# The rate of $\mathbb{F}$-convergence for Ricci flows with closed and smooth
tangent flows
_In dedication to the $70$th birthday of Professor Peter Li_

Pak-Yeung Chan, Zilu Ma, and Yongjia Zhang

###### Abstract

This article is a continuation of [CMZ21c], where we proved that a Ricci flow
with a closed and smooth tangent flow has unique tangent flow, and its
corresponding forward or backward modified Ricci flow converges in the rate of
$t^{-\beta}$ for some $\beta>0$. In this article, we calculate the
corresponding $\mathbb{F}$-convergence rate: after being scaled by a factor
$\lambda>0$, a Ricci flow with closed and smooth tangent flow is
$|\log\lambda|^{-\theta}$ close to its tangent flow in the $\mathbb{F}$-sense,
where $\theta$ is a positive number, $\lambda\gg 1$ in the blow-up case, and
$\lambda\ll 1$ in the blow-down case.

##  1 Introduction

In [Bam20a, Bam20b, Bam20c], the newly proposed and studied notion of
$\mathbb{F}$-convergence has greatly expanded the horizon of the field of
Ricci flow. Bamler’s notion of $\mathbb{F}$-distance can be regarded as the
flow version of the Gromov-$W_{1}$-Wasserstein distance studied by Sturm
[St06a, St06b]; the latter measures the closeness of two metric spaces endowed
with probability measures, whereas the former is a distance between two metric
flows (see [Bam20b]). As an effective version of weak convergence, the success
of the $\mathbb{F}$-convergence reveals that the Ricci flow and the Cheeger-
Colding theory [CC97, CC20a, CC20b] are united on the fundamental level, since
manifolds with Ricci curvature bounded from below can be regarded as super
Ricci flows, for which Bamler’s $\mathbb{F}$-compactness theorems [Bam20b] are
applicable.

One of the central problems in the field of Cheeger-Colding theory is that of
tangent cones, and, in particular, of their uniqueness. For instance, Cheeger-
Tian [CT94] proved the uniqueness of tangent cones at infinity for Ricci flat
manifolds with Euclidean volume growth under certain curvature and
integrability assumptions. This result was later strengthened by Colding-
Minicozzi [CM14], who also proved an analogous result for the mean curvature
flow [CM15]—that if a tangent flow at a point is a cylinder, then it is the
unique tangent flow at that point. For the uniqueness of tangent cones in the
field of minimal surface, refer to [AA81, S83, W83]. Recently, Colding-
Minicozzi [CM21] proved that the cylindrical singularities in the Ricci flow
are isolated singularities, and that their appearance implies the uniqueness
of the blow-up limit.

In the previous paper of the authors [CMZ21c], following the methods of [Se06,
Ac12, SW15], it is proved that if a Ricci flow has closed and smooth tangent
flow, then the tangent flow must be unique, and is the canonical form of a
closed shrinking gradient Ricci soliton. Here the tangent flow is either of an
ancient solution at infinity, or of a Ricci flow developing a finite-time
singularity at the singular time. Furthermore, we have also proved that the
corresponding modified forward or backward Ricci flow converges to the
shrinker metric at the rate of $t^{-\beta}$ as $t\to\infty$; see Section 2 for
more details.

The modified Ricci flow is the gradient flow of Perelman’s $\mu$-functional.
It differs from the Ricci flow by a $1$-parameter family of self-
diffeomorphisms. The convergence of the modified Ricci flow obviously implies
the Cheeger-Gromov-Hamilton convergence of the scaled sequence of the
corresponding Ricci flow, and therefore its convergence in the
$\mathbb{F}$-sense (c.f. [CMZ21a, Theorem 6.1]). Nevertheless, [CMZ21a,
Theorem 6.1] is only a qualitative result, and no comparison between the rates
of the two kinds of convergence is made. In this paper, we shall prove that
for a Ricci flow with smooth and closed tangent flow, the scaled metric flow
has a convergence rate of the form $|\log\lambda|^{-\beta}$, where $\lambda>0$
is the scaling factor, and $\lambda\nearrow\infty$ or $\lambda\searrow 0$ when
the tangent flow is a blow-up or blow-down limit, respectively.

Let us recall the settings of [CMZ21c], which is identical to that of the
present article. We consider a metric flow
$\displaystyle\left(\mathcal{X},\mathfrak{t},({\rm dist}_{t})_{t\in
I},(\nu_{x\,|\,s})_{x\in\mathcal{X},s\in I,s\leq\mathfrak{t}(x)}\right)$
induced by a smooth Ricci flow $(M^{n},g_{t})_{t\in I}$, where $M$ is a closed
manifold. See [Bam20b, Definition 5.1] for the definition of metric flow
pairs. For any $\lambda>0$, we shall use the notation
$\mathcal{X}^{0,\lambda}$ to represent the scaled metric flow ([Bam20b,
Section 6.8]):

$\mathcal{X}^{0,\lambda}=\left(\mathcal{X},\lambda^{2}\mathfrak{t},(\lambda{\rm
dist}_{\lambda^{-2}t})_{t\in\lambda^{2}I},(\nu_{x\,|\,\lambda^{-2}s})_{x\in\mathcal{X},s\in\lambda^{2}I,s\leq\mathfrak{t}(x)}\right),\quad
t\in\lambda^{2}I.$

  1. (1)

If $(M,g_{t})$ is ancient, then we assume that $I=(-\infty,0]$ and hence
$\mathcal{X}=M^{n}\times(-\infty,0]$. Let us fix a point $(p_{0},0)\in
M\times\\{0\\}=\mathcal{X}_{0}$ and let $\mu_{t}:=\nu_{p_{0},0\,|\,t}$ be the
conjugate heat kernel based at $(p_{0},0)$. By Bamler’s compactness theorems
[Bam20b], for any sequence $\\{\tau_{i}\\}_{i=1}^{\infty}$ with
$\tau_{i}\nearrow\infty$, we have

$\displaystyle\left(\mathcal{X}^{0,(\tau_{i})^{-\frac{1}{2}}},\left(\mu_{\tau_{i}t}\right)_{t\in(-\infty,0]}\right)\xrightarrow{\makebox[28.45274pt]{$\mathbb{F}$}}\left(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in-(\infty,0)}\right)$
(1.1)

after passing to a subsequence. Here $\mathcal{X}^{\infty}$ is a metric flow
over $(-\infty,0]$. If the original ancient solution $(M,g_{t})$ has uniformly
bounded Nash entropy, that is, if

$\displaystyle\text{there exist $(p,t)\in M^{n}\times(-\infty,0]$ and
$Y<\infty$ such that}\quad\mathcal{N}_{p,t}(\tau)\geq-Y\text{ for all
}\tau>0,$ (1.2)

then the singular set of $\mathcal{X}^{\infty}$ has parabolic Minkowski
codimension no smaller than $4$, and
$\left(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in-(\infty,0)}\right)$ is a
metric soliton; see [Bam20c].

  2. (2)

If $(M,g_{t})$ has finite-time singularity, then we assume that $I=[-T,0)$ and
$t=0$ is the singular time. In this case, the metric flow $\mathcal{X}$ is not
defined at $t=0$. We will let $(\mu_{t})_{t\in I}$ be a singular conjugate
heat kernel based at $t=0$ as defined in [MM15]. By Bamler’s compactness
theorem [Bam20b] again, for any sequence $\\{\tau_{i}\\}_{i=1}^{\infty}$ with
$\tau_{i}\searrow 0$, we also have that (1.1) holds after passing to a
subsequence. In this case, the sequence is automatically noncollapsed, and the
limit $\left(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in-(\infty,0)}\right)$
is a metric soliton whose singular set has parabolic Minkowski codimension no
smaller than $4$ [Bam20c].

The results of this paper are as follows.

######  Theorem 1.1.

Let $(M,g_{t})_{t\in(-\infty,0]}$ be an ancient Ricci flow satisfying (1.2),
where $M$ is a smooth and closed manifold. Suppose that there is a sequence
$\tau_{i}\nearrow\infty$, such that the metric flow $\mathcal{X}^{\infty}$
given in (1.1) is induced by a smooth and closed Ricci flow. Then the
following holds. For any $A>1$, there are constants $\overline{\lambda}(A)>0$
and $C(A)<\infty$, which also depend on $\mathcal{X}^{\infty}$, such that

${\rm
dist}_{\mathbb{F}}^{[-A,-1/A]}\left(\left(\mathcal{X}^{0,\lambda},\mu_{\lambda^{-2}t}\right),(\mathcal{X}^{\infty},\mu^{\infty}_{t})\right)\leq
C(-\log\lambda)^{-\beta}\quad\text{whenever}\quad\lambda\leq\overline{\lambda},$

where $\beta>0$ depends only on $\mathcal{X}^{\infty}$.

######  Theorem 1.2.

Let $(M,g_{t})_{t\in[-T,0)}$ be a Ricci flow, where $M$ is a smooth and closed
manifold, and $t=0$ is the singular time. Suppose that there is a sequence
$\tau_{i}\searrow 0$, such that the metric flow $\mathcal{X}^{\infty}$ given
in (1.1) is induced by a smooth and closed Ricci flow. Then the following
holds. For any $A>1$, there are constants $\underline{\lambda}(A)<\infty$ and
$C(A)<\infty$, which also depend on $\mathcal{X}^{\infty}$, such that

${\rm
dist}_{\mathbb{F}}^{[-A,-1/A]}\left(\left(\mathcal{X}^{0,\lambda},\mu_{\lambda^{-2}t}\right),(\mathcal{X}^{\infty},\mu^{\infty}_{t})\right)\leq
C(\log\lambda)^{-\beta}\quad\text{whenever}\quad\lambda\geq\underline{\lambda},$

where $\beta>0$ depends only on $\mathcal{X}^{\infty}$.

To summarize the structure of this article, recall that according to the
definition of $\mathbb{F}$-distance (c.f. Definition 2.3 below), the proofs of
Theorem 1.1 and Theorem 1.2 consist of two parts: one is to estimate the
$1$-Wasserstein distance between corresponding conjugate heat kernels on
$\mathcal{X}^{0,\lambda}$ and on $\mathcal{X}^{\infty}$, the other is to
estimate the $1$-Wasserstein between the referential conjugate heat flows
$\mu_{\lambda^{-2}t}$ and $\mu^{\infty}_{t}$; these two estimates are carried
out in Section 3 and Section 4, respectively. Finally, in Section 5, we
conclude the proofs of the above theorems.

##  2 Background and preparatory results

In this section, we briefly summarize the main results in [CMZ21c], establish
the current settings, and show some preliminary results for the proof of
Theorem 1.1 and Theorem 1.2. For more basic definitions and notations such as
the conjugate heat kernel and the Nash entropy, etc., refer to [CMZ21c].

###  2.1 Modified Ricci flow

Under the assumptions of Theorem 1.1 and Theorem 1.2, the tangent flow
$\mathcal{X}^{\infty}$ is induced by the canonical form of a shrinking
gradient Ricci soliton $(M^{n},g_{o},f_{o})$. We normalize the metric $g_{o}$
and the potential function $f_{o}$ in the way that

$\displaystyle\operatorname{Ric}_{g_{o}}+\nabla^{2}f_{o}$
$\displaystyle=\tfrac{1}{2}g_{o},$
$\displaystyle\int(4\pi)^{-\frac{n}{2}}e^{-f_{o}}dg_{o}$ $\displaystyle=1.$

We have also followed the idea of [SW15] to consider (backward) modified Ricci
flow instead. The modified Ricci flow is defined to be the gradient flow of
the functional

$\mu_{g}:=\inf\left\\{\int_{M}(|\nabla
f|^{2}+R_{g}+f-n)(4\pi)^{-\frac{n}{2}}e^{-f}dg\,\left|\,\int_{M}(4\pi)^{-\frac{n}{2}}e^{-f}dg=1\right\\}\right.=\mu(g,1),$

where $\mu(\cdot,\cdot)$ is Perelman’s $\mu$-functional. Specifically, the
modified Ricci flow and its backward version are respectively defined as

$\displaystyle\partial_{t}g_{t}$
$\displaystyle=\nabla\mu_{g_{t}}=-2\left(\operatorname{Ric}_{g_{t}}+\nabla^{2}f_{g_{t}}-\tfrac{1}{2}g_{t}\right),$
$\displaystyle\partial_{t}g_{t}$
$\displaystyle=-\nabla\mu_{g_{t}}=2\left(\operatorname{Ric}_{g_{t}}+\nabla^{2}f_{g_{t}}-\tfrac{1}{2}g_{t}\right).$

Here (and henceforth) we use $f_{g}$ to represent the minimizer of $\mu_{g}$,
should it exist and be unique.

In fact, if the minimizer of $\mu_{g}$ is unique and depends smoothly on $g$,
then one may easily convert a Ricci flow into a modified Ricci flow or a
backward modified Ricci flow. More precisely, let $(M,g_{t})_{t\in I}$, where
$M$ is a closed manifold and $I\subset(-\infty,0]$, be a Ricci flow, then
$\overline{g}_{s}$ defined as

$\displaystyle\left\\{\begin{array}[]{rl}\tilde{g}_{s}&=\ e^{s}g_{-e^{-s}},\\\
\partial_{s}\psi_{s}&=\
-\nabla_{\tilde{g}_{s}}f_{\tilde{g}_{s}}\circ\psi_{s},\\\ \overline{g}_{s}&=\
\psi_{s}^{*}\tilde{g}_{s},\end{array}\right.$ (2.4)

is a modified Ricci flow, and $\overline{g}_{s}$ defined as

$\displaystyle\left\\{\begin{array}[]{rl}\tilde{g}_{s}&=\ e^{-s}g_{-e^{s}},\\\
\partial_{s}\psi_{s}&=\
\nabla_{\tilde{g}_{s}}f_{\tilde{g}_{s}}\circ\psi_{s},\\\ \overline{g}_{s}&=\
\psi_{s}^{*}\tilde{g}_{s},\end{array}\right.$ (2.8)

is a backward modified Ricci flow.

###  2.2 Łojaciewicz argument

Although the flows in (2.4) and (2.8) are not necessarily well-defined, it is
proved in [CMZ21c] that, for the Ricci flow $g_{t}$ considered in Theorem 1.1
or Theorem 1.2, its corresponding backward modified Ricci flow or modified
Ricci flow is well defined, and converges to $g_{o}$. This is due to the
Łojaciewicz argument of Sun-Wang [SW15] which we shall briefly summarize in
this subsection.

####  2.2.1 Łojaciewicz inequality

Let us recall the Łojaciewicz inequality in [SW15]. Letting $\mathfrak{R}(M)$
be the space of Riemannian metrics on $M$, $k\,\in\mathbb{N}$,
$\gamma\in(0,1)$, the $C^{k,\gamma}$ neighborhood of $g_{o}$ is defined to be

$\mathcal{V}^{k,\gamma}_{\delta}:=\left\\{g\in\mathfrak{R}(M)\,\Big{|}\,\|g-g_{o}\|_{C^{k,\gamma}_{g_{o}}}\leq\delta\right\\}.$

Sung-Wang [SW15] proved the following Łojaciewicz inequality:

######  Theorem 2.1 (Lemma 3.1 in [SW15]).

There is a $C^{k,\gamma}$ ($k\gg 1$) neighborhood $\mathcal{U}$ of $g_{o}$,
called a _regular neighborhood_ , such that for all $g\in\mathcal{U}$, there
is a unique $f_{g}$, and the map $P:f\mapsto f_{g}$ is analytic on
$\mathcal{U}$. Furthermore, there are constants $C>0$ and
$\alpha\in[\frac{1}{2},1)$, such that for any $g\in\mathcal{U}$, we have

$\|\nabla\mu_{g}\|_{L^{2}_{g}}\geq C|\mu_{g_{o}}-\mu_{g}|^{\alpha},$

where $\|\,\cdot\,\|_{L^{2}_{g}}$ is the $L^{2}$ norm taken with respect to
$g$.

####  2.2.2 Rate of convergence

We recall some basic computations related to the Łojaciewicz arguments, which
are very useful in the present article. Assume that $\overline{g}_{s}$, where
$s\in[0,\infty)$, is an _either forward or backward_ modified Ricci flow on a
closed manifold $M^{n}$, which converges smoothly to a shrinker metric
$g_{o}$. We also assume that $\overline{g}_{s}$ is in a small neighborhood of
$g_{o}$ for all $s\geq 0$. By [CMZ21c, Corollary 1.3, Corollary 1.4], the
Ricci flow in Theorem 1.1 or Theorem 1.2 can be converted to such a backward
or forward modified Ricci flow through (2.8) or (2.4), respectively.

Since $\overline{g}_{s}$ is always in a neighborhood of $g_{o}$, it is always
regular, and one may estimate the curvature of $\overline{g}_{s}$ in terms of
the curvature of $g_{o}$. Then, the standard Shi’s estimates and the
regularity of the metric imply that

$\left|\nabla^{l}\operatorname{Rm}_{\overline{g}_{s}}\right|_{\overline{g}_{s}}\leq
C(l),\quad\left\|\operatorname{Ric}_{\overline{g}_{s}}+\nabla_{\overline{g}_{s}}^{2}f_{\overline{g}_{s}}-\tfrac{1}{2}\overline{g}_{s}\right\|_{C^{l,\gamma}_{\overline{g}_{s}}}\leq
C(l),$

for all $s\in[0,\infty)$ and $l\in\mathbb{N}$; see, for instance, the proof of
[SW15, Lemma 3.2]. Let $\alpha\in[\frac{1}{2},1)$ be the constant in the
statement of Theorem 2.1 and fix $\beta\in(2-1/\alpha,1)$. Now we consider the
forward and backward cases separately.

(1) Backward case. The argument below is from the appendix of [CMZ21c], which
is modeled after [SW15]. By the standard embedding and interpolation
inequality, for any integer $l$, there are integers $p\geq l$ and $N(p)\gg p$,
such that

$\displaystyle\left\|\partial_{s}\overline{g}_{s}\right\|_{C^{l,\gamma}_{\overline{g}_{s}}}\leq
C(l)\left\|\partial_{s}\overline{g}_{s}\right\|_{W^{p,2}_{\overline{g}_{s}}}$
$\displaystyle\leq$ $\displaystyle
C(l,p)\left\|\partial_{s}\overline{g}_{s}\right\|_{L^{2}_{\overline{g}_{s}}}^{\beta}\left\|\operatorname{Ric}_{\overline{g}_{s}}+\nabla_{\overline{g}_{s}}^{2}f_{\overline{g}_{s}}-\tfrac{1}{2}\overline{g}_{s}\right\|^{1-\beta}_{W^{N(p),2}_{\overline{g}_{s}}}$
$\displaystyle\leq$ $\displaystyle
C(l)\left\|\partial_{s}\overline{g}_{s}\right\|_{L^{2}_{\overline{g}_{s}}}^{\beta},$

for all $s\geq 0$. Therefore, by the definition of backward modified Ricci
flow and Theorem 2.1, we have

$\displaystyle\frac{d}{ds}\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{1-(2-\beta)\alpha}$
$\displaystyle=-2(1-(2-\beta)\alpha)\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{-(2-\beta)\alpha}\|\nabla\mu_{\overline{g}_{s}}\|^{2}_{L^{2}_{\overline{g}_{s}}}$
$\displaystyle\leq-C(\beta)\left\|\partial_{s}\overline{g}_{s}\right\|_{L^{2}_{\overline{g}_{s}}}^{\beta}.$

Then

$\left\|\partial_{s}\overline{g}_{s}\right\|_{C^{l,\gamma}_{g_{o}}}\leq
C(l)\left\|\partial_{s}\overline{g}_{s}\right\|_{C^{l,\gamma}_{\overline{g}_{s}}}\leq-C(\beta,l)\frac{d}{ds}\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{1-(2-\beta)\alpha}.$
(2.9)

Integrating the above inequality from $s$ to $\infty$, we have

$\displaystyle\|\overline{g}_{s}-g_{o}\|_{C^{l,\gamma}_{g_{o}}}\leq
C(\beta,l)\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{1-(2-\beta)\alpha}.$
(2.10)

On the other hand, we compute using Theorem 2.1 again

$\displaystyle\frac{d}{ds}\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{1-2\alpha}$
$\displaystyle=$
$\displaystyle-2(1-2\alpha)\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{-2\alpha}\left\|\nabla\mu_{\overline{g}_{s}}\right\|^{2}_{L^{2}_{\overline{g}_{s}}}$
$\displaystyle\geq$ $\displaystyle
2(2\alpha-1)\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{-2\alpha}\cdot
C\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{2\alpha}$
$\displaystyle\geq$ $\displaystyle 2C(2\alpha-1).$

Integrating the above inequality, we have

$\mu_{\overline{g}_{s}}-\mu_{g_{o}}\leq
C(\alpha)s^{-\frac{1}{2\alpha-1}}\quad\text{ for all }\quad s>0.$ (2.11)

Combining (2.10) and (2.11), we have

$\displaystyle\|\overline{g}_{s}-g_{o}\|_{C^{l,\gamma}_{g_{o}}}\leq
C(\beta,l)s^{-\frac{1-(2-\beta)\alpha}{2\alpha-1}}\quad\text{ for all }\quad
s\in(0,\infty).$ (2.12)

(2) Forward case. The forward case is similar to the backward case; more
detailed computations can be found in [SW15]. Note that in this case we
consider $\mu_{g_{o}}-\mu_{\overline{g}_{s}}$ instead, since
$\mu_{\overline{g}_{s}}\leq\mu_{g_{o}}$ for all $s\in[0,\infty)$. By the same
computation as before, we have

$\left\|\partial_{s}\overline{g}_{s}\right\|_{C^{l,\gamma}_{g_{o}}}\leq
C(l)\left\|\partial_{s}\overline{g}_{s}\right\|_{C^{l,\gamma}_{\overline{g}_{s}}}\leq-C(\beta,l)\frac{d}{ds}\left(\mu_{g_{o}}-\mu_{\overline{g}_{s}}\right)^{1-(2-\beta)\alpha}$

and

$\displaystyle\|\overline{g}_{s}-g_{o}\|_{C^{l,\gamma}_{g_{o}}}\leq
C(\beta,l)s^{-\frac{1-(2-\beta)\alpha}{2\alpha-1}}\quad\text{ for all }\quad
s\in(0,\infty),$ (2.13)

where $l$, $\gamma$, and $\beta$ are the same as before.

####  2.2.3 The derivative of the $P$ operator

Recall the map $P$ defined in Theorem 2.1 is analytic in a regular
neighborhood of a shrinker metric, and hence the minimizer $f_{g}$ of
$\mu_{g}$ depends on $g$ smoothly, if $g$ is close to a shrinker metric. In
the proof of the main theorem of the article, we also need an estimate on the
derivative of $P$. In this subsection, by applying the implicit function
theorem, we show that $DP$ is a bounded operator in a small neighborhood of a
shrinker metric.

Recall that in the proof of [SW15, Lemma 2.2], Sun-Wang consider the operator
$L:C^{k,\gamma}(\mathfrak{R}(M))\times C^{k,\gamma}(M;\mathbb{R})$
$\longrightarrow C^{k-4,\gamma}(M;\mathbb{R})$ defined by

$L(g,f):=2\Delta^{f}_{g}\left(2\Delta_{g}f+f-|\nabla
f|^{2}+R_{g}\right)+(4\pi)^{-\frac{n}{2}}\int_{M}e^{-f}dv_{g}-1,$

where $k\gg 1$, $\gamma\in(0,1)$, and $C^{k,\gamma}(\mathfrak{R}(M))$ stand
for all the $C^{k,\gamma}$ Riemannian metrics on $M$ and $\Delta^{f}_{g}$ is
the weighted laplacian $\Delta_{g}-\nabla_{g}f\cdot\nabla_{g}$. Direct
computation shows that the linearization of $L$ in the second component is
given by

$\displaystyle DL_{(g,f)}(0,h)=$ $\displaystyle\
2\Delta^{f}_{g}\left(2\Delta_{g}^{f}h+h\right)-(4\pi)^{-\frac{n}{2}}\int_{M}he^{-f}dv_{g}$
$\displaystyle-2\left\langle\nabla_{g}h,\nabla_{g}\left(2\Delta_{g}f+f-|\nabla
f|^{2}+R_{g}\right)\right\rangle.$

If $(g_{o},f_{o})$ is a normalized shrinking soliton on $M$, then
$L(g_{o},f_{o})=0$ and the above expression can be simplified to

$DL_{(g_{o},f_{o})}(0,h)=2\Delta^{f_{o}}_{g_{o}}\left(2\Delta^{f_{o}}_{g_{o}}h+h\right)-(4\pi)^{-\frac{n}{2}}\int_{M}he^{-f_{o}}\,dv_{g_{o}},$

which is a linear isomorphism from $C^{k,\gamma}(M;\mathbb{R})$ to
$C^{k-4,\gamma}(M;\mathbb{R})$ ([SW15, Lemma 3.5]). Therefore, by the real
analytic version of the implicit function theorem (see the proof of [SW15,
Lemma 2.2]), there exists a $C^{k,\gamma}$-neighborhood $\mathcal{U}$ of
$g_{o}$ in $\mathfrak{R}(M)$, and a real analytic mapping $P:\mathcal{U}\to
C^{k,\gamma}(M;\mathbb{R})$, such that

$L(g,P(g))=0,\quad\text{ for all }\quad g\in\mathcal{U}.$

Furthermore,

$DL_{(g,P(g))}(0,h)=2\Delta^{P(g)}_{g}\left(2\Delta^{P(g)}_{g}h+h\right)-(4\pi)^{-\frac{n}{2}}\int_{M}he^{-P(g)}\,dv_{g},\quad\text{
for all }\quad g\in\mathcal{U}.$ (2.14)

We shall then estimate the upper bound of $DP_{g}$ for all $g$ close to
$g_{o}$ in the $C^{k,\gamma}$ sense.

######  Theorem 2.2.

There is a $C^{k,\gamma}$ ($k\gg 1$) neighborhood
$\mathcal{V}\subset\mathcal{U}$ of $g_{o}$, such that the linearization of $P$

$DP_{g}:\left(C^{k,\gamma}(\mathcal{S}(M)),\|\cdot\|_{C^{k,\gamma}_{g_{o}}}\right)\longrightarrow\left(C^{k,\gamma}(M;\mathbb{R}),\|\cdot\|_{C^{k,\gamma}_{g_{o}}}\right)$

has uniformly bounded operator norm for any $g\,\in\mathcal{V}$. Here
$C^{k,\gamma}(\mathcal{S}(M))$ stands for the space of all $C^{k,\gamma}$
symmetric $2$-tensors on $M$, and $\mathcal{U}$ is the regular neighborhood of
$g_{o}$ defined in the statement of Theorem 2.1.

###### Proof.

Let us compute $DL_{(g,f)}(w,0)$. If $g_{s}$ is a variation of $g$ with
$\frac{\partial}{\partial s}g_{s}\big{|}_{s=0}=w\in
C^{k,\gamma}(\mathcal{S}(M))$, then, by a straightforward computation, we have

$\left.\frac{\partial}{\partial s}\right|_{s=0}\Delta_{g}^{f}=-\left\langle
w,\nabla^{2}_{g}\right\rangle_{g}-\left\langle\left({\rm
div}_{g}w-\tfrac{1}{2}\nabla_{g}\operatorname{tr}_{g}w\right),\nabla_{g}\right\rangle_{g}+w(\nabla_{g}f,\nabla_{g}).$

Similarly, we have

$\displaystyle\left.\frac{\partial}{\partial
s}\right|_{s=0}\left(2\Delta_{g}f+f-|\nabla_{g}f|^{2}_{g}+R_{g}\right)$
$\displaystyle=$ $\displaystyle-2\left\langle
w,\nabla^{2}_{g}f\right\rangle_{g}-\big{\langle}\left(2{\rm
div}_{g}w-\nabla_{g}\operatorname{tr}_{g}w\right),\nabla_{g}f\big{\rangle}_{g}+w(\nabla_{g}f,\nabla_{g}f)$
$\displaystyle-\Delta_{g}\operatorname{tr}_{g}w+{\rm div}_{g}{\rm
div}_{g}w-\langle w,\operatorname{Ric}_{g}\rangle_{g}.$

Hence if we let $u:=2\Delta_{g}f+f-|\nabla_{g}f|^{2}_{g}+R_{g}$, then

$\begin{split}DL_{(g,f)}(w,0)&=-2\left\langle
w,\nabla^{2}_{g}u\right\rangle_{g}-\left\langle\left(2{\rm
div}_{g}w-\nabla_{g}\operatorname{tr}_{g}w\right),\nabla_{g}u\right\rangle_{g}+2w(\nabla_{g}f,\nabla_{g}u)\\\
&\,+2\Delta_{g}^{f}\Big{(}-2\left\langle
w,\nabla^{2}_{g}f\right\rangle_{g}-\left\langle\left(2{\rm
div}_{g}w-\nabla_{g}\operatorname{tr}_{g}w\right),\nabla_{g}f\right\rangle_{g}+w(\nabla_{g}f,\nabla_{g}f)\\\
&\,-\Delta_{g}\operatorname{tr}_{g}w+{\rm div}_{g}{\rm div}_{g}w-\langle
w,\operatorname{Ric}_{g}\rangle_{g}\Big{)}+(4\pi)^{-\frac{n}{2}}\int_{M}\tfrac{1}{2}\operatorname{tr}_{g}w\,e^{-f}\,dv_{g}.\end{split}$
(2.15)

Now, by the implicit function theorem we have

$DP_{g}(w)=-\left(DL_{(g,P(g))}(0,\cdot)\right)^{-1}\circ DL_{(g,P(g))}(w,0).$
(2.16)

Since $P:\mathcal{U}\to C^{k,\gamma}(M;\mathbb{R})$ is analytic, we can find a
$C^{k,\gamma}$ neighborhood $\mathcal{V}$ of $g_{o}$ and a constant $C$
depending on $\mathcal{V}\subset\mathcal{U}$ but independent of
$g\in\mathcal{V}$ such that, by (2.15),

$\|DL_{(g,P(g))}(w,0)\|_{C^{k-4,\gamma}_{g_{o}}}\leq
C\|w\|_{C^{k,\gamma}_{g_{o}}}\quad\text{ for all }\quad w\in
C^{k,\gamma}(\mathcal{S}(M)).$ (2.17)

Note that here we have used the fact that if $f=P(g)$, then
$\|f\|_{C^{k,\gamma}_{g_{o}}}$ is bounded independent of $g\in\mathcal{V}$.

Claim: There exists a $C^{k,\gamma}$ neighborhood of
$\mathcal{V}^{\prime}\subseteq\mathcal{V}$ of $g_{o}$ such that
$DL_{(g,P(g))}(0,\cdot)$ is also a linear isomorphism from
$C^{k,\gamma}(M;\mathbb{R})$ to $C^{k-4,\gamma}(M;\mathbb{R})$, and

$\left\|\left(DL_{(g,P(g))}(0,\cdot)\right)^{-1}\right\|\leq
2\left\|\left(DL_{(g_{o},f_{o})}(0,\cdot)\right)^{-1}\right\|\quad\text{ for
all }\quad g\in\mathcal{V^{\prime}}.$

###### Proof of the claim.

Let $\mathcal{B}_{1}$ be the set of all bounded linear operators from
$C^{k,\gamma}(M;\mathbb{R})$ to $C^{k-4,\gamma}(M;\mathbb{R})$ endowed with
the operator norm, i.e.

$\|A\|_{\mathcal{B}_{1}}=\sup_{u\neq
0}\frac{\|A(u)\|_{C^{k-4,\gamma}_{g_{o}}}}{\|u\|_{C^{k,\gamma}_{g_{o}}}}.$

With the induced topology by this norm, $\mathcal{B}_{1}$ is a Banach space.
Similarly, we denote by $\mathcal{B}_{2}$ and $\mathcal{B}_{3}$, the Banach
spaces of all bounded linear operators from $C^{k-4,\gamma}(M;\mathbb{R})$ to
$C^{k,\gamma}(M;\mathbb{R})$, and of all bounded linear operators from
$C^{k,\gamma}(M;\mathbb{R})$ to $C^{k,\gamma}(M;\mathbb{R})$, respectively,
with the corresponding operator norms. We consider the map
$Q:\mathcal{V}\longrightarrow\mathcal{B}_{1}$ given by

$Q(g):=DL_{(g,P(g))}(0,\cdot).$

Since, $P:\mathcal{U}\to C^{k,\gamma}(M)$ is continuous (c.f. [J05, Theorem
10.1]), it can be seen from (2.14) that $Q$ is also continuous. For any
$g\in\mathcal{V}$, we have

$\begin{split}Q(g)&=Q(g_{o})+Q(g)-Q(g_{o})\\\
&=Q(g_{o})\circ\left[I+Q(g_{o})^{-1}\circ\left(Q(g)-Q(g_{o})\right)\right],\end{split}$
(2.18)

where $I$ is the identity map on $C^{k,\gamma}(M;\mathbb{R})$. As $Q(g_{o})$
is a linear isomorphism, we have that $(Q(g))^{-1}\in\mathcal{B}_{2}$ if and
only if $I+Q(g_{o})^{-1}\circ\left(Q(g)-Q(g_{o})\right)$ is an invertible
bounded linear map from $C^{k,\gamma}(M;\mathbb{R})$ to
$C^{k,\gamma}(M;\mathbb{R})$. It suffices to check that for $g$ sufficiently
close to $g_{o}$, $\|E(g)\|_{\mathcal{B}_{3}}<1$, where
$E(g):=Q(g_{o})^{-1}\circ\left(Q(g)-Q(g_{o})\right)$, since in this case [J05,
Lemma 7.22], we know that $I+E(g)$ is invertible and

$\left(I+E(g)\right)^{-1}=\sum_{k=0}^{\infty}(-1)^{k}\left(E(g)\right)^{k}\text{
and
}\left\|\left(I+E(g)\right)^{-1}\right\|_{\mathcal{B}_{3}}\leq\frac{1}{1-\left\|E(g)\right\|_{\mathcal{B}_{3}}}.$
(2.19)

By the continuity of $Q$, there is a $C^{k,\gamma}$ neighborhood of
$\mathcal{V}^{\prime}\subseteq\mathcal{V}$ of $g_{o}$ such that

$\left\|E(g)\right\|_{\mathcal{B}_{3}}\leq\left\|Q(g_{o})^{-1}\right\|_{\mathcal{B}_{2}}\left\|Q(g)-Q(g_{o})\right\|_{\mathcal{B}_{1}}<\tfrac{1}{2}\quad\text{
for all }\quad g\in\mathcal{V}^{\prime}.$

Hence on $\mathcal{V}^{\prime}$, $\left(I+E(g)\right)$ as well as $Q(g)$, are
invertible. Moreover by (2.18) and (2.19), we have

$\begin{split}\left\|\left(DL_{(g,P(g))}(0,\cdot)\right)^{-1}\right\|_{\mathcal{B}_{2}}&=\left\|Q(g)^{-1}\right\|_{\mathcal{B}_{2}}\leq\left\|\left(I+E(g)\right)^{-1}\right\|_{\mathcal{B}_{3}}\left\|Q(g_{o})^{-1}\right\|_{\mathcal{B}_{2}}\\\
&\leq
2\left\|Q(g_{o})^{-1}\right\|_{\mathcal{B}_{2}}=2\left\|\left(DL_{(g_{o},f_{o})}(0,\cdot)\right)^{-1}\right\|_{\mathcal{B}_{2}}.\end{split}$

This completes the proof of the claim. ∎

Hence, by replacing $\mathcal{V}$ by $\mathcal{V}^{\prime}$ if necessary,
there exists a $C$ independent of $g\in\mathcal{V}$, such that for all
$g\in\mathcal{V}$, we have

$\left\|\left(DL_{(g,P(g))}(0,\cdot)\right)^{-1}(h)\right\|_{C^{k,\gamma}_{g_{o}}}\leq
C\left\|h\right\|_{C^{k-4,\gamma}_{g_{o}}}\quad\text{ for all }\quad
h\in\mathcal{C}^{k-4,\gamma}(M).$ (2.20)

Finally, by (2.16), (2.17), and (2.20), the theorem follows.

∎

###  2.3 Definition of the $\mathbb{F}$-distance

The $\mathbb{F}$-distance is a generalization of the
Gromov-$W_{1}$-Wasserstein distance studied by Sturm [St06a, St06b]. To help
the reader grasp the main ingredients of the proof, we include the definition
of Bamler’s $\mathbb{F}$-distance in this subsection. Note that Definition 2.3
is slightly simpler than the original definition in [Bam20b, Section 5], but
it is more appropriate for application and is sufficient for our purpose.

The definition of $\mathbb{F}$-convergence is involved in the notions of
coupling and $1$-Wasserstein distance between probability measures. Let $X$
and $Y$ be metric spaces. We denote by $\mathcal{P}(X)$ the space of
probability measures on $X.$ For any $\mu\in\mathcal{P}(X)$ and
$\nu\in\mathcal{P}(Y),$ we denote by $\Pi(\mu,\nu)$ the space of _couplings_
between $\mu$ and $\nu$, namely, the set of all the probability measures
$q\in\mathcal{P}(X\times Y)$ satisfying

$q(A\times Y)=\mu(A),\quad q(X\times B)=\nu(B),$

for any measurable subsets $A\subset X$ and $B\subset Y$. The $1$-Wasserstein
distance between $\mu,\nu\in\mathcal{P}(X)$ is defined to be

${\rm dist}_{W_{1}}(\mu,\nu):=\inf_{q\in\Pi(\mu,\nu)}\int_{X\times X}{\rm
dist}(x,y)\,dq(x,y).$

By the Kantorovich-Rubinstein Theorem, this definition is equivalent to

${\rm dist}_{W_{1}}(\mu,\nu)=\sup_{f}\left(\int f\,d\mu-\int f\,d\nu\right),$

where the supremum is taken over all bounded $1$-Lipschitz functions $f$ on
$X.$

Let $I$ be an interval and $(\mathcal{X}^{i},(\mu^{i}_{t})),i=1,2,$ be metric
flow pairs defined over $I$.

######  Definition 2.3.

Let $J\subset I.$ The $\mathbb{F}$-distance between
$(\mathcal{X}^{1},(\mu^{1}_{t}))$ and $(\mathcal{X}^{2},(\mu^{2}_{t}))$
(uniform over $J$), denoted by

${\rm
dist}_{\mathbb{F}}^{J}\left((\mathcal{X}^{1},(\mu^{1}_{t})),(\mathcal{X}^{2},(\mu^{2}_{t}))\right),$

is defined to be the infimum of number $r>0$ with the following properties:
There are a measurable set $E\subset I$, couplings
$q_{t}\in\Pi(\mu_{t}^{1},\mu_{t}^{2}),$ and embeddings
$\phi^{i}_{t}:\mathcal{X}^{i}_{t}\to\left(Z_{t},{\rm dist}^{Z_{t}}\right)$ for
$i=1,2,t\in I\setminus E,$ where $\left(Z_{t},{\rm dist}^{Z_{t}}\right)$ is a
complete separable metric space, such that $J\subset I\setminus E$, and

  1. 1.

$|E|\leq r^{2}$;

  2. 2.

For any $s,t\in I\setminus E,s<t$,

$\int_{\mathcal{X}_{t}^{1}\times\mathcal{X}_{t}^{2}}{\rm
dist}^{Z_{s}}_{W_{1}}\left(\phi^{1}_{*s}\nu^{1}_{x_{1}|s},\phi^{2}_{*s}\nu^{2}_{x_{2}|s}\right)\,dq_{t}(x_{1},x_{2})\leq
r.$

The tuple $\left(Z_{t},{\rm
dist}^{Z_{t}},\\{\phi^{i}_{t}\\}_{i=1,2}\right)_{t\in I\setminus E}$ is called
a _correspondence_.

##  3 Almost monotonicity of the $1$-Wasserstein distance

As indicated by Definition 2.3, an important ingredient in the estimate of the
$\mathbb{F}$-distance is to compare the corresponding conjugate heat kernels
in the metric flows. In this section, we consider the $1$-Wasserstein distance
between two conjugate heat kernels on two different Ricci flows sufficiently
close to each other. It turns out that an almost monotonicity property similar
to [Bam20a, Lemma 2.7] is available. This almost monotonicity further implies
that two Ricci flows close to each other in the $C^{2}$-sense are also close
in the $\mathbb{F}$-sense, given that the referntial conjugate heat flows are
close to each other.

Throughout this section, we shall let $M^{n}$ be a closed $n$-dimensional
manifold and $(M^{n},g_{i,t})_{t\in I}$, $i=1,2$, be Ricci flows defined on
the same time interval $I$. Let $\overline{g}$ be a background metric on $M$.
We make the assumption that for all $t\in I$ and $i=1,2$, it holds that

$\displaystyle C_{0}^{-1}\overline{g}\leq g_{i,t}\leq
C_{0}\overline{g},\quad\operatorname{diam}(\overline{g})\leq
D,\quad\sup_{M\times I}|{\operatorname{Rm}}_{g_{i}}|\leq\Lambda,$ (3.1)

for some positive constants $C_{0}$, $D$, and $\Lambda$. The background metric
$\overline{g}$ can be taken as, for instance, $g_{1,t_{0}}$ for some $t_{0}\in
I$. Unless otherwise specified, the tensor norms in this section are always
computed using the metric $\overline{g}$, and we shall suppress the metric
specification in the norm notations, i.e.,
$\|\cdot\|_{C^{2}}=\|\cdot\|_{C^{2}_{\overline{g}}}$, etc.

Let us assume that

$\displaystyle\sup_{t\in
I}\left\|g_{1,t}-g_{2,t}\right\|_{C^{2}}\leq\varepsilon\ll 1.$ (3.2)

Then, following [Bam20b], we may easily construct a correspondence
$\left(Z_{t},{\rm dist}^{Z_{t}},\\{\phi^{i}_{t}\\}_{i=1,2}\right)_{t\in I}$ as
follows. Let $Z_{t}=M_{1}\sqcup M_{2}$ for $t\in I$, where $M_{1}=M_{2}=M$
and, for notational convenience, we have added the subindex to indicate that
$M_{i}$ corresponds to the underlying manifold of the Ricci flow $g_{i,t}$.
For all $z_{1}\in M_{1},z_{2}\in M_{2},$ we define

${\rm dist}^{Z_{t}}(z_{1},z_{2}):={\rm dist}^{Z_{t}}(z_{2},z_{1}):=\inf_{w\in
M}\left({\rm dist}_{g_{1,t}}(z_{1},w)+{\rm
dist}_{g_{2,t}}(w,z_{2})\right)+\varepsilon.$

Then it is straightforward to check that $\left(Z_{t},{\rm
dist}^{Z_{t}}\right)$ is a compact metric space. Let
$\phi^{i}_{t}:\left(M,{\rm dist}_{g_{i,t}}\right)\to\left(Z_{t},{\rm
dist}^{Z_{t}}\right)$ be the canonical embedding. Clearly, for any $x\in M$,
we have

${\rm dist}^{Z_{t}}(\phi^{1}_{t}(x),\phi^{2}_{t}(x))=\varepsilon.$ (3.3)

By a slight abuse of notations, we still write $M_{i}=\phi^{i}_{t}(M)\subset
Z_{t}$, $i=1,2$.

######  Proposition 3.1.

Let $(M^{n},g_{i,t})_{t\in I}$ be Ricci flows satisfying (3.1) and (3.2),
where $M$ is a closed manifold. For all $x_{1},x_{2}\in M$ and for all $s\leq
s^{\prime}\leq s_{1},s_{2},$ where $s,s^{\prime},s_{1},s_{2}\in I,$ if

$\mathcal{N}^{2}_{x_{2},s_{2}}(s_{2}-s)\geq-Y\quad\text{ or
}\quad\mathcal{N}^{1}_{x_{1},s_{1}}(s_{1}-s)\geq-Y,$

then

$\displaystyle{\rm dist}^{Z_{s}}_{{\rm
W}_{1}}\left(\phi^{1}_{s*}\nu^{1}_{x_{1},s_{1}\,|\,s},\phi^{2}_{s*}\nu^{2}_{x_{2},s_{2}|s}\right)$
(3.4) $\displaystyle\leq\ $ $\displaystyle(1+C\varepsilon){\rm
dist}^{Z_{s^{\prime}}}_{{\rm
W}_{1}}\left(\phi^{1}_{s^{\prime}*}\nu^{1}_{x_{1},s_{1}\,|\,s^{\prime}},\phi^{2}_{s^{\prime}*}\nu^{2}_{x_{2},s_{2}\,|\,s^{\prime}}\right)+C\varepsilon(1+{s^{\prime}-s}),$

where $\mathcal{N}^{i}$ and $\nu^{i}$ stand for the Nash entropy and the
conjugate heat kernel on the Ricci flow $(M,g_{i,t})_{t\in I}$, respectively,
and $C$ is a constant depending only on $C_{0}$, $D$, $Y$, $\Lambda$, and $n$.
In particular, setting $s_{1}=s_{2}=s^{\prime}=t$, we have

$\displaystyle{\rm dist}^{Z_{s}}_{{\rm
W}_{1}}\left(\phi^{1}_{s*}\nu^{1}_{x_{1},t\,|\,s},\phi^{2}_{s*}\nu^{2}_{x_{2},t\,|\,s}\right)\leq(1+C\varepsilon){\rm
dist}^{Z_{t}}(\phi^{1}_{t}(x_{1}),\phi^{2}_{t}(x_{2}))+C\varepsilon(1+{t-s}).$
(3.5)

###### Proof.

The idea of the proof is similar to that of [Bam20a, Lemma 2.7]. Fix a point
$o\in M$ and write

$d\nu^{i}_{t}:=d\nu^{i}_{x_{i},s_{i}\,|\,t}=v_{i,t}\,dg_{i,t},$

where $d\nu^{i}_{x_{i},s_{i}\,|\,t}$ is the conjugate heat kernel on the Ricci
flow $(M,g_{i,t})$ based at the point $(x_{i},s_{i})$. By the symmetry in the
statement of this proposition, we may assume that

$\mathcal{N}^{2}_{x_{2},s_{2}}(s_{2}-s)\geq-Y.$

Let $F:Z_{s}\to\mathbb{R}$ be an arbitrary $1$-Lipschitz function. By
replacing $F$ by $F-F(\phi^{1}_{s}(o))$ if necessary, we may assume that
$F\circ\phi^{1}_{s}(o)=0$. Then we write $u^{0}_{i}=F\circ\phi^{i}_{s}$ and
solve the heat equations

$\displaystyle\Box_{g_{i,t}}u_{i,t}=0\quad\text{ on }\quad
M\times[s,s^{\prime}],$ $\displaystyle u_{i,s}=u^{0}_{i},$

for $i=1$, $2$. In fact, we will not need $u_{2,t}$ in the following.
Obviously, $u^{0}_{i}$, $i=1,2$, are both $1$-Lipschitz, and hence [Bam20a,
Lemma 2.5] implies that $|\nabla u_{i,t}|_{g_{i,t}}\leq 1$ for all
$t\in[s,s^{\prime}]$ and $i=1,2$. By our assumption on $F$, we have
$u^{0}_{1}(o)=0$. Then the parabolic maximum principle and the bound of
diameter in (3.1) imply

$|u^{0}_{1}|\leq{\rm diam}(g_{1,t})\leq C_{0}D$

and

$\displaystyle\sup_{M}|u_{1,t}|\leq\sup_{M}|u^{0}_{1}|\leq C_{0}D\quad\text{
for }\quad t\in[s,s^{\prime}]\quad\text{ and }\quad i=1,2.$ (3.6)

Letting $\bar{s}=(s+s^{\prime})/2$, we clearly have

$\left.\int u_{1,t}d\nu^{2}_{t}\,\right|_{t=s}^{t=s^{\prime}}=\left.\int
u_{1,t}d\nu^{2}_{t}\,\right|_{t=\bar{s}}^{t=s^{\prime}}+\left.\int
u_{1,t}d\nu^{2}_{t}\,\right|^{t=\bar{s}}_{t=s}.$

We shall estimate the two terms above separately. First of all, we have

$\displaystyle\left.\int
u_{1,t}d\nu^{2}_{t}\,\right|_{t=\bar{s}}^{t=s^{\prime}}$
$\displaystyle=\int_{\bar{s}}^{s^{\prime}}\int_{M}\left(\Box_{g_{2,t}}u_{1,t}\right)d\nu^{2}_{t}dt.$

Note that for $t\in(s,s^{\prime}]$, we have

$\displaystyle\left|\Box_{g_{2,t}}u_{1,t}\right|$
$\displaystyle=\left|\Box_{g_{2,t}}u_{1,t}-\Box_{g_{1,t}}u_{1,t}\right|=\left|\Delta_{g_{2,t}}u_{1,t}-\Delta_{g_{1,t}}u_{1,t}\right|$
$\displaystyle\leq
C\left\|g_{1,t}-g_{2,t}\right\|_{C^{2}}\cdot\sup_{M}\left(|\nabla^{2}u_{1,t}|_{g_{1,t}}+|\nabla
u_{1,t}|_{g_{1,t}}\right)$ $\displaystyle\leq
C\varepsilon\frac{1+\sqrt{t-s}}{t-s},$

where $C$ depends on $C_{0}$, $D$, $\Lambda$, $n$, and we have applied the
standard Bando-Bernstein-Shi estimates for the heat equation (see, e.g.,
[Bam20b, Lemma 9.14]). It follows that

$\displaystyle\left|\left.\int
u_{1,t}d\nu^{2}_{t}\right|_{t=\bar{s}}^{t=s^{\prime}}\,\right|$
$\displaystyle\leq C\varepsilon(1+\sqrt{s^{\prime}-s})\leq
C\varepsilon(1+s^{\prime}-s).$ (3.7)

On the other hand, for any $t\in[s,s^{\prime}]$, we have

$\displaystyle\left|\int u_{1,t}d\nu^{2}_{t}-\int
u_{1,t}v_{2,t}\,dg_{1,t}\right|$ $\displaystyle\leq C_{0}D\int
v_{2,t}\,\left|dg_{2,t}-dg_{1,t}\right|$ $\displaystyle\leq
C(C_{0},D)\varepsilon\int v_{2,t}\,dg_{2,t}\leq C\varepsilon,$

where we have applied (3.6). Hence we have

$\displaystyle\left|\left.\int
u_{1,t}d\nu^{2}_{t}\,\right|^{t=\bar{s}}_{t=s}-\left.\int
u_{1,t}v_{2,t}\,dg_{1,t}\,\right|^{t=\bar{s}}_{t=s}\right|$ $\displaystyle\leq
C\varepsilon$ (3.8)

Next,

$\displaystyle\left.\int u_{1,t}v_{2,t}\,dg_{1,t}\,\right|^{t=\bar{s}}_{t=s}$
$\displaystyle=-\int_{s}^{\bar{s}}u_{1,t}\left(\Box^{*}_{g_{1,t}}v_{2,t}\right)\,dg_{1,t},$
(3.9)

and we also have

$\displaystyle\left|\Box^{*}_{g_{1,t}}v_{2,t}\right|$
$\displaystyle=\left|\Box^{*}_{g_{1,t}}v_{2,t}-\Box^{*}_{g_{2,t}}v_{2,t}\right|$
(3.10)
$\displaystyle\leq\left|\Delta_{g_{1,t}}v_{2,t}-\Delta_{g_{2,t}}v_{2,t}\right|+|R_{g_{1,t}}-R_{g_{2,t}}|v_{2,t}$
$\displaystyle\leq C\varepsilon\left(|\nabla^{2}v_{2,t}|_{g_{2,t}}+|\nabla
v_{2,t}|_{g_{2,t}}+v_{2,t}\right).$

Writing $v_{2,t}=(4\pi(s_{2}-t))^{-n/2}e^{-f_{t}}$, we have

$\nabla v_{2,t}=-v_{2,t}\nabla
f_{t},\quad\nabla^{2}v_{2,t}=v_{2,t}\left(\nabla f_{t}\otimes\nabla
f_{t}-\nabla^{2}f_{t}\right).$

Therefore, applying [Bam20c, Proposition 5.2] and (3.1), we have

$\displaystyle\int_{s}^{\bar{s}}\int_{M}\left(|\nabla^{2}v_{2,t}|_{g_{2,t}}+|\nabla
v_{2,t}|_{g_{2,t}}+v_{2,t}\right)\,dg_{1,t}dt$ (3.11) $\displaystyle\leq\ $
$\displaystyle C\int_{s}^{\bar{s}}\int_{M}\left(|\nabla^{2}f_{t}|+|\nabla
f_{t}|^{2}+|\nabla f_{t}|+1\right)v_{2,t}\,(1+\varepsilon)dg_{2,t}dt$
$\displaystyle\leq\ $ $\displaystyle
C\left(\int_{s}^{\bar{s}}\int_{M}(s_{2}-t)|\nabla^{2}f_{t}|^{2}d\nu^{2}_{t}dt\right)^{\frac{1}{2}}\left(\int_{s}^{\bar{s}}\int_{M}\frac{1}{s_{2}-t}d\nu^{2}_{t}dt\right)^{\frac{1}{2}}$
$\displaystyle+\,C\left(\int_{s}^{\bar{s}}\int_{M}(s_{2}-t)|\nabla
f_{t}|^{4}d\nu^{2}_{t}dt\right)^{\frac{1}{2}}\left(\int_{s}^{\bar{s}}\int_{M}\frac{1}{s_{2}-t}d\nu^{2}_{t}dt\right)^{\frac{1}{2}}$
$\displaystyle+\,C\left(\int_{s}^{\bar{s}}\int_{M}|\nabla
f_{t}|^{2}d\nu^{2}_{t}dt\right)^{\frac{1}{2}}\left(\int_{s}^{\bar{s}}\int_{M}1\,d\nu^{2}_{t}dt\right)^{\frac{1}{2}}+C(s^{\prime}-s)$
$\displaystyle\leq\ $ $\displaystyle
C(Y)(1+\sqrt{s^{\prime}-s}+s^{\prime}-s)\leq C(Y)(1+s^{\prime}-s).$

Combining (3.6), (3.8), (3.9), (3.10), and (3.11), we have

$\left|\left.\int u_{1,t}d\nu^{2}_{t}\,\right|^{t=\bar{s}}_{t=s}\right|\leq
C(C_{0},D,Y)\varepsilon(1+{s^{\prime}-s}).$ (3.12)

It then follows from (3.7) and (3.12) that

$\displaystyle\left|\left.\int
u_{1,t}d\nu^{2}_{t}\right|^{t=s^{\prime}}_{t=s}\right|\leq
C\varepsilon\left(1+{s^{\prime}-s}\right).$ (3.13)

By (3.1) again, we have

$\left||\nabla u_{1,s^{\prime}}|^{2}_{g_{2,s^{\prime}}}-|\nabla
u_{1,s^{\prime}}|^{2}_{g_{1,s^{\prime}}}\right|\leq C\varepsilon,$

and hence $u_{1,s^{\prime}}$ is a $(1+C\varepsilon)$-Lipschitz function on
$\left(M,{\rm dist}_{g_{2,s^{\prime}}}\right)$. Therefore, we may find a
$(1+C\varepsilon)$-Lipschitz function $U$ on $Z_{s^{\prime}}$, such that
$U\circ\phi^{i}_{s^{\prime}}=u_{1,s^{\prime}}$ for both $i=1,2.$ It follows
that

$\int u_{1,s^{\prime}}\,d\nu_{s^{\prime}}^{1}-\int
u_{1,s^{\prime}}\,d\nu_{s^{\prime}}^{2}\leq(1+C\varepsilon){\rm
dist}^{Z_{s^{\prime}}}_{{\rm
W}_{1}}\left(\phi^{1}_{s^{\prime}*}\nu^{1}_{s^{\prime}},\phi^{2}_{s^{\prime}*}\nu^{2}_{s^{\prime}}\right).$

Recall

$\int u_{1,s^{\prime}}\,d\nu_{s^{\prime}}^{1}=\int u_{1}^{0}\,d\nu_{s}^{1},$

thus, by (3.13), we have

$\displaystyle\int u_{1}^{0}\,d\nu^{1}_{s}-\int u_{1}^{0}\,d\nu^{2}_{s}$
$\displaystyle=\ $ $\displaystyle\int
u_{1,s^{\prime}}\,d\nu^{1}_{s^{\prime}}-\int
u_{1,s^{\prime}}\,d\nu^{2}_{s^{\prime}}+\left.\int
u_{1,t}d\nu^{2}_{t}\,\right|^{t=s^{\prime}}_{t=s}$ $\displaystyle\leq\ $
$\displaystyle(1+C\varepsilon){\rm dist}^{Z_{s^{\prime}}}_{{\rm
W}_{1}}\left(\phi^{1}_{s^{\prime}*}\nu^{1}_{s^{\prime}},\phi^{2}_{s^{\prime}*}\nu^{2}_{s^{\prime}}\right)+C\varepsilon\left(1+{s^{\prime}-s}\right).$

Since $F$ is $1$-Lipschitz on $Z_{s}$ and because of (3.3), for any $x\in M$
we have

$\left|u^{0}_{1}(x)-u^{0}_{2}(x)\right|=\left|F(\phi^{1}_{s}(x))-F(\phi^{2}_{s}(x))\right|\leq{\rm
dist}^{Z_{s}}(\phi^{1}_{s}(x),\phi^{2}_{s}(x))=\varepsilon.$

It follows that

$\displaystyle\int_{Z_{s}}F\,d(\phi^{1}_{s*}\nu^{1}_{s})-\int_{Z_{s}}F\,d(\phi^{2}_{s*}\nu^{2}_{s})$
$\displaystyle=\ $
$\displaystyle\int_{M}u^{0}_{1}\,d\nu^{1}_{s}-\int_{M}u_{2}^{0}d\nu^{2}_{s}\leq\int_{M}u^{0}_{1}\,d\nu^{1}_{s}-\int_{M}u_{1}^{0}d\nu^{2}_{s}+\varepsilon$
$\displaystyle\leq\ $ $\displaystyle(1+C\varepsilon){\rm
dist}^{Z_{s^{\prime}}}_{{\rm
W}_{1}}\left(\phi^{1}_{s^{\prime}*}\nu^{1}_{s^{\prime}},\phi^{2}_{s^{\prime}*}\nu^{2}_{s^{\prime}}\right)+C\varepsilon\left(1+{s^{\prime}-s}\right).$

Since $F$ is arbitrary, the almost monotonicity formula (3.4) is proved.

∎

Next, applying Proposition 3.1, we show the following corollary, which is
analogous to [Bam20b, Lemma 5.19].

######  Corollary 3.2.

Suppose that the same assumptions of the previous proposition hold. Let
$(\mu^{i}_{t})_{t\in I}$ be a conjugate heat flow on $(M,g_{i,t})_{t\in I}$,
where $i=1,2$, and let $J\subset I.$ Then

${\rm dist}_{\mathbb{F}}^{J}\big{(}((M,g_{1,t})_{t\in I},(\mu^{1}_{t})_{t\in
I}),((M,g_{2,t})_{t\in I},(\mu^{2}_{t})_{t\in
I})\big{)}\leq(1+C\varepsilon)r+C\varepsilon(1+|I|),$

where $r>0$ is any number with the property that there is a measurable subset
$E\subset I$ such that $J\subset I\setminus E$ and

$|E|<r^{2},\quad\sup_{t\in I\setminus E}{\rm dist}^{Z_{t}}_{{\rm
W}_{1}}(\phi^{1}_{t*}\mu_{t}^{1},\phi^{2}_{t*}\mu_{t}^{2})<r.$

###### Proof.

Let $r>0$ be a number satisfying the property stated in the proposition. For
any $t\in I\setminus E,$ let $q_{t}\in\Pi(\mu_{t}^{1},\mu_{t}^{2})$ be a
coupling satisfying

$\int_{M\times M}{\rm
dist}^{Z_{t}}(\phi^{1}_{t}(x),\phi^{2}_{t}(y))\,dq_{t}(x,y)<r.$

Then for any $s,t\in I\setminus E$, $s\leq t$, by applying Proposition 3.1, we
have

$\displaystyle\int_{M\times M}{\rm dist}^{Z_{s}}_{{\rm
W}_{1}}\left(\phi^{1}_{s*}\nu^{1}_{x,t|s},\phi^{2}_{s*}\nu^{2}_{y,t|s}\right)\,dq_{t}(x,y)$
$\displaystyle\leq$ $\displaystyle\ (1+C\varepsilon)\int_{M\times M}{\rm
dist}^{Z_{t}}(\phi^{1}_{t}(x),\phi^{2}_{t}(y))\,dq_{t}(x,y)+C\varepsilon(1+t-s)$
$\displaystyle<$ $\displaystyle\ (1+C\varepsilon)r+C\varepsilon(1+|I|).$

The conclusion follows from Definition 2.3. ∎

##  4 Convergence rate of the referential conjugate heat flow

It is obvious from Corollary 3.2 that, to estimate the
$\mathbb{F}$-convergence rate for the Ricci flows in Theorem 1.1 and Theorem
1.2, we need also to estimate the convergence rate of the referential
conjugate heat flow $\mu_{t}$; this is the goal of the present section.

Let $(M^{n},g_{t})_{t\in I}$ be the Ricci flow in the statement either of
Theorem 1.1, in which case $I=(-\infty,0]$, or, of Theorem 1.2, in which case
$I=(-T,0]$. Let

$d\mu_{t}:=u(\cdot,t)dg_{t}:=(4\pi|t|)^{-\frac{n}{2}}e^{-f_{t}}dg_{t},\quad
t\in I$ (4.1)

be the referential conjugate heat flow in the statement of Theorem 1.1 or
Theorem 1.2. Note that in the case of Theorem 1.1,
$\mu_{t}:=\nu_{p_{0},0\,|\,t}$ for some arbitrarily fixed $p_{0}\in M$; in the
case of Theorem 1.2, $u$ is a singular conjugate heat kernel constructed in
[MM15]. Let $(M,g_{o},f_{o})$ be the normalized shrinker that generates the
unique tangent flow in Theorem 1.1 or Theorem 1.2 (c.f. [CMZ21c, Corollary
1.3, Corollary 1.4]).

Let $\overline{g}_{s}$, where $s\in[0,\infty)$, be the (backward) modified
Ricci flow constructed from $g_{t}$, as demonstrated in Section 2.1.
Furthermore, $\overline{g}_{s}$ satisfies all the properties listed in Section
2.2.2. For the sake of simplicity, we write

$\|\overline{g}_{s}-g_{o}\|_{C^{k,\gamma}_{g_{o}}}\leq Cs^{-\theta}\quad\text{
for all }\quad s>0,$ (4.2)

where $\theta:=\frac{1-(2-\beta)\alpha}{2\alpha-1}$, $k\gg 1$, and
$\gamma\in(0,1)$; in particular, we choose $k$ and $\gamma$ to be the
constants in Theorem 2.1 and Theorem 2.2.

To simplify our argument, we may, by shifting $s$ if necessary, without loss
of generality, assume that $\overline{g}_{s}$ is very close to $g_{o}$, so
that it satisfies Theorem 2.1 and Theorem 2.2 for all $s\geq 0$. Furthermore,
if $\overline{g}_{s}$ is very close to $g_{o}$ for all $s\geq 0$, we also have
that $f_{\overline{g}_{s}}$ is very close to $f_{o}$ for all $s\geq 0$.
Therefore, in view of the fact that
$\lambda_{1}\left(-\Delta_{g_{o}}^{f_{o}}\right)>\frac{1}{2}$ (c.f. [SW15,
Lemma 3.5]), where

$\Delta_{g}^{f}:=\Delta_{g}-\langle\nabla_{g}f,\nabla_{g}\cdot\,\rangle$

is the drifted laplacian operator, and $\lambda_{1}$ is the first nonzero
eigenvalue, we also have that

$\lambda_{1}\left(-\Delta_{\overline{g}_{s}}^{f_{\overline{g}_{s}}}\right)>\frac{1}{2}\quad\text{
for all }\quad s\geq 0.$

As a consequence, we have the following Neumann type Poincaré inequality.

######  Lemma 4.1.

For any $s\geq 0$, if $v\in C^{\infty}(M)$ satisfies
$\displaystyle\int_{M}ve^{-f_{\overline{g}_{s}}}d\overline{g}_{s}=0$, then

$\int_{M}v^{2}e^{-f_{\overline{g}_{s}}}d\overline{g}_{s}\leq
2\int_{M}|\nabla_{\overline{g}_{s}}v|^{2}e^{-f_{\overline{g}_{s}}}d\overline{g}_{s}.$
(4.3)

Next, we also modify the referential conjugate heat flow. Let us define

$\displaystyle\tilde{f}_{s}$
$\displaystyle:=\left\\{\begin{array}[]{ll}f_{-e^{s}}&\text{in the backward
case}\\\ f_{-e^{-s}}&\text{in the forward case}\end{array}\right.$ (4.6)
$\displaystyle\overline{f}_{s}$ $\displaystyle:=\tilde{f}_{s}\circ\psi_{s}$

where $f_{t}$ is the function in (4.1), and $\psi_{s}$ is the $1$-parameter
family of self-diffeomorphism defined by (2.8) in the backward case, or by
(2.4) in the forward case. Perelman’s monotonicity formula obviously implies
that

$\overline{f}_{s}\longrightarrow f_{o}\quad\text{ smoothly, }\quad\text{when
}s\to\infty.$

Comsequently, we also have

$\left|\overline{f}_{s}-f_{\overline{g}_{s}}\right|\to 0\quad\text{
uniformly.}$ (4.7)

Our goal is to estimate the rate of these convergences. Now we split our
argument into two cases.

(1) Backward case. It follows from the definition of $u$ that

$\partial_{s}\tilde{f}_{s}-\Delta_{\tilde{g}_{s}}\tilde{f}_{s}+|\nabla_{\tilde{g}_{s}}\tilde{f}_{s}|^{2}-R_{\tilde{g}_{s}}+\frac{n}{2}=0.$
(4.8)

We will compare $\overline{f}_{s}$ to $f_{\overline{g}_{s}}$. To this end, we
start with deriving an evolution equation satisfied by
$\overline{f}_{s}-f_{\overline{g}_{s}}$. By virtue of (4.8), we have

$\displaystyle\partial_{s}\overline{f}_{s}$ $\displaystyle=$
$\displaystyle\left(\partial_{s}\tilde{f}_{s}\right)\circ\psi_{s}+\langle\nabla_{\tilde{g}_{s}}\tilde{f}_{s},\nabla_{\tilde{g}_{s}}f_{\tilde{g}_{s}}\rangle\circ\psi_{s}$
$\displaystyle=$
$\displaystyle\left(\partial_{s}\tilde{f}_{s}\right)\circ\psi_{s}+\langle\nabla_{\overline{g}_{s}}\overline{f}_{s},\nabla_{\overline{g}_{s}}f_{\overline{g}_{s}}\rangle$
$\displaystyle=$
$\displaystyle\Delta_{\overline{g}_{s}}\overline{f}_{s}-|\nabla_{\overline{g}_{s}}\overline{f}_{s}|^{2}+R_{\overline{g}_{s}}-\frac{n}{2}+\langle\nabla_{\overline{g}_{s}}\overline{f}_{s},\nabla_{\overline{g}_{s}}f_{\overline{g}_{s}}\rangle$
$\displaystyle=$
$\displaystyle\Delta_{\overline{g}_{s}}\left(\overline{f}_{s}-f_{\overline{g}_{s}}\right)-\langle\nabla_{\overline{g}_{s}}\overline{f}_{s},\nabla_{\overline{g}_{s}}\left(\overline{f}_{s}-f_{\overline{g}_{s}}\right)\rangle+\Delta_{\overline{g}_{s}}f_{\overline{g}_{s}}+R_{\overline{g}_{s}}-\frac{n}{2}$
$\displaystyle=$
$\displaystyle\Delta_{\overline{g}_{s}}^{f_{\overline{g}_{s}}}\left(\overline{f}_{s}-f_{\overline{g}_{s}}\right)-\left|\nabla_{\overline{g}_{s}}\left(\overline{f}_{s}-f_{\overline{g}_{s}}\right)\right|^{2}+\tfrac{1}{2}\text{tr
}_{\overline{g}_{s}}\left(\partial_{s}\overline{g}_{s}\right),$

where the norms and inner products are all computed with respect to the
evolving metric $\overline{g}_{s}$. Since $f_{g}=P(g)$, we have
$\partial_{s}f_{\overline{g}_{s}}=DP_{\overline{g}_{s}}\left(\partial_{s}\overline{g}_{s}\right)$,
where $DP$ is the linearization of the analytic map $P:g\mapsto f_{g}$. Hence

$\partial_{s}\left(f_{\overline{g}_{s}}-\overline{f}_{s}\right)=\Delta_{\overline{g}_{s}}^{f_{\overline{g}_{s}}}\left(f_{\overline{g}_{s}}-\overline{f}_{s}\right)+\left|\nabla_{\overline{g}_{s}}\left(\overline{f}_{s}-f_{\overline{g}_{s}}\right)\right|^{2}+\left(DP_{\overline{g}_{s}}-\tfrac{1}{2}\text{tr
}_{\overline{g}_{s}}\right)\left(\partial_{s}\overline{g}_{s}\right).$ (4.9)

Let

$v_{s}:=e^{f_{\overline{g}_{s}}-\overline{f}_{s}},\quad\mathcal{E}_{s}:=\left(DP_{\overline{g}_{s}}-\tfrac{1}{2}\text{tr
}_{\overline{g}_{s}}\right)\left(\partial_{s}\overline{g}_{s}\right),$ (4.10)

then (4.9) can be rewritten as

$\displaystyle\partial_{s}v_{s}=v_{s}\,\partial_{s}\left(f_{\overline{g}_{s}}-\overline{f}_{s}\right)$
$\displaystyle=$ $\displaystyle
v_{s}\,\Delta_{\overline{g}_{s}}^{f_{\overline{g}_{s}}}\left(f_{\overline{g}_{s}}-\overline{f}_{s}\right)+v_{s}\,\left|\nabla_{\overline{g}_{s}}\left(\overline{f}_{s}-f_{\overline{g}_{s}}\right)\right|^{2}+\mathcal{E}_{s}v_{s}$
$\displaystyle=$
$\displaystyle\Delta_{\overline{g}_{s}}^{f_{\overline{g}_{s}}}v_{s}+\mathcal{E}_{s}v_{s}.$

By (4.7), it is clear that $v_{s}\to 1$ uniformly. We shall then estimate the
rate of this convergence. Let us define

$d\bar{\nu}_{s}:=(4\pi)^{-n/2}e^{-f_{\overline{g}_{s}}}\,d\overline{g}_{s},\quad
Z(s):=\left(\int(v_{s}-1)^{2}\,d\bar{\nu}_{s}\right)^{\frac{1}{2}}.$ (4.11)

Then $d\bar{\nu}_{s}$ is a probability measure with

$\partial_{s}d\bar{\nu}_{s}=-\mathcal{E}_{s}\,d\bar{\nu}_{s}.$

######  Lemma 4.2.

There is a positive constant $C_{1}$ such that for all $s\geq 0$,

$C_{1}^{-1}\leq v_{s}\leq C_{1}.$ (4.12)

###### Proof.

Since the original Ricci flow $g_{t}$ is of Type I, applying the gaussian
upper and lower estimates of [X17, Theorem 3.5], in combination with the
radius bound

$\operatorname{diam}_{\overline{g}_{s}}(M)\leq
2\operatorname{diam}_{g_{o}}(M)\leq C\quad\text{ for all }\quad s\geq 0,$

we have that, there exists a constant $C^{\prime}>0$, such that

$-C^{\prime}\leq\overline{f}_{s}\leq C^{\prime}\quad\text{ for all }\quad
s\geq 0.$

On the other hand, since $\overline{g}_{s}$ and $g_{o}$ are close and
consequently $f_{\overline{g}_{s}}$ and $f_{o}$ are close, we also have

$-C^{\prime}\leq f_{\overline{g}_{s}}\leq C^{\prime}\quad\text{ for all }\quad
s\geq 0.$

The lemma then follows from the definition of $v_{s}$. ∎

######  Lemma 4.3.

For $Z(s)$ defined in (4.11), we have

$Z(s)\leq Cs^{-\theta},$

for $s>0,$ where $\theta:=\frac{1-(2-\beta)\alpha}{2\alpha-1}$ is the same
constant as in (4.2).

###### Proof.

$\displaystyle(Z^{2})^{\prime}(s)$ $\displaystyle=\int
2(v_{s}-1)\partial_{s}v_{s}\,d\bar{\nu}_{s}-\int(v_{s}-1)^{2}\mathcal{E}_{s}\,d\bar{\nu}_{s}$
(4.13) $\displaystyle=\int
2(v_{s}-1)\left(\Delta_{\overline{g}_{s}}^{f_{\overline{g}_{s}}}v_{s}+\mathcal{E}_{s}v_{s}\right)\,d\bar{\nu}_{s}-\int(v_{s}-1)^{2}\mathcal{E}_{s}\,d\bar{\nu}_{s}$
$\displaystyle=-2\int|\nabla_{\overline{g}_{s}}(v_{s}-1)|^{2}\,d\bar{\nu}_{s}+\int(v_{s}^{2}-1)\mathcal{E}_{s}\,d\bar{\nu}_{s}$
$\displaystyle\leq-\int(v_{s}-1)^{2}\,d\bar{\nu}_{s}+\sup_{M}\left(|v_{s}+1|\cdot|\mathcal{E}_{s}|\right)\int|v_{s}-1|\,d\bar{\nu}_{s}$
$\displaystyle=-Z^{2}(s)+C\sup_{M}|\mathcal{E}_{s}|\cdot Z(s),$

where we have applied (4.3), the fact that $\int v_{s}d\bar{\nu}_{s}\equiv 1$,
and Lemma 4.2. For the $\mathcal{E}_{s}$ term, we may estimate using (4.10)
and Theorem 2.2:

$\sup_{M}|\mathcal{E}_{s}|\leq
C\|\partial_{s}\overline{g}_{s}\|_{C^{0}_{g_{o}}}\leq-C\tfrac{d}{ds}\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{1-(2-\beta)\alpha}.$
(4.14)

Consequently, (4.13) becomes

$Z^{\prime}(s)\leq-\tfrac{1}{2}Z(s)-C\tfrac{d}{ds}\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{1-(2-\beta)\alpha}.$
(4.15)

Let us define

$\zeta(s):=C_{0}\left(\mu_{\overline{g}_{s}}-\mu_{g_{o}}\right)^{1-(2-\beta)\alpha},$

where $C_{0}$ is some large constant depending on $v_{0}$ and $g_{o}$ to be
determined. We may first choose $C_{0}\geq C$, where $C$ is the constant in
(4.15), so that

$(Z+\zeta)^{\prime}(s)\leq-\tfrac{1}{2}Z(s).$ (4.16)

Recall that (2.11) implies that

$\zeta(s)\leq C_{0}Cs^{-\theta}\quad\text{ for all }\quad s>0.$ (4.17)

On the other hand, $\mu_{\overline{g}_{s}}-\mu_{g_{o}}$ is strictly positive,
for otherwise the Ricci flow in question is trivial, so we may enlarge $C_{0}$
such that $Z(0)\leq\zeta(0)/2.$

Let us now proceed to estimate $Z(s)$. We fix an arbitrary $s\geq 1$. If
$Z(s)\leq\zeta(s)$, then, by (4.17), we are done. Let us assume
$Z(s)>\zeta(s)$, and define

$s_{1}:=\inf\\{s^{\prime}>0:Z\geq\zeta\text{ on the interval
}[s^{\prime},s]\\}.$

Since $Z(0)\leq\zeta(0)/2$, we must have $s_{1}\in(0,s)$. Hence,
$Z(s_{1})=\zeta(s_{1})$. Then, on the interval $[s_{1},s]$, the fact
$Z\geq\zeta$ together with (4.16) implies

$(Z+\zeta)^{\prime}\leq-\tfrac{1}{2}Z\leq-\tfrac{1}{4}(Z+\zeta).$

Integrating the above inequality from $s_{1}$ to $s$, we have

$(Z+\zeta)(s)\leq(Z+\zeta)(s_{1})e^{-\frac{1}{4}(s-s_{1})}=2\zeta(s_{1})e^{-\frac{1}{4}(s-s_{1})}\leq
Cs_{1}^{-\theta}e^{-\frac{1}{4}(s-s_{1})},$

where we have also applied (4.17). If $s_{1}\geq\frac{1}{2}s$, then

$Z(s)\leq(Z+\zeta)(s)\leq Cs_{1}^{-\theta}\leq C2^{-\theta}s^{-\theta}.$

If $s_{1}\leq\frac{1}{2}s$, then

$Z(s)\leq 2\zeta(s_{1})e^{-s/8}\leq Cs^{-\theta};$

note that $\zeta$ is uniformly bounded. ∎

Now we summarize the conclusion in the backward case.

######  Proposition 4.4.

Let $(M^{n},g_{t})_{t\in(-\infty,0]}$ be the ancient Ricci flow in the
statement of Theorem 1.1, where $M$ is a closed manifold. Let
$d\mu_{t}:=(4\pi|t|)e^{-f_{t}}dg_{t}$ be the referential conjugate heat flow.
Let $f^{*}_{g_{t}}$ be the minimizer of $\mu(g_{t},|t|)$, where $\mu$ is
Perelman’s $\mu$-functional. Then for any $\theta^{\prime}\in(0,\theta)$,
there is a positive constant $C$, such that

$\|f_{t}-f^{*}_{g_{t}}\|_{C^{0}}\leq C(\log|t|)^{-\theta^{\prime}}\quad\text{
for all }\quad t\leq-1.$

Note that the $C^{0}$ norm is independent of the choice of the metric.

###### Proof.

Indeed, we need only to estimate
$\left|\overline{f}_{s}-f_{\overline{g}_{s}}\right|$. By Lemma 4.2, we have

$\displaystyle\left|\overline{f}_{s}-f_{\overline{g}_{s}}\right|\leq\left(\sup_{x\in[C_{1}^{-1},C_{1}]}\left|(\log
x)^{\prime}\right|\right)\cdot\left|v_{s}-1\right|\leq C|v_{s}-1|.$

Hence, we have

$\displaystyle\left\|\overline{f}_{s}-f_{\overline{g}_{s}}\right\|_{L^{2}_{g_{o}}}\leq
C\|v_{s}-1\|_{L^{2}_{g_{o}}}\leq
C\|v_{s}-1\|_{L^{2}_{\bar{\nu}_{s}}}=CZ(s)\leq Cs^{-\theta}.$

Furthermore, since, by the standard parabolic derivative estimates and the
regularity of $\overline{g}_{s}$, the higher derivatives of $\overline{f}_{s}$
and $f_{\overline{g}_{s}}$ are bounded uniformly in $s$, we may apply the
standard Sobolev embedding and interpolation formula (c.f. [Ham82]).
Specifically, for any $\theta^{\prime}\in(0,\theta)$, we may find an integer
$N\gg 1$, such that

$\displaystyle\left\|\overline{f}_{s}-f_{\overline{g}_{s}}\right\|_{C^{0}}\leq
C(\theta^{\prime})\left\|\overline{f}_{s}-f_{\overline{g}_{s}}\right\|^{\theta^{\prime}/\theta}_{L^{2}_{g_{o}}}\cdot\left\|\overline{f}_{s}-f_{\overline{g}_{s}}\right\|^{1-\theta^{\prime}/\theta}_{W^{N,2}_{g_{o}}}\leq
Cs^{-\theta^{\prime}}.$

The conclusion of the proposition follows from the definition of
$\overline{f}_{s}$ and $f_{\overline{g}_{s}}$, and the change of variable
$s=\log(-t)=\log|t|$. ∎

(2) Forward case. The forward case is almost identical to the backward one. We
will omit most of the computational details and will focus on the points where
it is different from the former case. Defining $v_{s}$, $\mathcal{E}_{s}$ as
in (4.10), and $\bar{\nu}_{s}$, $Z(s)$ as in (4.11), we have

$\displaystyle\partial_{s}v_{s}=-\Delta_{\overline{g}_{s}}^{f_{\overline{g}_{s}}}v_{s}+\mathcal{E}_{s}v_{s},$
$\displaystyle\partial_{s}d\bar{\nu}_{s}=-\mathcal{E}_{s}\,d\bar{\nu}_{s},$
$\displaystyle\int_{M}v_{s}d\bar{\nu}_{s}\equiv 1.$

Then, we may compute as in (4.13)

$\displaystyle(Z^{2})^{\prime}(s)$ $\displaystyle=\int
2(v_{s}-1)\left(-\Delta_{\overline{g}_{s}}^{f_{\overline{g}_{s}}}v_{s}+\mathcal{E}_{s}v_{s}\right)\,d\bar{\nu}_{s}-\int(v_{s}-1)^{2}\mathcal{E}_{s}\,d\bar{\nu}_{s}$
$\displaystyle=2\int|\nabla_{\overline{g}_{s}}(v_{s}-1)|^{2}\,d\bar{\nu}_{s}+\int(v_{s}^{2}-1)\mathcal{E}_{s}\,d\bar{\nu}_{s}$
$\displaystyle\geq\int(v_{s}-1)^{2}\,d\bar{\nu}_{s}-\sup_{M}\left(|v_{s}+1|\cdot|\mathcal{E}_{s}|\right)\int|v_{s}-1|\,d\bar{\nu}_{s}$
$\displaystyle=Z^{2}(s)-C\sup_{M}|\mathcal{E}_{s}|\cdot Z(s),$

where we have applied Lemma 4.3 and (4.12); obviously, the latter formula is
also valid in this case due to the gaussian estimates of [MM15, Proposition
2.7, Proposition 2.8]. Arguing as in (4.14) and using (2.13), we have

$\sup_{M}|\mathcal{E}_{s}|\leq
C\|\partial_{s}\overline{g}_{s}\|_{C^{0}_{g_{o}}}\leq-C\tfrac{d}{ds}\left(\mu_{g_{o}}-\mu_{\overline{g}_{s}}\right)^{1-(2-\beta)\alpha}.$

Therefore, we have

$\displaystyle
Z^{\prime}(s)\geq\tfrac{1}{2}Z(s)+C\tfrac{d}{ds}\left(\mu_{g_{o}}-\mu_{\overline{g}_{s}}\right)^{1-(2-\beta)\alpha},$

and

$\displaystyle\tfrac{d}{ds}\left(Z(s)-C\left(\mu_{g_{o}}-\mu_{\overline{g}_{s}}\right)^{1-(2-\beta)\alpha}\right)\geq\tfrac{1}{2}Z(s)\geq\tfrac{1}{2}\left(Z(s)-C\left(\mu_{g_{o}}-\mu_{\overline{g}_{s}}\right)^{1-(2-\beta)\alpha}\right).$

Defining

$\xi(s):=Z(s)-C\left(\mu_{g_{o}}-\mu_{\overline{g}_{s}}\right)^{1-(2-\beta)\alpha},$

we have

$\displaystyle\xi^{\prime}(s)\geq\tfrac{1}{2}\xi(s)\quad\text{ for all }\quad
s\geq 0.$ (4.18)

If there is a $s_{0}\geq 0$ such that $\xi(s_{0})>0$, then, integrating (4.18)
from $s_{0}$ to $s\in(s_{0},\infty)$, we have

$\xi(s)\geq\xi(s_{0})\exp\left(\tfrac{1}{2}(s-s_{0})\right)\to\infty\quad\text{
as }\quad s\to\infty;$

this obviously is a contradiction.

In conclusion, we have that $\xi(s)\leq 0$ for all $s\in[0,\infty)$, and hence

$Z(s)\leq
C\left(\mu_{g_{o}}-\mu_{\overline{g}_{s}}\right)^{1-(2-\beta)\alpha}\leq
Cs^{-\theta}\quad\text{ for all }\quad s>0,$

where $\theta:=\frac{1-(2-\beta)\alpha}{2\alpha-1}\in(0,1)$. The rest of the
argument is identical to the forward case. We summarize the above results in
the following proposition.

######  Proposition 4.5.

Let $(M^{n},g_{t})_{t\in[-T,0)}$ be the Type I Ricci flow in the statement of
Theorem 1.2, where $M$ is a closed manifold. Let
$d\mu_{t}:=(4\pi|t|)e^{-f_{t}}dg_{t}$ be the referential conjugate heat flow,
which is a singular conjugate heat kernel defined in [MM15]. Let
$f^{*}_{g_{t}}$ be the minimizer of $\mu(g_{t},|t|)$, where $\mu$ is
Perelman’s $\mu$-functional. Then for any $\theta^{\prime}\in(0,\theta)$,
there is a positive constant $C$, such that

$\|f_{t}-f^{*}_{g_{t}}\|_{C^{0}}\leq C(-\log|t|)^{-\theta^{\prime}}\quad\text{
for all }\quad t\in[-T/2,0).$

Note that the $C^{0}$ norm is independent of the choice of the metric.

##  5 $\mathbb{F}$-distance estimate

Finally, we prove Theorem 1.1 and Theorem 1.2 in this section. The following
result due to Bahuaud-Guenther-Isenberg [BGI20] will be applied to convert the
closeness between $\overline{g}_{s}$ and $g_{o}$ to the closeness between the
scaled ancient Ricci flow and the canonical form of the shrinker.

######  Theorem 5.1 (Theorem A in [BGI20]).

Let $M^{n}$ be a closed manifold and $(M,g_{0}(t))_{t\in[0,\tau_{0})}$ be the
maximal solution to the Ricci flow with initial metric $g_{0}(0)=\bar{g}_{0},$
where $\tau_{0}\in(0,\infty].$ Then for any $\tau\in(0,\tau_{0})$, any integer
$k\geq 4$, and any $\gamma\in(0,1),$ there are $r>0$ and $C<\infty$ depending
on $\bar{g}_{0}$ and $\tau$ such that if $\bar{g}_{1}$ is another smooth
metric on $M$ satisfying

$\|\bar{g}_{1}-\bar{g}_{0}\|_{C^{k,\gamma}}\leq r,$

then the maximal Ricci flow $g_{1}(t)$ starting at $\bar{g}_{1}$ exist on
$[0,\tau]$ and

$\|g_{1}(t)-g_{0}(t)\|_{C^{k-2,\gamma}}\leq
C\|\bar{g}_{1}-\bar{g}_{0}\|_{C^{k,\gamma}},$

for all $t\in[0,\tau].$ Here, all the Hölder norms are induced from some fixed
background metric on $M.$

Note that in their statements in [BGI20], the authors used the notation
$h^{k,\gamma}$ to denote the completion of smooth sections with respect to the
Hölder norm $C^{k,\gamma},$ which is strictly contained in the usual Hölder
space. We shall not make the explicit distinction here as we only need the
Hölder norms.

The proof of Theorem 1.1 and Theorem 1.2 are almost identical, we shall only
consider the former. Let $(M^{n},g_{t})_{t\in(-\infty,0]}$ be the Ricci flow
in the statement of Theorem 1.1. Let $(M^{n},g_{o},f_{o})$ be the Ricci
shrinker whose canonical form $(M,g^{o}_{t},f^{o}_{t})_{t\in(-\infty,0)}$ is
the unique tangent flow at infinity of $(M^{n},g_{t})_{t\in(-\infty,0]}$.
Specifically, let $\Phi_{t}$ be the group of $1$-parameter family of
diffeomorphisms generated by $\nabla_{g_{o}}f_{o},$ then we have

$g^{o}_{t}=|t|\Phi^{*}_{-\log|t|}g_{o},\quad f^{o}_{t}=\Phi^{*}_{-\log|t|}f.$

Let $A>1$ be an arbitrarily fixed constant and $I_{A}:=[-A,-1/A].$ For any
$\lambda\ll A^{-1},$ write $g^{\lambda}_{t}:=\lambda^{2}g_{t/\lambda^{2}}$ and
$s_{-}:=\log(A/\lambda^{2}).$ Then, by (2.12), we have

$\|\Phi_{-\log
A}^{*}\psi_{s_{-}}^{*}g^{\lambda}_{-A}-g^{o}_{-A}\|_{C_{g_{o}}^{k,\gamma}}=\|\Phi_{-\log
A}^{*}(A\overline{g}_{s_{-}}-Ag_{o})\|_{C_{g_{o}}^{k,\gamma}}\leq
C(A)(-\log\lambda)^{-\theta},$

where $\psi_{s}$ is defined in (2.8). By Theorem 5.1, if
$\lambda\leq\overline{\lambda}(A)$, then we have

$\sup_{t\in
I_{A}}\|\Psi_{\lambda}^{*}g^{\lambda}_{t}-g^{o}_{t}\|_{C_{g_{o}}^{k-2,\gamma}}\leq
C(A)(-\log\lambda)^{-\theta},$ (5.1)

where $\Psi_{\lambda}:=\psi_{s_{-}}\circ\Phi_{-\log A}$ is a diffeomorphism,
and we are using $g_{o}$ as the fixed background metric. In the following, we
may assume that $\Psi_{\lambda}={\rm id}$ by considering the pullback flow.

Let $d\mu_{t}:=u_{t}dg_{t}:=(4\pi|t|)e^{-f_{t}}$ be the referential conjugate
heat flow in the statement of Theorem 1.1, and
$d\mu^{\lambda}_{t}:=u^{\lambda}_{t}dg^{\lambda}_{t}:=(4\pi|t|)e^{-f^{\lambda}_{t}}dg^{\lambda}_{t}$
its scaled version, where $f^{\lambda}_{t}=f_{t/\lambda^{2}}$. Let
$d\mu^{o}_{t}:=u^{o}_{t}dg^{o}_{t}:=(4\pi|t|)e^{-f^{o}_{t}}dg^{o}_{t}$ be the
referential conjugate heat kernel of $(M,g^{o}_{t})$. We shall now compare
$\mu^{\lambda}_{t}$ and $\mu^{o}_{t}$.

By (5.1) (note that $\Psi^{*}_{\lambda}$ is taken to be $\operatorname{id}$),
we have

$\displaystyle\Big{\|}|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}-g_{o}\Big{\|}_{C^{k-2,\gamma}_{\Phi^{*}_{\log|t|}g_{o}}}$
$\displaystyle=|t|^{-1}\Big{\|}|t|\Phi^{*}_{-\log|t|}(|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}-g_{o})\Big{\|}_{C_{g_{o}}^{k-2,\gamma}}$
(5.2) $\displaystyle\leq C(A)(-\log\lambda)^{-\theta}\quad\text{ for all
}\quad t\in I_{A}.$

Now we measure the first norm with the metric $g_{o}$ instead. Since $I_{A}$
is a compact interval, we have

$C(A)^{-1}g_{o}\leq\Phi^{*}_{\log|t|}g_{o}\leq C(A)g_{o}\quad\text{ for all
}\quad t\in I_{A}.$ (5.3)

Let $h:=|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}-g_{o}$ and
$\tilde{g}:=\Phi^{*}_{\log|t|}g_{o}$. By (5.3), we have

$|h|^{2}_{g_{o}}=g_{o}^{ik}g_{o}^{jl}h_{ij}h_{kl}\leq
C(A)^{2}\tilde{g}^{ik}\tilde{g}^{jl}h_{ij}h_{kl}\leq C(A)|h|^{2}_{\tilde{g}}.$

Hence, by (5.2), we have

$\displaystyle\Big{\|}|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}-g_{o}\Big{\|}_{C^{0}_{g_{o}}}$
$\displaystyle\leq
C(A)\Big{\|}|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}-g_{o}\Big{\|}_{C^{0}_{\Phi^{*}_{\log|t|}g_{o}}}$
(5.4) $\displaystyle\leq C(A)(-\log\lambda)^{-\theta}\quad\text{ for all
}\quad t\in I_{A}.$

Furthermore, by the standard Shi’s estimates for $g^{\lambda}_{t}$ and the
fact that $I_{A}$ is a compact interval, we obviously have

$\displaystyle\Big{\|}|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}-g_{o}\Big{\|}_{C^{l}_{g_{o}}}\leq
C(A,l)\quad\text{ for all }\quad t\in I_{A},$ (5.5)

so long as we take $\lambda\leq\overline{\lambda}(A)$ for
$\overline{\lambda}(A)$ small enough. Applying the standard interpolation
formula with (5.4) and (5.5), we have that, for any
$\theta^{\prime}\in(0,\theta)$,

$\Big{\|}|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}-g_{o}\Big{\|}_{C^{k,\gamma}_{g_{o}}}\leq
C(k,\theta^{\prime})\Big{\|}|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}-g_{o}\Big{\|}^{\theta^{\prime}/\theta}_{C^{0}_{g_{o}}}\leq
C(A,\theta^{\prime})(-\log\lambda)^{-\theta^{\prime}}\quad\text{ for all
}\quad t\in I_{A}.$

Taking $\overline{\lambda}(A)>0$ to be small enough and letting
$\lambda\leq\overline{\lambda}(A)$, we may apply Theorem 2.2 to obtain

$\left\|f_{|t|^{-1}\Phi^{*}_{\log|t|}g^{\lambda}_{t}}-f_{o}\right\|_{C^{0}}\leq
C(A,\theta^{\prime})(-\log\lambda)^{-\theta^{\prime}}\quad\text{ for all
}\quad t\in I_{A},$

and hence

$\left\|f^{*}_{g^{\lambda}_{t}}-f^{o}_{t}\right\|_{C^{0}}\leq
C(A,\theta^{\prime})(-\log\lambda)^{-\theta^{\prime}}\quad\text{ for all
}\quad t\in I_{A},$ (5.6)

where $f^{*}_{g_{t}^{\lambda}}$ is the minimizer of
$\mu(g_{t}^{\lambda},|t|)$. Note that the $C^{0}$ norm is independent of the
choice of the metric, and $f_{o}=f_{|t|^{-1}\Phi^{*}_{\log|t|}g_{t}^{o}}$.
Combining (5.6) with Proposition 4.4, we have

$\|f^{\lambda}_{t}-f^{o}_{t}\|_{C^{0}}\leq
C(A,\theta^{\prime})(-\log\lambda)^{-\theta^{\prime}}\quad\text{ for all
}\quad t\in I_{A},$

and consequently

$\|u^{\lambda}_{t}-u^{o}_{t}\|_{C^{0}}\leq
C(A,\theta^{\prime})(-\log\lambda)^{-\theta^{\prime}}\quad\text{ for all
}\quad t\in I_{A}.$ (5.7)

Henceforth, we fix a $\theta^{\prime}\in(0,\theta)$.

Finally, we are ready to apply Corollary 3.2 with (5.1) and (5.7) to conclude
the proof of Theorem 1.1. Define
$\varepsilon:=C(A,\theta^{\prime})(-\log\lambda)^{-\theta^{\prime}}$. Let
$\mathfrak{C}=(Z_{t},(\phi^{\lambda}_{t},\phi^{o}_{t}))_{t\in I_{A}}$ be the
correspondence between $(M,g^{\lambda}_{t},\mu^{\lambda}_{t})_{t\in I_{A}}$
and $(M,g^{o}_{t},\mu^{o}_{t})_{t\in I_{A}}$ constructed at the beginning of
Section 3. Fix $t\in I_{A}$ and let $F:Z_{t}\to\mathbb{R}$ be any
$1$-Lipschitz function. Write $F_{\lambda}=F\circ{\phi^{\lambda}_{t}}$ and
$F_{o}=F\circ{\phi^{o}_{t}}$. Since $F$ is $1$-Lipschitz, for any $x\in M,$ we
have

$|F_{\lambda}(x)-F_{o}(x)|=|F(\phi^{\lambda}_{t}(x))-F(\phi^{o}_{t}(x))|\leq{\rm
dist}^{Z_{t}}(\phi_{t}^{\lambda}(x),\phi^{o}_{t}(x))=\varepsilon.$

By adding a constant to $F$, we may assume that $F_{\lambda}$ vanishes at one
point on $M$. It follows that

$\displaystyle\int_{Z_{t}}F\,d\phi^{\lambda}_{t*}\mu^{\lambda}_{t}-\int_{Z_{t}}F\,d\phi^{o}_{t*}\mu^{o}_{t}=\int_{M}F_{\lambda}\,d\mu^{\lambda}_{t}-\int_{M}F_{o}\,d\mu^{o}_{t}$
$\displaystyle\leq$ $\displaystyle\
\int_{M}F_{\lambda}\,d\mu^{\lambda}_{t}-\int_{M}F_{\lambda}\,d\mu^{o}_{t}+\varepsilon=\int_{M}F_{\lambda}u^{\lambda}_{t}\,dg^{\lambda}_{t}-\int_{M}F_{\lambda}\,u^{o}_{t}\,dg^{o}_{t}+\varepsilon$
$\displaystyle\leq$ $\displaystyle\
\int_{M}F_{\lambda}(u^{\lambda}_{t}-u^{o}_{t})\,dg^{\lambda}_{t}+\int_{M}F_{\lambda}\,u^{o}_{t}\left(dg_{t}^{\lambda}-dg^{o}_{t}\right)+\varepsilon$
$\displaystyle\leq$ $\displaystyle\ C(A)(-\log\lambda)^{-\theta^{\prime}}.$

Note that the bound of $F_{\lambda}$ comes from the fact that $F_{\lambda}$
attains $0$ at some point on $M$, that $F_{\lambda}$ is $1$-Lipschitz, and
that $(M,g^{\lambda}_{t})$ has uniformly bounded diameter for $t\in I_{A}$.
Since $F$ is arbitrary, we have

$\sup_{t\in I_{A}}{\rm dist}^{Z_{t}}_{{\rm
W}_{1}}\left(\phi^{\lambda}_{t*}\mu^{\lambda}_{t},\phi^{o}_{t*}\mu^{o}_{t}\right)\leq
C(A)(-\log\lambda)^{-\theta^{\prime}},$

and Theorem 1.1 follows from Corollary 3.2.

## References

  * [Ac12] Ache, Antonio G. _On the uniqueness of asymptotic limits of the Ricci flow._ arXiv preprint arXiv:1211.3387 (2012).
  * [AA81] Allard, William K.; Almgren, Frederick J., Jr. _On the radial behavior of minimal surfaces and the uniqueness of their tangent cones_. Ann. of Math. (2) 113 (1981), no. 2, 215–265.
  * [Bam20a] Richard H. Bamler, _Entropy and heat kernel bounds on a Ricci flow background_ , https://arxiv.org/abs/2008.07093 (2020).
  * [Bam20b] , _Compactness theory of the space of super Ricci flows_ , https://arxiv.org/abs/2008.09298 (2020).
  * [Bam20c] , _Structure theory of non-collapsed limits of Ricci flows_ , https://arxiv.org/abs/2009.03243 (2020).
  * [BGI20] Bahuaud, Eric, Christine Guenther, and James Isenberg. _Convergence stability for Ricci flow._ The Journal of Geometric Analysis 30.1 (2020): 310-336.
  * [CT94] Cheeger, Jeff; Tian, Gang. _On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay._ Invent. Math. 118 (1994), no. 3, 493–571.
  * [CC97] Cheeger, Jeff, and Tobias H. Colding. _On the structure of spaces with Ricci curvature bounded below. I._ Journal of Differential Geometry 46.3 (1997): 406-480.
  * [CC20a] Cheeger, Jeff, and Tobias H. Colding. _On the structure of spaces with Ricci curvature bounded below. II._ Journal of Differential Geometry 54.1 (2000): 13-35.
  * [CC20b] Cheeger, Jeff, and Tobias H. Colding. _On the structure of spaces with Ricci curvature bounded below. III._ Journal of Differential Geometry 54.1 (2000): 37-74.
  * [CM14] Colding, Tobias Holck, William P. Minicozzi. _On uniqueness of tangent cones for Einstein manifolds._ Invent. math. 196.3 (2014): 515-588.
  * [CM15] Colding, Tobias Holck, William P. Minicozzi. _Uniqueness of blowups and Łojasiewicz inequalities._ Ann. of math. (2015): 221-285.
  * [CMZ21a] Chan, Pak-Yeung, Zilu Ma, and Yongjia Zhang. _Ancient Ricci flows with asymptotic solitons._ arXiv preprint arXiv:2106.06904 (2021).
  * [CMZ21c] Chan, Pak-Yeung, Zilu Ma, and Yongjia Zhang. _On Ricci flows with closed and smooth tangent flows._ arXiv preprint arXiv:2109.14763 (2021).
  * [CM21] Colding, Tobias Holck, William P. Minicozzi. _Singularities of Ricci flow and diffeomorphisms._ arXiv:2109.06240
  * [Ham82] Hamilton, Richard S. _Three-manifolds with positive Ricci curvature._ Journal of Differential geometry 17.2 (1982): 255-306.
  * [J05] Jost, Jürgen. _Postmodern analysis_. Springer Science & Business Media, 2005.
  * [MM15] Mantegazza, Carlo; Müller, Reto, _Perelman’s entropy functional at Type I singularities of the Ricci flow_. J. Reine Angew. Math. 703 (2015), 173–199.
  * [Se06] Sesum, Natasa. _Convergence of the Ricci flow toward a soliton._ Communications in Analysis and Geometry 14.2 (2006): 283-343.
  * [S83] Simon, Leon, _Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems._ Ann. of Math. (2) 118 (1983), no. 3, 525–571.
  * [St06a] Sturm, Karl-Theodor, _On the geometry of metric measure spaces_. I, Acta Math. 196 (2006), no. 1, 65–131.
  * [St06b] , _On the geometry of metric measure spaces_. II, Acta Math. 196 (2006), no. 1, 133–177.
  * [SW15] Sun, Song; Wang, Yuanqi, _On the Kähler-Ricci flow near a Kähler-Einstein metric_. J. Reine Angew. Math. 699 (2015), 143–158.
  * [W83] White, Brian. _Tangent cones to two-dimensional area-minimizing integral currents are unique._ Duke Math. J. 50 (1983), no. 1, 143–160.
  * [X17] Xu, Guoyi. _An equation linking $\mathcal{W}$-entropy with reduced volume._ Journal für die reine und angewandte Mathematik (Crelle’s Journal) 2017.727 (2017): 49-67.

Department of Mathematics, University of California, San Diego, CA 92093, USA
E-mail address<EMAIL_ADDRESS>`

Department of Mathematics, University of California, San Diego, CA 92093, USA
E-mail address<EMAIL_ADDRESS>

School of Mathematics, University of Minnesota, Twin Cities, MN 55414, USA
E-mail address<EMAIL_ADDRESS>