl}A_{il}-\frac{\partial S}{\partial
x_{l}}g^{ls}g_{sk}\Gamma^{k}_{ij}\\\
=&\nabla_{i}\nabla_{j}S-SA_{jl}A_{il},\end{split}$

which gives (2.6). $\blacksquare$

###  2.2. The Differential Operator $\mathcal{L}$

Colding and Minicozzi introduced the differential operator

$\mathcal{L}=\Delta-\frac{1}{2}\langle x,\nabla\cdot\rangle$

which satisfies the condition that

$\int_{M}f(\mathcal{L}g)e^{-\frac{|x|^{2}}{4}}d\mu=\int_{M}(\mathcal{L}f)ge^{-\frac{|x|^{2}}{4}}d\mu=-\int_{M}\nabla
f\nabla ge^{-\frac{|x|^{2}}{4}}d\mu$

for every $f,g$ suitably differentiable. It is sometimes referred to as the
“stability operator” and will be vital during our calculations of the
evolution of energy functionals as it shows up in many evolution equations and
is self-adjoint.

Next we shall recall some well known formulas for shrinkers, using the
equation $H=-\frac{x^{\perp}}{2}$, which we reframe for our setting where the
codimension is one as follows

$\mathcal{L}|A|^{2}=2|\nabla A|^{2}+2|A|^{2}(\frac{1}{2}-|A|^{2})$
$\mathcal{L}H^{2}=H^{2}-2H^{2}|A|^{2}+2|\nabla H|^{2}=2|\nabla
H|^{2}+2H^{2}(\frac{1}{2}-|A|^{2}),$

see, for example Ding and Lin [3] equations (2.6), (2.8) for the more general
form.

We will attempt to produce similar fomulas but for general hypersurfaces
rather than merely for shrinkers. We follow their derivation, but modify the
procedure by starting instead from the equations $H=-\frac{x\cdot\nu}{2}+S$
and $\frac{x\cdot\nu}{2}=S-H$. We collect our results in the following lemma.

######  Lemma 2.2.

The following calculations hold true for any hypersurface
$M\subset\mathbb{R}^{n+1}$

$\nabla_{i}\nabla_{j}H=\frac{1}{2}A_{ij}+(S-H)A_{ik}A_{jk}+\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{i}A_{jk}+\nabla_{i}\nabla_{j}S$ (2.7)
$\mathcal{L}|A|^{2}=2|\nabla
A|^{2}+2|A|^{2}(\frac{1}{2}-|A|^{2})+2S\text{tr}(A^{3})+2A_{ij}\nabla_{i}\nabla_{j}S$
(2.8) $\mathcal{L}H=\frac{1}{2}H+(S-H)|A|^{2}+\Delta S.$ (2.9)

_Proof:_ First observe that we can use the Gauss-Weingarten relations to
perform covariant differentiation on $H=-\frac{x\cdot\nu}{2}+S$ to get

$\nabla_{j}H=\frac{1}{2}\langle x,e_{k}\rangle A_{jk}+\nabla_{j}S.$

Differentiating and using Gauss-Weingarten once more, we calulate

$\nabla_{i}\nabla_{j}H=\frac{1}{2}A_{ij}+\frac{1}{2}\langle x,\nu\rangle
A_{ik}A_{jk}+\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{i}A_{jk}+\nabla_{i}\nabla_{j}S,$

and then substituting in $\frac{x\cdot\nu}{2}=S-H$ gives

$\nabla_{i}\nabla_{j}H=\frac{1}{2}A_{ij}+(S-H)A_{ik}A_{jk}+\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{i}A_{jk}+\nabla_{i}\nabla_{j}S,$

which is (2.7).

We also immediately get another useful equation to refer back to later

$\nabla_{i}\nabla_{j}S=\nabla_{i}\nabla_{j}H-\frac{1}{2}A_{ij}-(S-H)A_{ik}A_{jk}-\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{i}A_{jk}$ (2.10)

from (2.7).

At this point, we refer to a standard calculation (which holds for general
hypersurfaces), see Ding and Lin [3] equation (2.2) for details, which in
codimension $1$ says that

$\Delta|A|^{2}=2|\nabla
A|^{2}+2A_{ij}\nabla_{i}\nabla_{j}H+2H\text{tr}(A^{3})-2|A|^{4}.$

Substituting in for $\nabla_{i}\nabla_{j}H$ using (2.7) gives

$\begin{split}\mathcal{L}|A|^{2}=&\Delta|A|^{2}-\frac{1}{2}\langle\nabla|A|^{2},x\rangle\\\
=&2|\nabla
A|^{2}+2A_{ij}\Big{[}\frac{1}{2}A_{ij}+(S-H)A_{ik}A_{jk}+\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{i}A_{jk}+\nabla_{i}\nabla_{j}S\Big{]}+2H\text{tr}(A^{3})-2|A|^{4}-\frac{1}{2}\langle\nabla|A|^{2},x\rangle\\\
=&2|\nabla
A|^{2}+2|A|^{2}(\frac{1}{2}-|A|^{2})+2S\text{tr}(A^{3})+2A_{ij}\nabla_{i}\nabla_{j}S\end{split}$

and this shows (2.8).

We can also contract (14) with $g^{ij}$ to get

$\begin{split}\mathcal{L}H=&g^{ij}\Big{[}\frac{1}{2}A_{ij}+(S-H)A_{ik}A_{jk}+\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{i}A_{jk}+\nabla_{i}\nabla_{j}S\Big{]}-\frac{1}{2}\langle
x,\nabla H\rangle\\\ =&\frac{1}{2}H+(S-H)|A|^{2}+\Delta S\end{split}$

which is (2.9). $\blacksquare$

Remark In the next subsection, we will compute the evolution of some geoemtric
quantities with the geometric relationships we have established in this
subsection. The unique usefulness of $\mathcal{L}$ does not end here, however,
as its self-adjointness will be vital to establish the needed control of the
evolution of our energy functional.

###  2.3. Evolution of Geometric Quantities

In this subsection we will use our calculations up to this point to compute
the evolution equations of key geometric quantities for a family of
hypersurfaces moving by (2.1) in the form that will be most beneficial for us
throughout this paper. Key for us will be to express as much of the quantity
as possible in terms of $S$ and its covariant derivatives $\nabla^{l}S$, which
will be most useful for us later on as these quantities all converge to $0$ as
$t\to\infty$.

####  2.3.1. The Normalizer Lemma

We shall prove a quick preliminary lemma that we shall refer to as the
_normalizer lemma_ which allows us to easily transition from time evolutions
of scalar geometric quantities with respect to (1.1), to time evolutions of
scalar geometric quantities with respect to (2.1).

######  Lemma 2.3.

(Normalizer Lemma) If $\varphi(p,t)$ is a smooth family of embeddings of a
hypersurface $M$ moving according to (1.1), and we have that for some function
$f(p,t)$ which is defined on $M$ and we have that $\psi(p,t)$ is a smooth
family of diffeomorphisms such that $\varphi(\psi(p,t),t)$ is a smooth family
of embeddings moving by (2.1), where

$D_{p}\varphi(\psi(p,t),t)(\frac{\partial\psi}{\partial
t}(p,t))=-\frac{\varphi(\psi(p,t),t)^{T}}{2},$ (2.11)

then

$\frac{d}{dt}f(\psi(p,t),t)=\frac{\partial}{\partial
t}f(\psi(p,t),t)-\frac{1}{2}\langle x,\nabla f\rangle.$

_Proof:_ This is a straightforward computation that follows by computing the
total deriative of $f(\psi(p,t),t)$. Note that (2.11) defines diffeomorphisms
up to which the flows (1.1) and (2.1) are equivalent.$\blacksquare$

####  2.3.2. Evolution Equations

We are now ready to prove the following lemma.

######  Lemma 2.4.

The following calculations hold true for any family of hypersurfaces
$M_{t}\subset\mathbb{R}^{n+1}$ moving by (2.1)

$\frac{\partial}{\partial
t}A_{ij}=\mathcal{L}A_{ij}+(|A|^{2}-\frac{1}{2})A_{ij}-2SA_{ik}A_{jk}$ (2.12)
$\frac{d}{dt}|A|^{2}=2A_{ij}\nabla_{i}\nabla_{j}S+2Str(A^{3})$ (2.13)
$\frac{d}{dt}S=\mathcal{L}S+(|A|^{2}+\frac{1}{2})S$ (2.14)
$\frac{\partial}{\partial
t}e^{-\frac{|x|^{2}}{4}}d\mu=-S^{2}e^{-\frac{|x|^{2}}{4}}d\mu$ (2.15)

_Proof:_ Equation (2.6) can be combined with (2.8) to calculate that

$\frac{\partial}{\partial
t}A_{ij}=\nabla_{i}\nabla_{j}H-\frac{1}{2}A_{ij}-(S-H)A_{ik}A_{jk}-\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{i}A_{jk}-SA_{jl}A_{il}$

which, after we use Simon’s Identity

$\nabla_{i}\nabla_{j}H=\Delta A_{ij}+|A|^{2}A_{ij}-HA_{il}A_{lj},$

yields

$\begin{split}\frac{\partial}{\partial t}A_{ij}=&\Delta
A_{ij}+(|A|^{2}-\frac{1}{2})A_{ij}-\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{k}A_{ij}-2SA_{ik}A_{jk}\\\
=&\mathcal{L}A_{ij}+(|A|^{2}-\frac{1}{2})A_{ij}-2SA_{ik}A_{jk},\end{split}$

which is (2.12).

One may refer to standard calculations for the rescaled mean curvature flow
which give that for a family of hypersurfaces moving by (1.1) we have

$\frac{d}{dt}|A|^{2}=-|A|^{2}+\Delta|A|^{2}-2|\nabla A|^{2}+2|A|^{4}$

see e.g. [11], and we can combine this with (2.8) and Simon’s identity to
calculate that

$\begin{split}2A_{ij}\nabla_{i}\nabla_{j}S=&2A_{ij}\nabla_{i}\nabla_{j}H-|A|^{2}-2(S-H)tr(A^{3})-\frac{1}{2}\langle
x,\nabla|A|^{2}\rangle\\\ =&2A_{ij}\Delta
A_{ij}+2|A|^{4}-|A|^{2}-2Str(A^{3})-\frac{1}{2}\langle
x,\nabla|A|^{2}\rangle\\\ =&\frac{d}{dt}|A|^{2}-2Str(A^{3})-\frac{1}{2}\langle
x,\nabla|A|^{2}\rangle,\end{split}$

where in _this above_ equation $\frac{d}{dt}|A|^{2}$ is performed with respect
to the rescaled mean curvature for (1.3) as opposed to the normal rescaled
mean curvature flow (2.1). We apply the normalizer lemma to this equation and
calculate that for a family of hypersurfaces moving by (2.1)

$\begin{split}\frac{d}{dt}|A|^{2}=&2A_{ij}\nabla_{i}\nabla_{j}S+2Str(A^{3})\end{split}$

which is (2.13).

One may again refer to standard calculations for the rescaled mean curvature
flow which give that for a family of hypersurfaces moving by (1.3) we have

$\frac{d}{dt}H=-\frac{1}{2}H+\Delta H+H|A|^{2}$

see e.g. [11]. We combine this with (11) and Simon’s identity to calculate
that

$\begin{split}\Delta S=&g^{ij}\nabla_{i}\nabla_{j}S\\\
=&g^{ij}\Big{[}\nabla_{i}\nabla_{j}H-\frac{1}{2}A_{ij}-(S-H)A_{ik}A_{jk}-\frac{1}{2}\langle
x,e_{k}\rangle\nabla_{i}A_{jk}\Big{]}\\\
=&-\frac{1}{2}H-(S-H)|A|^{2}-\frac{1}{2}\langle x,\nabla H\rangle\\\
&+g^{ij}\Big{[}\Delta A_{ij}+|A|^{2}A_{ij}-HA_{il}A_{lj}\Big{]}\\\
=&-\frac{1}{2}H-(S-H)|A|^{2}-\frac{1}{2}\langle x,\nabla H\rangle\\\ &+\Delta
H+|A|^{2}H-H|A|^{2}\\\ =&\frac{d}{dt}H-\frac{1}{2}\langle x,\nabla
H\rangle-S|A|^{2},\end{split}$

where here again we mean $\frac{d}{dt}H$ with respect to the rescaled mean
curvature flow (1.3). We apply Lemma 2.3 to this equation to calculate that,
for a family of hypersurfaces moving by (2.1), we have

$\begin{split}\frac{d}{dt}H=&\Delta S+|A|^{2}S.\end{split}$

(2.1) and (2.5) combined give

$\frac{\partial}{\partial
t}\frac{x\cdot\nu}{2}=\frac{S}{2}-\langle\frac{x}{2},\nabla S\rangle.$

Combining our last two calculations gives

$\begin{split}\frac{\partial}{\partial t}S=&\frac{\partial}{\partial
t}[H+\frac{x\cdot\nu}{2}]\\\ =&\Delta
S+|A|^{2}S+\frac{S}{2}-\langle\frac{x}{2},\nabla S\rangle\\\
=&\mathcal{L}S+(|A|^{2}+\frac{1}{2})S,\end{split}$

which is (2.14).

We calculate, using a standard formula for the rate of change of the measure
$d\mu$, that

$\frac{d}{dt}d\mu=\frac{1}{2}g^{ij}\frac{d}{dt}g_{ij}d\mu=-SHd\mu.$

This can be used to calculate that

$\begin{split}\frac{\partial}{\partial
t}e^{-\frac{|x|^{2}}{4}}d\mu=&(-SH)e^{-\frac{|x|^{2}}{4}}d\mu+\frac{\partial}{\partial
t}[e^{-\frac{|x|^{2}}{4}}]d\mu\\\ =&(-SH-\frac{1}{4}\frac{\partial}{\partial
t}[\big{<}x,x\big{>}])e^{-\frac{|x|^{2}}{4}}d\mu\\\
=&(-SH-\frac{1}{2}\big{<}H\nu+\frac{x^{\perp}}{2},x\big{>})e^{-\frac{|x|^{2}}{4}}d\mu\\\
=&(-SH-\frac{1}{2}S\big{<}\nu,x\big{>})e^{-\frac{|x|^{2}}{4}}d\mu\\\
=&-S^{2}e^{-\frac{|x|^{2}}{4}}d\mu,\end{split}$

which is (2.15). $\blacksquare$

###  2.4. Second Derivatives

The final thing we will need to move on is a second derivative of $S$, as this
will be necessary to compute the second derivative of our energy functional
$E(t)$ and this will be necessary for the analysis of our energy functional.

######  Lemma 2.5.

We have for a family of hypersurfaces $M_{t}\subset\mathbb{R}^{n+1}$ moving by
(2.1) that

$\begin{split}\frac{d^{2}}{dt^{2}}S=&2SA_{ij}\nabla_{i}\nabla_{j}S+2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}+Sg^{kl}\nabla_{l}S\nabla_{k}H\\\
&+S(2A_{ij}\nabla_{i}\nabla_{j}S+2S\text{tr}(A^{3}))\\\
&+\mathcal{L}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}+(|A|^{2}+\frac{1}{2})\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}\end{split}$
(2.16)

_Proof:_ We saw in the previous subsection, equation (2.14), that

$\frac{d}{dt}S=\mathcal{L}S+(|A|^{2}+\frac{1}{2})S,$

so to compute $\frac{d^{2}}{dt^{2}}S$, we have all we need with the exception
of $\frac{d}{dt}\Delta S$. As such, we calculate that

$\begin{split}\frac{\partial}{\partial t}\Delta S=&\frac{\partial}{\partial
t}\Big{[}g^{ij}\nabla_{i}\nabla_{j}S\Big{]}\\\ =&\frac{\partial}{\partial
t}\Big{[}g^{ij}\big{(}\frac{\partial}{\partial x_{i}}\frac{\partial}{\partial
x_{j}}S-\Gamma^{k}_{ij}\nabla_{k}S\big{)}\Big{]}\\\ =&\frac{\partial}{\partial
t}g^{ij}\big{(}\frac{\partial}{\partial x_{i}}\frac{\partial}{\partial
x_{j}}S-\Gamma^{k}_{ij}\nabla_{k}S\big{)}\\\ &+\Delta\frac{\partial}{\partial
t}S\\\ &-g^{ij}\frac{\partial}{\partial
t}\Gamma^{k}_{ij}\nabla_{k}S.\end{split}$

Using Lemma 2.1, equations (2.3), (2.14), Simon’s identity, Codazzi’s
equation, and normal coordinates at a point we compute,

$\begin{split}\frac{\partial}{\partial t}\Delta S=&\frac{\partial}{\partial
t}g^{ij}\big{(}\nabla_{i}\nabla_{j}S\big{)}+\Delta\frac{\partial}{\partial
t}S-g^{ij}\frac{\partial}{\partial t}\Gamma^{k}_{ij}\nabla_{k}S\\\
=&2SA^{ij}\nabla_{i}\nabla_{j}S\\\
&+\Delta\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}\\\
&+g^{ij}g^{kl}\Big{[}\frac{\partial}{\partial
x_{j}}\big{(}A_{il}S\big{)}+\frac{\partial}{\partial
x_{i}}\big{(}A_{jl}S\big{)}-\frac{\partial}{\partial
x_{l}}\big{(}A_{ij}S\big{)}\Big{]}\nabla_{k}S\\\
=&2SA_{ij}\nabla_{i}\nabla_{j}S\\\
&+\Delta\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}\\\
&+A^{jk}\nabla_{j}S\nabla_{k}S+A^{ik}\nabla_{i}S\nabla_{k}S-H|\nabla S|^{2}\\\
&+2Sg^{ij}\nabla_{j}A_{il}\nabla_{l}S-Sg^{kl}\nabla_{l}S\nabla_{k}H\\\
=&2SA_{ij}\nabla_{i}\nabla_{j}S+2A^{jk}\nabla_{j}S\nabla_{k}S\\\ &-H|\nabla
S|^{2}+Sg^{kl}\nabla_{l}S\nabla_{k}H\\\
&+\Delta\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}.\\\ \end{split}$
(2.17)

With (2.17) computed, we can use it to plug in, calculating
$\frac{d^{2}}{dt^{2}}S$ as follows

$\begin{split}\frac{\partial}{\partial
t}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}=&\frac{\partial}{\partial
t}\Delta S-\frac{1}{2}\frac{\partial}{\partial t}\langle x,\nabla
S\rangle+S\frac{\partial}{\partial
t}|A|^{2}+(|A|^{2}+\frac{1}{2})\frac{\partial}{\partial t}S\\\
=&2SA_{ij}\nabla_{i}\nabla_{j}S+2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}+Sg^{kl}\nabla_{l}S\nabla_{k}H\\\
&+\Delta\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}-\frac{1}{2}\frac{\partial}{\partial
t}\langle x,\nabla S\rangle\\\
&+S(2A_{ij}\nabla_{i}\nabla_{j}S+2S\text{tr}(A^{3}))\\\
&+(|A|^{2}+\frac{1}{2})\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}\\\
=&2SA_{ij}\nabla_{i}\nabla_{j}S+2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}+Sg^{kl}\nabla_{l}S\nabla_{k}H\\\
&+S(2A_{ij}\nabla_{i}\nabla_{j}S+2S\text{tr}(A^{3}))\\\
&+\mathcal{L}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}+(|A|^{2}+\frac{1}{2})\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}\\\
\end{split}$

and this is (2.16). $\blacksquare$

##  3\. Energy Functional

In this section we introduce our energy functional and analyze it to control
its long time behavior under the flow (2.1).

######  Definition 3.1.

We define our energy functional $E(t)$ for a family of hypersurfaces
$M_{t}\subset\mathbb{R}^{n+1}$ via the following formula

$E(t)=\int_{M_{t}}S^{2}e^{-\frac{|x|^{2}}{4}}d\mu.$ (3.1)

One might recognize this as the negative right hand side of the rescaled
Huisken’s monotonicity formula (3), that is, we can write

$\int_{M_{t}}e^{-\frac{|x|^{2}}{4}}d\mu_{t}-\int_{M_{s}}e^{-\frac{|x|^{2}}{4}}d\mu_{s}=\int_{t}^{s}E(\xi)d\xi,$
(3.2)

but it is, in a sense, physical as well, regarding $S$ as the “speed” of (2.1)
and $e^{-\frac{|x|^{2}}{4}}d\mu$ as the “weight”.

###  3.1. Evolution of the Energy Functional

In this section, we will continue to work under the assumption that $M_{t}$,
or better yet $\varphi(M,t)$ is a smooth family of hypersurfaces evolving by
the normal rescaled mean curvature flow (2.1). In what follows we compute the
evolution equation for $E(t)$. Much of the hard work has already been done for
us in section 2.

As this quantity makes frequent appearences we will frequently refer to the
weighted measure as

$d\tilde{\mu}=e^{-\frac{|x|^{2}}{4}}d\mu$

for cleanliness in our computations.

######  Lemma 3.2.

For a family of hypersurfaces $M_{t}\subset\mathbb{R}^{n+1}$ moving by the
normal rescaled mean curvature flow (2.1), we have that

$\begin{split}\frac{\partial}{\partial t}E(t)=&-\int S^{4}d\tilde{\mu}+2\int
S\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\end{split}$
(3.3)

_Proof:_ We calculate directly from the definition of $E(t)$ that

$\frac{\partial}{\partial t}E(t)=2\int_{M_{t}}S\frac{\partial}{\partial
t}Se^{-\frac{|x|^{2}}{4}}d\mu+\int_{M_{t}}S^{2}\frac{\partial}{\partial
t}[e^{-\frac{|x|^{2}}{4}}d\mu]$

and use (2.14) to plug in for $\frac{\partial}{\partial t}S$ and use (2.15) to
plug in for $\frac{\partial}{\partial t}[e^{-\frac{|x|^{2}}{4}}d\mu]$.

We obtain

$\begin{split}\frac{\partial}{\partial t}E(t)=&-\int S^{4}d\tilde{\mu}+2\int
S\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\end{split}$

which gives (3.3). $\blacksquare$

We also would like to compute the second derivative of the energy functional,
which we do in the following lemma.

######  Lemma 3.3.

For a family of hypersurfaces $M_{t}\subset\mathbb{R}^{n+1}$ moving by the
normal rescaled mean curvature flow (2.1), we have that

$\begin{split}\ddot{E}(t)=&\int\Big{[}-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}\Big{]}^{2}d\tilde{\mu}\\\
&-2\int
S^{3}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}+2\int
S\Big{[}4SA_{ij}\nabla_{i}\nabla_{j}S+Sg^{kl}\nabla_{l}S\nabla_{k}H+2S^{2}\text{tr}(A^{3})\Big{]}d\tilde{\mu}\\\
&+2\int S\Big{[}2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}\Big{]}d\tilde{\mu}\end{split}$ (3.4)

_Proof:_ We begin by reexamining the equation (3.3)

$\begin{split}\frac{\partial}{\partial t}E(t)=&-\int S^{4}d\tilde{\mu}+2\int
S\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\end{split}$

and from this, and (2.14), (2.15), we get, as a first expanded expression

$\begin{split}\ddot{E}(t)=&\int S^{6}d\tilde{\mu}-4\int
S^{3}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\\\ &-2\int
S^{3}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\\\
&+2\int\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}^{2}d\tilde{\mu}\\\
&+2\int S\frac{\partial}{\partial
t}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}.\\\
\end{split}$

From here we recall equation (2.16), which we re-express as

$\begin{split}\frac{d}{dt}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}=&2SA_{ij}\nabla_{i}\nabla_{j}S+2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}+Sg^{kl}\nabla_{l}S\nabla_{k}H\\\
&+S(2A_{ij}\nabla_{i}\nabla_{j}S+2S\text{tr}(A^{3}))\\\
&+\mathcal{L}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}+(|A|^{2}+\frac{1}{2})\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]},\end{split}$

and we will use (2.16) to rewrite

$\begin{split}\int S\frac{\partial}{\partial
t}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}=&\int
S\Big{[}\mathcal{L}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}+(|A|^{2}+\frac{1}{2})\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}\Big{]}d\tilde{\mu}\\\
&+\int S^{2}(2A_{ij}\nabla_{i}\nabla_{j}S+2S\text{tr}(A^{3}))d\tilde{\mu}\\\
&+\int
S\Big{[}2SA_{ij}\nabla_{i}\nabla_{j}S+2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}+Sg^{kl}\nabla_{l}S\nabla_{k}H\Big{]}d\tilde{\mu}.\end{split}$

The self-adjointness of $\mathcal{L}$ allows us to write

$\int
S\Big{(}\mathcal{L}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}+(|A|^{2}+\frac{1}{2})\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}\Big{)}d\tilde{\mu}=\int\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}^{2}d\tilde{\mu}.$

Combining our calculations to this point gives

$\begin{split}\ddot{E}(t)=&4\int\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}^{2}d\tilde{\mu}\\\
&+2\int
S\Big{[}4SA_{ij}\nabla_{i}\nabla_{j}S+Sg^{kl}\nabla_{l}S\nabla_{k}H+2S^{2}\text{tr}(A^{3})\Big{]}d\tilde{\mu}\\\
&+2\int S\Big{[}2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}\Big{]}d\tilde{\mu}\\\ &+\int S^{6}d\tilde{\mu}-6\int
S^{3}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\\\ =&\int
S^{6}d\tilde{\mu}-6\int
S^{3}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}+4\int\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}^{2}d\tilde{\mu}\\\
&+2\int
S\Big{[}4SA_{ij}\nabla_{i}\nabla_{j}S+Sg^{kl}\nabla_{l}S\nabla_{k}H+2S^{2}\text{tr}(A^{3})\Big{]}d\tilde{\mu}\\\
&+2\int S\Big{[}2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}\Big{]}d\tilde{\mu}\\\
=&\int\Big{[}-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}\Big{]}^{2}d\tilde{\mu}\\\
&-2\int
S^{3}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}+2\int
S\Big{[}4SA_{ij}\nabla_{i}\nabla_{j}S+Sg^{kl}\nabla_{l}S\nabla_{k}H+2S^{2}\text{tr}(A^{3})\Big{]}d\tilde{\mu}\\\
&+2\int S\Big{[}2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}\Big{]}d\tilde{\mu}\end{split}$

which is (3.4). $\blacksquare$

In the next section we will define the Dirichlet-Einstein quotient and explore
its significance for the long term behavior of our energy functional.

##  4\. Dirichlet-Einstein Quotient

In this section we introduce the Dirichlet-Einstein quotient and analyze its
evolution in time in an attempt to better understand the long term behavior of
(2.1). In 1 we have motivated the definition of the Dirichelt-Einstein
quotient, which we recall is defined as follows: for a family of hypersurfaces
moving by the normal rescaled mean curvatire flow (2.1) or the recaled mean
curvature flow (1.3) (which is not a shrinker), we define the _Dirichlet-
Einstein quotient_ $N(t)$ by the following formula

$N(t)=\frac{\dot{E}(t)}{E(t)}=\frac{d}{dt}\text{ln}(E(t)).$ (4.1)

###  4.1. Evolution of the Dirichlet-Einstein Quotient

In this subsection we investigate the time-evolution of the Dirichlet-Einstein
quotient. We prove the following lemma.

######  Lemma 4.1.

For a family of hypersurfaces $M_{t}\subset\mathbb{R}^{n+1}$ moving by the
normal rescaled mean curvature flow (2.1), N(t) which is defined by (4.1)
evolves by

$\begin{split}\dot{N}(t)=&\\\ &\frac{\int
S^{2}d\tilde{\mu}\int\Big{[}-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}\Big{]}^{2}d\tilde{\mu}-\Big{[}\int
S\Big{[}-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}\Big{]}d\tilde{\mu}\Big{]}^{2}}{E(t)^{2}}\\\
+\frac{1}{E(t)}\Big{[}&-2\int
S^{3}\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}d\tilde{\mu}+2\int
S\Big{(}4SA_{ij}\nabla_{i}\nabla_{j}S+Sg^{kl}\nabla_{l}S\nabla_{k}H+2S^{2}\text{tr}(A^{3})\Big{)}d\tilde{\mu}\\\
&+\int S^{2}\Big{(}\langle\nabla H,\nabla
S\rangle+H\mathcal{L}S\Big{)}d\tilde{\mu}-2\int
S^{2}\Big{(}\nabla_{j}A^{jk}\nabla_{k}S+A^{jk}\nabla_{j}\nabla_{k}S-2\frac{x_{j}}{2}A^{jk}\nabla_{k}S\Big{)}d\tilde{\mu}\Big{]}\end{split}$
(4.2)

for all $t$.

_Proof:_ We begin by the obvious calculation

$\dot{N}=\frac{\ddot{E}E-\dot{E}^{2}}{E^{2}}$

which can be computed directly from the definition of $N(t)$.

This calculation allows us to appeal to equations (3.3) and (3.4) to calculate
that

$\begin{split}\dot{N}(t)=\frac{1}{E^{2}(t)}&\Big{[}\int
S^{2}d\tilde{\mu}\int\Big{[}-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}\Big{]}^{2}d\tilde{\mu}\\\
&-2\int S^{2}d\tilde{\mu}\int
S^{3}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\\\ &+2\int
S^{2}d\tilde{\mu}\int
S\Big{[}4SA_{ij}\nabla_{i}\nabla_{j}S+Sg^{kl}\nabla_{l}S\nabla_{k}H+2S^{2}\text{tr}(A^{3})\Big{]}d\tilde{\mu}\\\
&+2\int S^{2}d\tilde{\mu}\int S\Big{[}2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}\Big{]}d\tilde{\mu}\\\ &-\Big{(}\int S^{4}d\tilde{\mu}+2\int
S\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\Big{)}^{2}\Big{]}.\end{split}$

Now we shall investigate some of these terms. Consider the quantity

$\int S\Big{[}2A^{jk}\nabla_{j}S\nabla_{k}S-H|\nabla
S|^{2}\Big{]}d\tilde{\mu}.$

We may integrate this by parts, term by term as follows. We have

$\begin{split}2\int A^{jk}S\nabla_{j}S\nabla_{k}Sd\tilde{\mu}=&\int
A^{jk}\nabla_{j}S^{2}\nabla_{k}Sd\tilde{\mu}\\\
=&\int\nabla_{j}S^{2}A^{jk}\nabla_{k}Se^{-\frac{|x|^{2}}{4}}d\mu\\\ =&-\int
S^{2}\Big{[}\nabla_{j}A^{jk}\nabla_{k}Se^{-\frac{|x|^{2}}{4}}+A^{jk}\nabla_{j}\nabla_{k}Se^{-\frac{|x|^{2}}{4}}+A^{jk}\nabla_{k}S(-\frac{x_{j}}{2})e^{-\frac{|x|^{2}}{4}}\Big{]}d\mu\\\
=&-\int
S^{2}\Big{[}\nabla_{j}A^{jk}\nabla_{k}S+A^{jk}\nabla_{j}\nabla_{k}S-\frac{x_{j}}{2}A^{jk}\nabla_{k}S\Big{]}d\tilde{\mu}.\end{split}$

Next, we write

$\begin{split}-\int HS|\nabla
S|^{2}d\tilde{\mu}=&-\frac{1}{2}\int\nabla_{j}S^{2}\big{(}H\nabla_{j}Se^{-\frac{|x|^{2}}{4}}\big{)}d\mu\\\
=&\frac{1}{2}\int S^{2}\Big{[}\langle\nabla H,\nabla S\rangle
e^{-\frac{|x|^{2}}{4}}+H\Delta
Se^{-\frac{|x|^{2}}{4}}-H\frac{x_{j}}{2}\nabla_{j}Se^{-\frac{|x|^{2}}{4}}\Big{]}d\mu\\\
=&\frac{1}{2}\int S^{2}\Big{[}\langle\nabla H,\nabla
S\rangle+H\mathcal{L}S\Big{]}d\tilde{\mu}.\end{split}$

Combining these calculations gives

$\begin{split}\dot{N}(t)=\frac{1}{E^{2}(t)}&\Big{[}\int
S^{2}d\tilde{\mu}\int\Big{[}-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}\Big{]}^{2}d\tilde{\mu}\\\
&-2\int S^{2}d\tilde{\mu}\int
S^{3}\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\\\ &+2\int
S^{2}d\tilde{\mu}\int
S\Big{[}4SA_{ij}\nabla_{i}\nabla_{j}S+Sg^{kl}\nabla_{l}S\nabla_{k}H+2S^{2}\text{tr}(A^{3})\Big{]}d\tilde{\mu}\\\
&-2\int S^{2}d\tilde{\mu}\int
S^{2}\Big{[}\nabla_{j}A^{jk}\nabla_{k}S+A^{jk}\nabla_{j}\nabla_{k}S-\frac{x_{j}}{2}A^{jk}\nabla_{k}S\Big{]}d\tilde{\mu}\\\
&+\int S^{2}d\tilde{\mu}\int S^{2}\Big{[}\langle\nabla H,\nabla
S\rangle+H\mathcal{L}S\Big{]}d\tilde{\mu}\\\ &-\Big{(}\int
S^{4}d\tilde{\mu}+2\int
S\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\Big{)}^{2}\Big{]}\end{split}$

which is (4.2). $\blacksquare$

With the evolution of the Dirichlet-Einstein quotient computed, we can now
begin to work on bounding it in accordance with our plan laid out in the
introduction (section 1).

###  4.2. Order of Terms in $\dot{N}$

In this subsection we prove a bound on $\dot{N}$ that will be central to our
proof. We will bound $\dot{N}$ in terms of the various covariant derivatives
$\nabla^{l}S$ of $S$. As we approach a shrinker, since $S\equiv 0$ on a
shrinker, we obviously have that $\nabla^{l}S\to 0$ on $M_{t}$. The hope is
that later on we can prove some bound of the form

$\int_{T_{0}}^{\infty}||\nabla^{(l)}S(p,t)||_{\infty,M_{t}}\leq
C_{l}(\Sigma,\int_{M_{0}}e^{-\frac{|x|^{2}}{4}}d\mu-\int_{\Sigma}e^{-\frac{|x|^{2}}{4}}d\mu).$

Such bounds would enable us to control

$N(t)=N(T_{0})+\int_{T_{0}}^{t}\dot{N}(s)ds$

as $t\to\infty$.

From our work above the following corollary is immediate.

######  Corollary 4.2.

For a family of hypersurfaces $M_{t}\subset\mathbb{R}^{n+1}$ moving by the
normal rescaled mean curvature flow (2.1), if $N(t)$ is the Dirichlet-Einstein
quotient, we have that

$\dot{N}(t)\geq-C(||S||_{\infty}+||\nabla
S||_{\infty}+||\nabla^{2}S||_{\infty})$ (4.3)

where $C=C(S,|A|,|\nabla A|,H,\nabla H,x)$.

_Proof:_ Consider term by term on the right hand side of equation (4.2).

We have

$-\frac{1}{E(t)}\int
S^{3}\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}d\tilde{\mu}\geq-\Big{(}\|S\mathcal{L}S\|_{\infty}+\||A|^{2}+\frac{1}{2}\|_{\infty}\|S^{2}\|_{\infty}\Big{)}$

so this term is suitably bounded by the uniform boundedness of $|A|^{2}$ and
$|\nabla^{l}S|$ along the flow.

Next we have that

$\begin{split}\frac{1}{E(t)}\int
S&\Big{(}4SA_{ij}\nabla_{i}\nabla_{j}S+Sg^{kl}\nabla_{l}S\nabla_{k}H+2S^{2}\text{tr}(A^{3})\Big{)}d\tilde{\mu}\geq\\\
&-\Big{(}4(\|A\|_{\infty}\|\nabla^{2}S\|_{\infty}+\|\nabla
H\|_{\infty}\|\nabla
S\|_{\infty}+2\|A^{3}\|_{\infty}\|S^{2}\|_{\infty}\Big{)}\end{split}$

which again shows this term gives us an estimate that is claimes in (4.3). We
also have that

$\frac{1}{E(t)}\int S^{2}\Big{(}\langle\nabla H,\nabla
S\rangle+H\mathcal{L}S\Big{)}d\tilde{\mu}\geq-\Big{(}\|\nabla
H\|_{\infty}\|\nabla
S\|_{\infty}+\|H\|_{\infty}(\|\frac{x}{2}\|_{\infty}\|\nabla
S\|_{\infty}+\|\nabla^{2}S\|_{\infty})\Big{)}$

which is suitably bounded as $H$ is another geometric quantity over which we
have uniform control as $t\to\infty$, as well as the fact that $x$ stays
bounded along the flow. Furthermore,

$\begin{split}-\frac{1}{E(t)}\int&S^{2}\Big{(}\nabla_{j}A^{jk}\nabla_{k}S+A^{jk}\nabla_{j}\nabla_{k}S-2\frac{x_{j}}{2}A^{jk}\nabla_{k}S\Big{)}d\tilde{\mu}\geq\\\
&-\Big{(}\|\nabla A\|_{\infty}\|\nabla
S\|_{\infty}+\|A\|_{\infty}\|\nabla^{2}S\|_{\infty}+\|x\|_{\infty}\|A\|_{\infty}\|\nabla
S\|_{\infty}\Big{)}\end{split}$

which again is suitably bounded for all the reasons we have discussed prior.

Finally, we use the Cauchy-Schwartz inequality with $S$ and
$-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}$ to show that

$\int
S^{2}d\tilde{\mu}\int\Big{[}-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}\Big{]}^{2}d\tilde{\mu}-\Big{[}\int
S\Big{[}-S^{3}+2\Big{(}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{)}\Big{]}d\tilde{\mu}\Big{]}^{2}\geq
0.$

Combining the results from above yields

$\dot{N}(t)\geq-C(S,|A|,|\nabla A|,H,\nabla H,x)(||S||_{\infty}+||\nabla
S||_{\infty}+||\nabla^{2}S||_{\infty})$

which proves (4.3). $\blacksquare$

##  5\. Controlling $\nabla^{l}S$

In order to use (4.3) we need to get a better control of $S$ and its covariant
derivatives. We take care of that in the following Proposition.

######  Proposition 5.1.

Let $\Sigma$ be a compact shrinker. Then if $M_{t}$ is some family of
hypersurface moving by the normal rescaled mean curvature flow (2.1)
satisfying

$\Omega(M_{t})\geq\Omega(\Sigma)$

for all $t\geq 0$, then we may find $\delta$ and $T_{0}$ such that if

$dist(M_{0},\Sigma)<\delta,$ (5.1)

then we may write $M_{t}$ as a normal graph of $v(p,t)$ over $\Sigma$ for
$t\geq T_{0}$ and such that for every $l\geq 0$,

$\int_{T_{0}}^{\infty}||\nabla^{(l)}S(p,t)||_{\infty,M_{t}}\,dt\leq
C_{l}(\Sigma,\int_{M_{0}}e^{-\frac{|x|^{2}}{4}}d\mu-\int_{\Sigma}e^{-\frac{|x|^{2}}{4}}d\mu).$
(5.2)

_Proof:_ First, suppose that $\varphi:M\times[0,\infty)\to\mathbb{R}^{n+1}$ is
a normal rescaled mean curvature flow moving by (2.1). Then we may find a
family of diffeomorphisms $\psi(t):M\to M$ such that we have for every $t$
that

$\frac{d}{dt}\varphi(\psi(t)(p),t)=H(\psi(t)(p),t)\nu(\psi(t)(p),t)+\frac{\varphi(\psi(t)(p),t)}{2}$

in other words, that $\tilde{\varphi}(p,t)=\varphi(\psi(t)(p),t)$ moves by the
(original) rescaled mean curvature flow (1.3). Let us write $\tilde{M}_{t}$
for this family of hypersurfaces moving by (1.3), keeping in mind that for
every $t$, we have $\tilde{M}_{t}=\psi(t)(M_{t})$, that is, our underlying
geometric object is the same. Since we are moving by (1.3), if we define the
new variable $\tau=-e^{-t}$, and
$\bar{\varphi}:M\times[-1,0)\to\mathbb{R}^{n+1}$ by the equation

$\bar{\varphi}(\cdot,\tau)=e^{-\frac{t}{2}}\tilde{\varphi}(\cdot,t),$

then $\bar{\varphi}(M_{\tau})$ moves by the mean curvature flow (1.1). We
write $\bar{M}_{\tau}$ for this family of hypersurfaces, and have that
$\bar{M}_{-1}=\tilde{M}_{0}=M_{0}$, as hypersurfaces. Note also that $\Sigma$
is a stationary solution to (2.1), and that the mean curvature flow
$\Sigma_{\tau}=\sqrt{-\tau}\Sigma$, up to tangential diffeomorphisms satisfies
(1.1). By our assumption we have that

$\Omega(M_{0})\geq\Omega(\Sigma),$

and hence by the continuity of the dependence of the mean curvature flow on
its initial conditions, we may find $\tau_{2}<0$, and some $\delta_{0}$ such
that if $dist(M_{0},\Sigma)<\delta_{0}$, then the flow
$(\bar{M}_{\tau})_{\tau\in(-1,\tau_{2})}$ is sufficiently close in measure to
the flow $\Sigma_{\tau}$. Thus, by Theorem 1.1 in [13] we have the solutions
$\tilde{M_{t}}$ of the rescaled flow (1.3), may be written as normal graphs of
a function $v$ over $\Sigma$ for $\frac{-1+\tau_{2}}{2}<\tau<0$. Hence we may
write $\tilde{M}_{t}$ as a normal graph of a function $v(p,t)$ over $\Sigma$
for all $t>-\text{log}(-\frac{-1+\tau_{2}}{2})=:T_{0}$.

Again, by the continuous dependence, we may also choose $\delta_{0}$ small
enough that

$\|v(T_{0})\|_{C^{2,\alpha}}<\sigma_{0}$

where $\sigma_{0}$ and $\theta\in(\frac{1}{2},1)$ are as in [13], such that if
$v$ is a $C^{2,\alpha}$ section of $T^{\perp}\Sigma$ with
$\|v\|_{C^{2,\alpha}}<\sigma_{0}$ then

$||\nabla\Omega(v)||_{L^{2}(\Sigma)}\geq|\Omega(v)-\Omega(0)|^{\theta}.$ (5.3)

Now, for $t>T_{0}$, we consider the quantity $Q(t)$ defined by

$\tilde{Q}(t)=\Omega(\tilde{M}_{t})-\Omega(\Sigma).$

Obviously, by Huisken’s monotonicity formula, $Q(t)$ is monotone decreasing,
and $Q(t)\geq 0$. This together with monotonicity of $Q(t)$ implies that if
$Q(t_{0})=0$, we must have $Q(t)\equiv 0$ for all $t\geq t_{0}$, meaning, of
course, that

$\frac{d}{dt}\Omega(\tilde{M}_{t})\equiv 0$

for $t\geq t_{0}$, which would mean that $\tilde{M}_{t}$ is a shrinker for
$t\geq t_{0}$ by Huisken’s monotonicity formula. In fact we may apply the
backwards uniqueness results in [8] to show that $\tilde{M}_{t}$ is a shrinker
for all times. Obviously, in this case, we have (5.2) for free since
$\nabla^{l}S\equiv 0$ on any shrinker.

As such, we will now consider the case when $Q(t)>0$ for all $t>T_{0}$. We
proceed similarly to [10]. Fixing $\beta\in(2-\frac{1}{\theta},1)$, gives that
$0<(2-\beta)\theta<1$, so we calculate directly that

$-\frac{d}{dt}\tilde{Q}^{1-(2-\beta)\theta}=(1-(2-\beta)\theta)\tilde{Q}^{-(2-\beta)\theta}\frac{d\tilde{Q}}{dt}$

and Huisken’s monotonicity formula gives

$-\frac{d}{dt}\tilde{Q}^{1-(2-\beta)\theta}=(1-(2-\beta)\theta)||S||^{2}_{L^{2}(e^{-\frac{|x|^{2}}{4}}d\mu_{M_{t}})}\tilde{Q}^{-(2-\beta)\theta}.$

The Lojasiewicz inequality (5.3) allows us to write that

$\tilde{Q}^{-(2-\beta)\theta}||S||^{2-\beta}_{L^{2}(e^{-\frac{|x|^{2}}{4}}d\mu_{M_{t}})}=\tilde{Q}^{-(2-\beta)\theta}||\nabla\Omega(M_{t})||^{2-\beta}_{L^{2}(d\mu_{M_{t}})}\geq
1,$

which implies that

$-\frac{d}{dt}\tilde{Q}^{1-(2-\beta)\theta}\geq
C_{0}^{-1}||S||^{\beta}_{L^{2}(e^{-\frac{|x|^{2}}{4}}d\mu_{M_{t}})}$

for some $C_{0}=C_{0}(\beta,\theta)$.

At this point we should note that every calculation we have performed for
$\tilde{Q}(t)$, assuming that our family of hypersurfaces is moving by the
rescaled mean curvature flow (2) also holds true for $Q(t)$ if we define
$Q(t)$ as

$Q(t)=\Omega(M_{t})-\Omega(\Sigma)$

where $M_{t}$ is the corresponding family of hypersurfaces moving via the
normal rescaled mean curvature flow (2.1), since

$\Omega(M_{t})=\Omega(\psi(t)(M_{t}))$

for all $t$ for any family $\psi$ of tangential diffeomorphisms. As such, from
here on we simply use $Q(t)$ and $\tilde{Q}(t)$ interchangeably, writing
$Q(t)$ for either.

Because of this, we can write that for $0<a,s<\infty$,

$\int_{a}^{s}||\partial_{t}\varphi(p,t)||^{\beta}_{L^{2}(e^{-\frac{|x|^{2}}{4}}d\mu_{M_{t}})}dt\leq
C(C_{0},\beta,\theta,C_{L})\Big{[}Q(a)^{\gamma}-Q(s)^{\gamma}\Big{]}$ (5.4)

for $\gamma=1-(2-\beta)\theta$.

Derivative estimates for the rescaled mean curvature flow (2), are obviously
invariant under tangential diffeomorphism and allow us to write

$\|\nabla^{l}S\|_{\infty,M_{t}}\leq K_{l}$

on $M_{t}$ uniformly. We will now explore the implications of inequality
(5.4).

Returning now to the matter at hand, we shall break the proof into two cases
depending on the relationship between $\beta$ and $1-\beta$.
_Case 1:_ $\beta<1-\beta$

In this case, given any $p$, we may choose $N=N(p)$ such that

$\frac{p}{N}\leq\beta$

and therefore, of course, $1-\frac{p}{N}\geq 1-\beta$. We may use
interpolation inequalities to write that for $0\leq q\leq p$ we have

$\int|\nabla^{q}(S)|^{2}d\mu\leq
C\Big{[}\int|\nabla^{N}S|^{2}d\mu\Big{]}^{\frac{q}{N}}\Big{[}\int|S|^{2}d\mu\Big{]}^{1-\frac{q}{N}}$
(5.5)

or

$\|\nabla^{q}S\|\leq C\|\nabla^{N}S\|^{\frac{q}{N}}\|S\|^{1-\frac{q}{N}}.$

In what follows we consider the quantity

$s:=1-\frac{q}{N}-(1-\beta)=\beta-\frac{q}{N},$

which by our choice of $N$ guarantees that $s\geq 0$.

We obviously have, by the definition of $s$,

$\|S\|^{1-\frac{q}{N}}=\|S\|^{1-\beta}\|S\|^{s}.$

Since $\|S\|\leq\|S\|_{L^{2,N}}$, $\|\nabla^{N}S\|\leq\|S\|_{L^{2,N}}$, and
$s+\frac{q}{N}=\beta$ we can write

$\|\nabla^{q}S\|\leq C\|S\|^{1-\beta}\|S\|^{\beta}_{L^{2,N}}$

for every $q\leq p$. This allows us the find $C^{\prime}(p,m)$ such that we
may write

$\|S\|_{L^{2,p}}\leq C^{\prime}\|S\|^{1-\beta}\|S\|^{\beta}_{L^{2,N}}.$

Since $\beta<1-\beta$ and $\|S\|\leq\|S\|_{L^{2,N}}$, we can write that

$\|S\|_{L^{2,p}}\leq C^{\prime}\|S\|^{\beta}\|S\|^{1-\beta}_{L^{2,N}}.$

I still have a problem with above, how do you use $\beta<1-\beta$? This is not
true the way it is written. I do not know how you intend to use it later.
_Case 2:_ $1-\beta\leq\beta$

In this case, given any $p$, we may choose $N=N(p)$ such that

$\frac{p}{N}\leq 1-\beta$

and therefore, of course, $1-\frac{p}{N}\geq\beta$.

In what follows we consider the quantity

$s:=1-\frac{q}{N}-\beta=(1-\beta)-\frac{q}{N},$

which by our choice of $N$ guarantees that $s\geq 0$.

We obviously have, by the definition of $s$,

$\|S\|^{1-\frac{q}{N}}=\|S\|^{\beta}\|S\|^{s}.$

Since $\|S\|\leq\|S\|_{L^{2,N}}$, $\|\nabla^{N}S\|\leq\|S\|_{L^{2,N}}$, and
$s+\frac{q}{N}=1-\beta$, by the interpolation inequality (5.5), similarly as
in the previous case we have

$\|\nabla^{q}S\|\leq C\|S\|^{\beta}\|S\|^{1-\beta}_{L^{2,N}}$

for every $q\leq p$. This allows us to find $C^{\prime}(p,m)$ such that we may
write

$\|S\|_{L^{2,p}}\leq C^{\prime}\|S\|^{\beta}\|S\|^{1-\beta}_{L^{2,N}}.$

Thus either way, since $|\nabla^{l}S|$ is uniformly bounded for every $l$, and
since $M_{0}$ is compact, we may write that

$\|S\|_{L^{2,p}}\leq
C^{\prime\prime}(\beta,p,\Sigma,M_{0})\|S\|^{\beta}_{L^{2}(e^{-\frac{|x|^{2}}{4}})}.$

The Sobolev embedding theorem says that if $n$ is odd, we have a natural
embedding $W^{k,2}\subset C^{k-\frac{n+1}{2},\frac{1}{2}}$, and if $n$ is even
then we have natural embeddings $W^{k,2}\subset C^{k-1-\frac{n}{2},\gamma}$,
for every $\gamma\in(0,1)$. Given that we have

$\|S\|_{L^{2,k}}\leq
C^{\prime\prime}(\beta,k,\Sigma,M_{0})\|S\|^{\beta}_{L^{2}(e^{-\frac{|x|^{2}}{4}})}$

for any $k$, we may simply enlarge $k$ as much as possible to claim that due
to the Sobolev embedding theorems we may find a $C^{\prime\prime}$ such that

$\|S\|_{C^{l}}\leq
C^{\prime\prime}(\beta,l,n,\Sigma,M_{0})\|S\|^{\beta}_{L^{2}(e^{-\frac{|x|^{2}}{4}})}$
(5.6)

for any $l$.

Recall that for $t>T_{0}$ we may express $M_{t}$ as the normal graph of a
function $v(t)$ over $\Sigma$. We have, for $T_{0}\leq a\leq s<\infty$,

$\|v(s)-v(a)\|_{C^{l}_{\Sigma}}\leq
C(l)\int_{a}^{s}\|\partial_{t}\varphi(p,t)\|_{C^{l}_{\Sigma}}dt$

and

$\int_{a}^{s}\|\partial_{t}\varphi(p,t)\|_{C^{l}_{\Sigma}}dt\leq
C(l,\Sigma,M_{0},C^{\prime\prime})\int_{a}^{s}\|\partial_{t}\varphi(p,t)\|_{C^{l}_{M_{t}}}dt.$

Recall that $\partial_{t}\varphi=S\nu$ so

$\|S\nu\|_{C^{l}}\leq\sum_{k=0}^{l}\Big{(}\sum_{p+q=k}\|\nabla^{p}S\|_{\infty}\|\nabla^{q}\nu\|_{\infty}\Big{)}\leq
C\|S\|_{C^{l}}$

where $C$ is finite as it is dependant on the $\nu$ and its covariant
derivatives which are dependant on the covariant derivatives of the second
fundamental form, which are all bounded along the flow. And since

$\|S\|_{C^{l}}=\|S\|_{\infty}+\|\nabla
S\|_{\infty}+...+\|\nabla^{l}S\|_{\infty},$

we have that inequality (32) combined with inequality (34) give

$\int_{a}^{s}\Big{[}\|S(t)\|_{\infty}+\|\nabla
S(t)\|_{\infty}+...+\|\nabla^{l}S(t)\|_{\infty}\Big{]}dt\leq
C(C_{0},\beta,\theta,C_{L},l,\Sigma,M_{0})\Big{[}Q(a)^{\gamma}-Q(s)^{\gamma}\Big{]}$
(5.7)

and the monotonicity of $Q(t)$ allows us to simply write

$\int_{a}^{s}\Big{[}\|S(t)\|_{\infty}+\|\nabla
S(t)\|_{\infty}+...+\|\nabla^{l}S(t)\|_{\infty}\Big{]}dt\leq
C(C_{0},\beta,\theta,C_{L},l,\Sigma,M_{0})\Big{[}Q(T_{0})^{\gamma}\Big{]}$
(5.8)

for all $a,s\in[T_{0},\infty)$ and this, pushing $s\to\infty$ and $a\to
T_{0}$, implies (30), which completes the proof of the proposition.
$\blacksquare$

##  6\. Rate of convergence

In this section we will prove Theorem 1.2. Our approach to the proof will be
through contradiction. We shall first prove the following lemma, which we
refer to as _the rate lemma_ , and which concerns the $C^{2,\alpha}$ norm of
the graph-function which represents $M_{t}$ as a graph of $v(t)$ over
$\Sigma$.

######  Lemma 6.1.

If for every $m\geq 0$, we have $C_{m}^{\prime}$ such that for $t_{i}>T_{0}$

$||v(\cdot,t_{i})||_{C^{2,\alpha}}<C^{\prime}_{m}e^{-mt_{i}},$

then we may also find constants $C_{m}^{\prime\prime}$ such that for
$t>T^{\prime}$ (where $T^{\prime}$ is the first $t_{i}>T_{0}$)

$\int_{M_{t_{i}}}e^{-\frac{|x|^{2}}{4}}d\mu-\int_{\Sigma}e^{-\frac{|x|^{2}}{4}}d\mu\leq
C^{{}^{\prime\prime}}_{m}e^{-mt_{i}}.$

_Proof:_ Suppose, in accordance with our hypotheses, that for every $m>0$, we
could find $C^{\prime}_{m}$ such that for $t>T_{0}$ we have

$\|{v}\|_{C^{2,\alpha}}<C^{\prime}_{m}e^{-mt_{i}}$

for all $t_{i}$.

We now take the time to rewrite $\int_{M_{t_{i}}}e^{-\frac{|x|^{2}}{4}}d\mu$,
as Schulze does in [13], as an integral over $\Sigma$ as follows

$\int_{M_{t_{i}}}e^{-\frac{|x|^{2}}{4}}d\mu=\int_{\Sigma}e^{-\frac{|y+v(y,t_{i})\,\nu|^{2}}{4}}|J|(y,v,\nabla^{\Sigma}v)d\mu(y)$

where $|J|(y,v,\nabla^{\Sigma}v)$ is the Jacobian of the transformation $y\to
y+v(y,t_{i})\nu$. We shall now rewrite
$\int_{M_{t_{i}}}e^{-\frac{|x|^{2}}{4}}d\mu-\int_{\Sigma}e^{-\frac{|x|^{2}}{4}}d\mu$
as

$\int_{\Sigma}\Big{(}e^{-\frac{|y+v(y,t_{i})\nu|^{2}}{4}}|J|(y,v,\nabla^{\Sigma}v)-e^{-\frac{|y|^{2}}{4}}\Big{)}d\mu(y).$

We further simplify this expression by writing

$e^{-\frac{|y+v(y,t_{i})\nu|^{2}}{4}}=e^{-\frac{|y|^{2}}{4}}\cdot
e^{-\frac{|v(y,t_{i})\nu|^{2}}{4}}\cdot e^{-\frac{\langle
y,v(y,t_{i})\nu\rangle}{2}}$

and then we rewrite our key quantity
$\int_{\Sigma}\Big{(}e^{-\frac{|y+v(y,t_{i})\nu|^{2}}{4}}|J|(y,v,\nabla^{\Sigma}v)-e^{-\frac{|y|^{2}}{4}}\Big{)}d\mu(y)$,
breaking it up as follows, writing our Jacobian as simply $|J|$

$\begin{split}\int_{\Sigma}\Big{(}e^{-\frac{|y+v(y,t_{i})\nu|^{2}}{4}}|J|(y,v,\nabla^{\Sigma}v)-e^{-\frac{|y|^{2}}{4}}\Big{)}d\mu(y)=&\int_{\Sigma}\Big{(}e^{-\frac{|y+v(y,t_{i})\nu|^{2}}{4}}|J|-e^{-\frac{|y|^{2}}{4}}\cdot
e^{-\frac{|v(y,t_{i})\nu|^{2}}{4}}|J|\Big{)}d\mu(y)\\\
&+\int_{\Sigma}\Big{(}e^{-\frac{|y|^{2}}{4}}\cdot
e^{-\frac{|v(y,t_{i})\nu|^{2}}{4}}|J|-e^{-\frac{|y|^{2}}{4}}|J|\Big{)}d\mu(y)\\\
&+\int_{\Sigma}\Big{(}e^{-\frac{|y|^{2}}{4}}|J|-e^{-\frac{|y|^{2}}{4}}\Big{)}d\mu(y)\\\
=&\int_{\Sigma}\Big{(}e^{-\frac{\langle
y,v(y,t_{i})\nu\rangle}{2}}-1\Big{)}e^{-\frac{|y|^{2}}{4}}\cdot
e^{-\frac{|v(y,t_{i})\nu|^{2}}{4}}|J|d\mu(y)\\\
&+\int_{\Sigma}\Big{(}e^{-\frac{|v(y,t_{i})\nu|^{2}}{4}}-1\Big{)}e^{-\frac{|y|^{2}}{4}}|J|d\mu(y)\\\
&+\int_{\Sigma}\Big{(}|J|-1\Big{)}e^{-\frac{|y|^{2}}{4}}d\mu(y)\\\
=&:A+B+D.\end{split}$

Since we assumed our flow is compact, we know that $y$ is bounded, and using
that $v(y,t_{i})$ is small, there exists a uniform constant $C$ that may
change from line to line in a uniform way, so that

$\begin{split}A=&\int_{\Sigma}\Big{(}e^{-\frac{\langle
y,v(y,t_{i})\nu\rangle}{2}}-1\Big{)}e^{-\frac{|y|^{2}}{4}}\cdot
e^{-\frac{|v(y,t_{i})\nu|^{2}}{4}}|J|d\mu(y)\\\
\leq&\int_{\Sigma}\Big{|}\langle
y,v(y,t_{i})\nu\rangle\Big{|}e^{-\frac{|y|^{2}}{4}}\cdot
e^{-\frac{|v(y,t_{i})\nu|^{2}}{4}}|J|d\mu(y)\\\
\leq&C\|v\|_{C_{0}}\int_{\Sigma}e^{-\frac{|y|^{2}}{4}}d\mu(y)\\\ \end{split}$

since $|J|$ is bounded. We also have a uniform constant $C$ which may change
uniformly from line to line such that

$\begin{split}B=&\int_{\Sigma}\Big{(}e^{-\frac{|v(y,t_{i})|^{2}}{4}}-1\Big{)}e^{-\frac{|y|^{2}}{4}}|J|d\mu(y)\\\
\leq&\int_{\Sigma}\frac{|v(y,t_{i})|^{2}}{2}e^{-\frac{|y|^{2}}{4}}|J|d\mu(y)\\\
\leq&C\|v\|_{C_{0}}^{2}\int_{\Sigma}e^{-\frac{|y|^{2}}{4}}d\mu(y)\\\
\leq&C\|v\|_{C_{0}}\int_{\Sigma}e^{-\frac{|y|^{2}}{4}}d\mu(y)\end{split}$

where we used the Talor expansion of the exponential function and the
smallness of $v$. Finally, we have $D$. Our Jacobian $|J|$ is the Jacobian of
the transformation $y\to y+v(y,t_{i})\nu(y)$, with the entries of its
corresponding matrix $J$ given by

$J_{jk}=\partial_{j}(y+v(y,t_{i})\nu(y))_{k}=\delta_{jk}+\partial_{j}v(y,t_{i})\nu_{k}-v(y,t_{i})h_{jl}\big{(}\frac{\partial\varphi}{\partial
x_{l}}\big{)}_{k}.$

One can see that every term of $|J|-1$ contains $v$ or its first derivatives
(or the entries of the second fundamental form which are of course
bounded),and so it is also only dependant on $\|{v}_{C^{1}}$, and we can see,
by writing out the entries of its corresponding matrix, that $J-1$ depends on
$v$ and its first derivatives in a sub-linear way, as such, we can write.

$\int_{M_{t_{i}}}e^{-\frac{|x|^{2}}{4}}d\mu-\int_{\Sigma}e^{-\frac{|x|^{2}}{4}}d\mu\leq\tilde{C}\|v\|_{C^{2,\alpha}}$

where $\tilde{C}$ is a uniform constant dependant only on quantities bounded
along the flow. $\blacksquare$

We are now ready to continue to the proof of Theorem 1.2.

###### Proof of Theorem 1.2.

We begin by stating Theorem 1.1 of [13], which says

Let $\mathcal{M}=(\mu_{t})_{t\in(t_{1},0)}$ with $t<0$ be an integral
$n$-Brakke flow such that

  1. i)

$(\mu_{t})_{t\in(t_{1},t_{2})}$ is sufficiently close in measure to
$\mathcal{M}_{\Sigma}$ for some $t_{1}<t_{2}<0$.

  2. ii)

$\int_{M_{t}}\frac{1}{(-4\pi
t)^{\frac{n}{2}}}e^{-\frac{|x|^{2}}{4}}d\mathcal{H}^{n}\geq\int_{\Sigma}\frac{1}{(-4\pi
t)^{\frac{n}{2}}}e^{-\frac{|x|^{2}}{4}}d\mathcal{H}^{n}$.

Then $\mathcal{M}$ is a smooth flow for $t\in[(t_{1}+t_{2})/2,0)$, and the
rescaled surfaces $\tilde{M}_{t}$ can be rewritten as normal graphs over
$\Sigma$, given by smooth sections $v(t)$ of the normal bundle with
$\|v(t)\|_{C^{m}(T^{\perp}\Sigma)}$ uniformly bounded for all
$t\in[(t_{1}+t_{2})/2,0)$ and all $m\in\mathbb{N}$. Furthermore, there exists
a self-similarly shrinking surface $\Sigma^{\prime}$ with

$\Sigma^{\prime}=\text{graph}_{\Sigma}(v^{\prime})$

and

$\|v(t)-v^{\prime}\|_{C^{m}}\leq c_{m}(\text{log}(-1/t))^{-\alpha_{m}}$

for some constants $c_{m}>0$ and exponents $\alpha_{m}>0$ for all
$m\in\mathbb{N}$.

If $\tilde{\varphi}:M\times[0,\infty)\to\mathbb{R}^{n+1}$ is a rescaled mean
curvature flow moving by (1.3), then as before we define $\tau=-e^{-t}$, and
$\varphi:M\times[-1,0)\to\mathbb{R}^{n+1}$ is moving by (1.1) via rescaling
$\varphi(\cdot,\tau)=e^{-\frac{t}{2}}\widetilde{\varphi}(\cdot,t).$ In the
non-rescaled flow, per our hypothesis, we can find some
$\tau_{1},\tau_{2}\in[-1,0)$ ($\tau_{1}<\tau_{2}$) such that our restriction
of the mean curvature flow
$\varphi:M\times[\tau_{1},\tau_{2})\to\mathbb{R}^{n+k}$ is sufficiently close
in measure to the homothetically shrinking flow produced by the shrinker
$\Sigma$. Thus we can apply Theorem 1.1 from [13], to get that our solutions
moving by (1.3) can be expressed as normal graphs of a function
$v(\cdot,\tau)$ for $\frac{\tau_{1}+\tau_{2}}{2}<\tau<0$.

As such, we may define $v(\cdot,t)$ (we use the same $v$, just change its time
variable) for all $t>-\text{log}(-\frac{\tau_{1}+\tau_{2}}{2})=:T_{0}$ such
that our original flow $\tilde{\varphi}$ is expressible as a normal graph of
$v(\cdot,t)$ over $\Sigma$ for all $t>T_{0}$. This establishes the needed
graphicality.

We will now approach the proof by contradiction. Suppose that the theorem were
false, that is, suppose that $M_{t}$ is a rescaled mean curvature flow such
that for every $m>0$, we have $C^{\prime}_{m}$ such that for $t_{i}>T_{0}$, we
have

$\|{v(\cdot,t_{i})}_{C^{2,\alpha}}<C^{\prime}_{m}e^{-mt_{i}}$

for every $i$, but $M_{t}$ is never a shrinker.

By Lemma 3.1 equation (23), we have that

$\begin{split}\dot{E}(t)=&-\int S^{4}d\tilde{\mu}+2\int
S\Big{[}\mathcal{L}S+(|A|^{2}+\frac{1}{2})S\Big{]}d\tilde{\mu}\\\ =&-\int
S^{4}d\tilde{\mu}-2\int|\nabla
S|^{2}d\tilde{\mu}+2\int(|A|^{2}+\frac{1}{2})S^{2}d\tilde{\mu}\\\
\leq&2\int(|A|^{2}+\frac{1}{2})S^{2}d\tilde{\mu}\\\
\leq&K(|A|^{2})E(t)\end{split}$

so we can write for $t>T_{0}$ that

$E(t)\leq E(T_{0})e^{K(t-T_{0})}.$

Now we recall the quantity

$N(t)=\frac{\dot{E}}{E}=\frac{d}{dt}\text{ln}(E).$

We claim that we are able to produce some $K^{\prime}>0$, such that we have
that $E(t)\geq E(T_{0})e^{-K^{\prime}(t-T_{0})}$. To show this, we will prove
that $\text{ln}(E(t))\geq-K^{\prime}(t-T_{0})+\text{ln}(E(T_{0}))$ by proving
that for all $t$, we have

$N(t)\geq-K^{\prime}$

i.e. that $N$ is bounded from below for all time. Corollary 8.1 gives us that

$\dot{N}(t)\geq-C(||S||_{\infty}+||\nabla
S||_{\infty}+||\nabla^{2}S||_{\infty}).$

We then apply Proposition 5.1, inequality (33) for the cases $l=0,1,2$ to get
that for for all $t$,

$\begin{split}N(t)=&\int_{0}^{t}\dot{N}(\xi)d\xi\\\
\geq&-C\int_{0}^{\infty}(\|{S(\xi)}_{\infty}+\|{\nabla
S(\xi)}_{\infty}+\|{\nabla^{2}S(\xi)}_{\infty})d\xi\\\
=:&K^{\prime}>-\infty,\end{split}$ (6.1)

which gives us the appropriate $K^{\prime}$, and allows us to write that

$E(T_{0})e^{-K^{\prime}(t-T_{0})}\leq E(t)\leq E(T_{0})e^{K(t-T_{0})}$ (6.2)

for all $t>T_{0}$. Huisken’s monotonicity formula gives

$\frac{d}{dt}\Omega(M_{t})=-E(t)$

so we can use inequality (38) to write that for $s\geq t$

$\begin{split}\Omega(M_{t})-\Omega(M_{s})=&\int_{t}^{s}E(\zeta)d\zeta\\\
\geq&\int_{t}^{s}E(T_{0})e^{-K^{\prime}(\zeta-T_{0})}d\zeta\\\
=&\frac{E(T_{0})}{K^{\prime}}\Big{[}e^{-K^{\prime}(t-T_{0})}-e^{-K^{\prime}(s-T_{0})}\Big{]}.\end{split}$

Pushing $s\to\infty$ gives

$\Omega(M_{t})-\Omega(\Sigma)\geq\frac{E(T_{0})}{K^{\prime}}e^{-K^{\prime}(t-T_{0})}.$
(6.3)

We apply the rate lemma, Lemma 6.1, to conclude that for every $m>0$, we must
also have $C^{\prime\prime}_{m}$ such that for $t_{i}>T_{0}$, we have

$\int_{M_{t_{i}}}e^{-\frac{|x|^{2}}{4}}d\mu-\int_{\Sigma}e^{-\frac{|x|^{2}}{4}}d\mu\leq
C^{\prime\prime}_{m}e^{-mt_{i}}$ (6.4)

for every $i$.

Combining inequalities (39) and (40) yields

$E(T_{0})\leq Ce^{(K^{\prime}-m)t_{i}}$

for every $m$ and $i$. Since we may increase $m$ and $i$ arbitrarily, we
conclude that we must have

$E(T_{0})=0,$ (6.5)

so $M_{T_{0}}$ must be a shrinker. By the backwards uniqueness results for
mean curvature flow proven in [8], we have that our flow is a shrinker for all
time, which is of course a contradiction. We are left to conclude that either
there exist $C,m>0$ such that $\|{v(\cdot,t_{i})}_{C^{2,\alpha}(\Sigma)}\geq
Ce^{-mt_{i}}$ or $M_{t}$ is a shrinker moving by diffeomorphisms.
$\blacksquare$

∎

## References

  * [1] K. Brakke. The Motion of a Surface by its Mean Curvature, Princeton Legacy Library
  * [2] T. Colding, W. Minicozzi. Generic Mean Curvature Flow 1; Generic Singularities, Annals of Mathematics (1.1) 175 (2012), no. 2 755-833
  * [3] Q. Ding, Y.L. Lin. The Rigidity Theorems of Self Shrinkers, Transactions of the American Mathematical Society, Vol. 366, No. 10 (October 2014), pp. 5067-5085
  * [4] K. Ecker. Regularity Theory for Mean Curvature Flow, Birkhauser
  * [5] K. Ecker, G. Huisken. Mean Curvature Evolution of Entire Graphs. The Annals of Mathematics, 2nd Ser., Vol. 130, No. 3. (Nov., 1989), pp. 453-471.
  * [6] S. Friedberg, A. Insel, L. Spence. Linear Algebra, 4th Edition, Prentice Hall (2003)
  * [7] R. Hamilton. Three-manifolds of positive Ricci curvature, J. Differential Geom. 17(1.1): 255-306 (1982). DOI: 10.4310/jdg/1214436922
  * [8] H. Huang. Backwards uniqueness of the mean curvature flow, arXiv:0907.0862v4
  * [9] G. Huisken. Asymptotic Behavior for Singularities of the Mean Curvature Flow, JDG 31 (1990), 285-299
  * [10] B. Kotschwar. On the Maximal Rate of Convergence Under the Ricci Flow, International Mathematics Research Notices (2022) 4, 2484-2512
  * [11] C. Mantegazza. Lecture Notes on Mean Curvature Flow, Birkhauser
  * [12] B. O’Niell. Semi-Riemannian Geometry with Applications to Relativity, Academic Press
  * [13] F. Schultze. Uniqueness of compact tangent flows in Mean Curvature Flow, Journal für die reine und angewandte Mathematik
  * [14] S. Sun, Y. Wang. On the Kähler-Ricci flow near a Kähler-Einstein metric, J. Reine Angew. Math. 699 (2015), 143–158
  * [15] L. Tu. An Introduction to Manifolds, Second Edition, Springer Universitext