# Ancient Ricci flows with asymptotic solitons

Pak-Yeung Chan, Zilu Ma, and Yongjia Zhang

###### Abstract

We study ancient Ricci flows which admit asymptotic solitons in the sense of
Perelman [Per02, Proposition 11.2]. We prove that the asymptotic solitons must
coincide with Bamler’s tangent flows at infinity [Bam20c]. Furthermore, we
show that Perelman’s $\nu$-functional is uniformly bounded on such ancient
solutions; this fact leads to logarithmic Sobolev inequalities and Sobolev
inequalities. Lastly, as an important tool for the proofs of the above
results, we also show that, for a complete Ricci flow with bounded curvature,
the bound of the Nash entropy depends only on the local geometry around an
$H_{n}$-center of its base point.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Conjugate heat kernel estimates on noncompact manifolds
  4. 4 A pseudolocality for the Nash entropy
  5. 5 Properties of ancient solutions satisfying Assumption B
  6. 6 Smooth convergence and $\mathbb{F}$-convergence
  7. 7 Conergence of the Nash entropy and Smooth tangent flow at infinity
  8. 8 Independence of the base point
  9. 9 Perelman’s $\nu$-functional on ancient solutions satisfying Assumption B
  10. 10 Synthesis with the classical cases
  11. 11 Ancient Ricci flows with bounded entropy and smooth tangent flows
  12. A Logarithmic Sobolev inequalities for noncompact Ricci flows
  13. B Convergence of heat kernels.

##  1 Introduction

Through Hamilton’s program [Ham93b], the study of ancient solutions emerged
into the field. Perelman’s seminal work [Per02] manifests not only that the
understanding of ancient solutions is crucial to the analysis of the
singularity formation, but also that ancient solutions are interesting in
their own right. The ancient solution can be approached from different
directions: one is the classification (c.f. [DHS12], [Bre20], [Li20], etc.),
which, in the higher-dimensional non-Kähler case, is impossible without any
strong curvature assumption; another is the construction of examples ([BKN12],
[Lau13], [LW16], etc.), whose immense quantity with their diverse behaviors
discloses the difficulty of the whole field; a third approach is the least
specific—to study the geometric and analytic properties of ancient solutions,
among which Perelman’s monotonicity formulas are very attracting ones. Works
in this category include [Y09], [N10], [CZ11], [Zh21], to list but a few.

Bamler’s recent ground-breaking works [Bam20a]—[Bam20c] have shed more light
on the field of Ricci flow. The estimates therein within the smooth category
can be regarded as some good improvements of Perelman’s entropy techniques,
whereas the weak version of the Ricci flow (called the _metric flow_) provides
an intriguing object for the future study. While the latter is fascinating and
promising, it is yet beyond the scope of the present article, and we shall
focus on the former, just as the last two authors have done in [MZ21].

As is well-known now, for ancient solutions under certain curvature
conditions, Perelman’s monotonicity formulas [Per02] imply the existence of an
asymptotic shrinker (c.f. [Per02, Proposition 11.2] and [N10]). On the other
hand, Bamler [Bam20c] has proved that an ancient solution has a tangent flow
at infinity, and that, if we further assume that the Nash entropy is uniformly
bounded and that the underlying manifold is closed, then the tangent flow is a
metric soliton with a singular set of co-dimension 4 (see [Bam20c, Section
1.8]; c.f. [Bam20c, Theorem 7.8]). A natural question arises that how
Perelman’s asymptotic shrinker and Bamler’s tangent flow at infinity are
related. Bamler himself remarks that, should the ancient solution be a
$\kappa$-solution, they two would then coincide. But what about other cases?
What is the weakest possible condition for this to be true? These are the
questions that motivated the current study. In fact, one of our results is
that the asymptotic shrinker and the tangent flow at infinity coincide if the
former exists. We do not assume any other curvature condition for the ancient
solution except for its time-wise boundedness.

Let $(M^{n},g(t))_{t\in(-\infty,0]}$ be an ancient Ricci flow. Throughout this
article, we shall make a technical assumption that $g(t)$ has bounded
curvature within each compact time interval, namely,

$\displaystyle\sup_{M\times[t_{1},t_{2}]}\big{|}{\operatorname{Rm}}_{g_{t}}\big{|}<\infty\quad\text{
for all }\quad-\infty<t_{1}\leq t_{2}\leq 0.$ (1.1)

Note that the curvature bound may depend on the interval $[t_{1},t_{2}]$, and
is not uniform on $M\times(-\infty,0]$. This assumption is only for the
validity of the classical formulas (especially the integration by parts at
infinity) concerning conjugate heat kernels and entropies. In particular, we
do not assume the ancient solution in question to be a $\kappa$-solution or to
be Type I.

Let $p_{0}\in M$ be a fixed point. Then, for a sequence
$\\{\tau_{i}\\}_{i=1}^{\infty}$ with $\tau_{i}\nearrow\infty$, we may find
$\\{p_{i}\\}_{i=1}^{\infty}$ such that $(p_{i},-\tau_{i})$ are $\ell$-centers
of $(p_{0},0)$, namely, $\ell_{p_{0},0}(p_{i},\tau_{i})\leq\frac{n}{2}$; see
section 2.3 for more details of the definitions. For most of the time in this
article, we make the following assumption for the ancient solutions which we
consider.

Assumption B: For the fixed point $(p_{0},0)$ and the sequences
$\\{\tau_{i}\\}_{i=1}^{\infty}$ and $\\{p_{i}\\}_{i=1}^{\infty}$ as described
above, there exists a smooth and complete Ricci flow
$\big{(}M_{\infty},g_{\infty}(t),p_{\infty}\big{)}_{t\in[-2,-1]}$, such that

$\displaystyle\big{(}M,g_{i}(t),p_{i}\big{)}_{t\in[-2,-1]}\xrightarrow{\makebox[28.45274pt]{}}\big{(}M_{\infty},g_{\infty}(t),p_{\infty}\big{)}_{t\in[-2,-1]}$
(1.2)

in the smooth Cheeger-Gromov-Hamilton sense, where the Ricci flow $g_{i}(t)$
is obtained by the Type I scaling

$\displaystyle g_{i}(t):=\tau_{i}^{-1}g(\tau_{i}t).$ (1.3)

The statement of Assumption B is involved with a base point $(p_{0},0)$, a
sequence of positive scales $\\{\tau_{i}\\}_{i=1}^{\infty}$, and the choices
of $\ell$-centers $(p_{i},-\tau_{i})$. Hence, if necessary, we shall refer to
an ancient solution as “satisfying Assumption B with respect to
$\big{(}p_{0},0,\\{\tau_{i}\\}_{i=1}^{\infty},\\{p_{i}\\}_{i=1}^{\infty}\big{)}$”.
For the limit Ricci flow $(M_{\infty},g_{\infty}(t))$ in (1.2), we do not
assume any shrinker structure, neither do we make any geometric assumption
except for its completeness. However, it will soon be clear that, because of
[CZ20], this limit is Perelman’s asymptotic shrinker; see the statement of
Theorem 1.1(1) below.

We are now ready to state the first of our main theorems.

######  Theorem 1.1.

Under Assumption B, the following hold (see section 2 for all the definitions
involved).

  1. (1)

$\big{(}M_{\infty},g_{\infty}(t),p_{\infty}\big{)}_{t\in[-2,-1]}$ admits a
shrinker structure, which makes it an asymptotic shrinker in the sense of
Perelman ([Per02, Proposition 11.2]).

  2. (2)

We have

$\displaystyle\lim_{\tau\rightarrow\infty}\mathcal{N}_{p_{0},0}(\tau)=\mu_{\infty},$

where $\mathcal{N}_{p_{0},0}$ is the Nash entropy based at $(p_{0},0)$ and
$\mu_{\infty}$ is the entropy of the asymptotic shrinker. In particular,
$\mu_{\infty}$ is the infimum of $\mathcal{N}_{p_{0},0}(\tau)$, $\tau>0$.

  3. (3)

Any $\mathbb{F}$-limit of the sequence
$\left\\{\left((M,g_{i}(t))_{t\in[-2,-1]},(\nu_{t}^{i})_{t\in[-2,-1]}\right)\right\\}_{i=1}^{\infty}$,
where $\nu^{i}_{t}:=\nu_{p_{0},0\,|\,\tau_{i}t}$, given by Bamler’s
compactness theorem [Bam20b, Theorem 7.6] is of the form

$\left((M_{\infty},g_{\infty}(t))_{t\in[-2,-1]},(\nu^{\infty}_{t})_{t\in[-2,-1)}\right),$

up to isometry, where $(\nu_{t}^{\infty})_{t\in[-2,-1)}$ is a conjugate heat
flow made of a shrinker potential function.

The proof of the above theorem consists of several elements. First of all, the
smooth convergence (1.2) in Assumption B implies locally uniform geometry
bounds for $g_{i}(t)$ around points $p_{i}$, which, after scaling back,
implies that $(M,g(t))$ is locally uniformly Type I (see Definition 2.8) along
$\\{(p_{i},-\tau_{i})\\}_{i=1}^{\infty}$. Hence, by [CZ20], the limit
$(M_{\infty},g_{\infty}(t))$ admits a shrinker structure, and it is an
asymptotic shrinker in the sense of Perelman. Then, applying Theorem 1.7 to
the the $\ell$-centers $(p_{i},-\tau_{i})$, we obtain a uniform lower bound
for the Nash entropy. Thirdly, the Nash entropy bound provides a Gaussian
upper bound for the conjugate heat kernel [Bam20a, Theorem 7.2], which is
applied to show Theorem 1.1(2). Finally, the Gaussian upper bound of the
conjugate heat kernel also implies that an $H_{n}$-center is always not far
from an $\ell$-center; this is the main idea in the proof of Theorem 1.1(3).

The last two authors [MZ21] have observed that, on an ancient solution with
bounded curvature within each compact time interval, the limit of the Nash
entropy $\lim_{\tau\rightarrow\infty}\mathcal{N}_{x,t}(\tau)$ and the limit of
Perelman’s entropy $\lim_{\tau\rightarrow\infty}\mathcal{W}_{x,t}(\tau)$ are
independent of the base point $(x,t)$. What is new in the corollary below is
that the limit of the reduced volume
$\lim_{\tau\rightarrow\infty}\mathcal{V}_{x,t}(\tau)$ is also independent of
the base point $(x,t)$. This fact follows not from [MZ21], but from an
observation that in the statement of Assumption B, the base point $(p_{0},0)$
is not so important as the sequence of positive scales
$\\{\tau_{i}\\}_{i=1}^{\infty}$.

######  Corollary 1.2.

Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution satisfying (1.1) and
Assumption B with respect to
$\big{(}p_{0},0,\\{\tau_{i}\\}_{i=1}^{\infty},\\{p_{i}\\}_{i=1}^{\infty}\big{)}$.
Then the following holds.

$\displaystyle\lim_{\tau\rightarrow\infty}\mathcal{N}_{p^{\prime}_{0},t^{\prime}_{0}}(\tau)=\lim_{\tau\rightarrow\infty}\mathcal{W}_{p^{\prime}_{0},t^{\prime}_{0}}(\tau)=\lim_{\tau\rightarrow\infty}\log\mathcal{V}_{p^{\prime}_{0},t^{\prime}_{0}}(\tau)=\mu_{\infty},$

where $\mathcal{W}$ and $\mathcal{V}$ stand for Perelman’s entropy and reduced
volume, respectively, $(p^{\prime}_{0},t^{\prime}_{0})$ is an arbitrary point
in $M$, (in particular, $(p^{\prime}_{0},t^{\prime}_{0})$ is not necessarily
$(p_{0},0)$), and $\mu_{\infty}$ is the entropy of the asymptotic shrinker
$(M_{\infty},g_{\infty}(t))$ (see section 2.2 and section 2.3 for the
definitions).

As a complement of Theorem 1.1, we also consider its reciprocal problem: is a
tangent flow of an ancient solution also Perelman’s asymptotic shrinker? This,
of course, is not true if the tangent flow is not smooth. However, if the
tangent flow is smooth, then this problem can be easily answered with Bamler’s
results in [Bam20c], which are currently only proved to be true for a sequence
of Ricci flows on closed manifolds and with bounded entropies. Hereby we would
like to clarify that the following theorem is true only if the results in
[Bam20c] (especially Theorem 1.6 therein) are valid for noncompact Ricci flows
with bounded geometry within each compact time interval (as it is
conventional, by bounded geometry we always mean that the sectional curvature
is bounded from above and from below, and that the volumes of unit balls are
uniformly bounded away from $0$). This generalization is not among the goals
of the present article. However, its validity, with relatively strong
assurance, is not only our speculation, but expectation.

######  Theorem 1.3 (The reciprocal of Theorem 1.1).

Let $(M^{n},g(t))_{t\in(-\infty,0]}$ be an ancient solution with bounded
geometry within each compact time interval and with bounded Nash entropy, that
is, there is a point $p_{0}\in M$ such that
$\mathcal{N}_{p_{0},0}(\tau)\geq-Y$ for all $\tau>0$, where $Y$ is a constant.
If a tangent flow of $(M,g(t))_{t\in(-\infty,0]}$ at infinity (c.f. Definition
2.17) is smooth, then it is also an asymptotic shrinker in the sense of
Perelman.

Remark: Of course, if we assume the underlying manifold $M^{n}$ (not
necessarily the underlying manifold of the tangent flow) to be closed, then
the proof of the above theorem can easily go through, since the techniques of
[Bam20c] do apply in this case.

Under Assumption B, we also consider Perelman’s $\nu$-functional (see section
2.2 for the definition) on the ancient solution. We prove that on such an
ancient solution, the $\nu$-functional is uniformly bounded everywhere, and a
lower bound (indeed, the infimum) is the entropy of the asymptotic shrinker.

######  Theorem 1.4.

Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient Ricci flow satisfying (1.1) and
Assumption B. Then we have

$\displaystyle\inf_{t\leq 0}\nu(g(t))=\mu_{\infty}>-\infty,$ (1.4)

where $\mu_{\infty}$ is the entropy of the asymptotic shrinker in Theorem
1.1(1). In particular, this result is true for all noncollapsed Type I ancient
solutions and all $\kappa$-solutions.

Once the $\nu$-functional is known to be bounded, the following logarithmic
Sobolev inequalities and Sobolev inequalities are simply consequences of
straightforward computations (c.f. [Zhq07, LW20]).

######  Corollary 1.5 (The logarithmic Sobolev and Sobolev inequalities).

Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient Ricci flow satisfying (1.1) and
Assumption B. Then, for any $t\in(-\infty,0]$, the following are true.

  1. (1)

Logarithmic Sobolev inequality: for any compactly supported locally Lipschitz
function $u$ and positive scale $\tau>0$, we have

$\displaystyle\int_{M}u^{2}\log
u^{2}dg_{t}-\left(\int_{M}u^{2}dg_{t}\right)\int_{M}u^{2}dg_{t}+\left(\mu_{\infty}+n+\frac{n}{2}\log(4\pi\tau)\right)\int_{M}u^{2}dg_{t}$
$\displaystyle\leq\tau\int_{M}(4|\nabla u|^{2}+Ru)dg_{t}$ .

  2. (2)

Sobolev inequality: for any compactly supported locally Lipschitz function
$u$, we have

$\displaystyle\left(\int_{M}|u|^{\frac{2n}{n-2}}dg_{t}\right)\leq
C(n)e^{-\frac{2\mu_{\infty}}{n}}\int_{M}(4|\nabla u|^{2}+Ru)dg_{t}.$

Here $\mu_{\infty}$ is the entropy of the asymptotic shrinker in Theorem
1.1(1). In particular, this result is true for all noncollapsed Type I ancient
solutions and all $\kappa$-solutions.

Obviously, we may also use Theorem 1.4 to obtain a finer version of the main
results in [Zh20] and [MZ21]. We include the following corollary, and the
details of the proof are merely combinations of Theorem 1.4 with [Zh20] and
[MZ21].

######  Corollary 1.6.

Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution satisfying (1.1).
Furthermore, assume _either one_ of the following conditions is true.

  1. (1)

$g(t)$ satisfies a Type I curvature bound, that is, there is a constant $C$
such that

$|\operatorname{Rm}_{g_{t}}|\leq\frac{C}{|t|}\quad\text{ for all }\quad
t\in(-\infty,0).$

  2. (2)

$g(t)$ satisfies Hamilton’s trace Harnack

$\frac{\partial R}{\partial t}+2\langle X,\nabla
R\rangle+2\operatorname{Ric}(X,X)\geq 0\quad\text{ for all vector field }X,$

and there is a constant $C$ such that

$|{\operatorname{Rm}}|\leq CR\quad\text{ everywhere on }\quad
M\times(-\infty,0].$

Then, $(M,g(t))_{t\in(-\infty,0]}$ is strongly $\kappa$-noncollapsed on all
scales if and only if

$\inf_{t\leq 0}\nu(g(t))\geq-\beta,$

where $\kappa$ and $\beta$ are mutually dependent on.

The next two results imply that the Nash entropy depends only on the local
geometry around the $\ell$-centers or the $H_{n}$-centers, and does not depend
on the geometry near the base point. These theorems are particularly useful in
the long-time analysis of the Nash entropy.

######  Theorem 1.7.

Let $(M^{n},g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature
within each time interval compact in $I$. Let $s$, $t\in I$, $s<t$, and assume
that $s-(t-s)\in I$. Let $x\in M$ and let $(z,s)$ be an $\ell$-center of
$(x,t)$. Furthermore, assume that

$\displaystyle|{\operatorname{Ric}}|\leq\frac{C_{0}}{t-s}\quad\text{ on }\quad
B_{s}(z,\sqrt{t-s})\times[s-(t-s),s],$
$\displaystyle\operatorname{Vol}_{g_{s}}\big{(}B_{s}(z,\sqrt{t-s})\big{)}\geq\alpha(t-s)^{\frac{n}{2}}.$

Then we have

$\displaystyle\mathcal{N}_{x,t}(t-s)\geq-\beta,$

where $\beta$ is a positive constant depending only on $n$, $\alpha$, and
$C_{0}$. Consequently, under the same assumption of the theorem, we have

$\displaystyle
K(x,t\,|\,y,s)\leq\frac{C}{(t-s)^{\frac{n}{2}}}\exp\left(-\frac{{\rm
dist}_{s}^{2}(z,y)}{C(t-s)}\right)\quad\text{ for all }\quad y\in M,$

where $C$ depends only on $n$, $\alpha$, and $C_{0}$, and $K$ is the conjugate
heat kernel (see section 2.1 for the definition.)

######  Theorem 1.8.

Let $(M^{n},g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature
within each time interval compact in $I$. Let $s$, $t\in I$, $s<t$, and assume
that $s-(t-s)\in I$. Let $x\in M$ and let $(z,s)$ be an $H_{n}$-center of
$(x,t)$. Furthermore, assume that

$\displaystyle\operatorname{Vol}_{g_{s}}\big{(}B_{s}(z,\sqrt{t-s})\big{)}\geq\alpha(t-s)^{\frac{n}{2}}.$

Then all the conclusions in Theorem 1.7 are true, with constants $\beta$ and
$C$ depending on $n$ and $\alpha$.

Remarks:

  1. 1.

In Theorem 1.7, the assumptions can be replaced by

$\displaystyle|{\operatorname{Ric}}|\leq\frac{C_{0}}{\varepsilon^{2}(t-s)}\quad\text{
on }\quad B_{s}(z,\varepsilon\sqrt{t-s})\times[s-\varepsilon^{2}(t-s),s],$
$\displaystyle\operatorname{Vol}_{g_{s}}\big{(}B_{s}(z,\varepsilon\sqrt{t-s})\big{)}\geq\alpha\varepsilon^{n}(t-s)^{\frac{n}{2}},$
$\displaystyle s-\varepsilon^{2}(t-s)\in I;$

in Theorem 1.8, the assumptions can be replaced by

$\displaystyle\operatorname{Vol}_{g_{s}}\big{(}B_{s}(z,\varepsilon\sqrt{t-s})\big{)}\geq\alpha\varepsilon^{n}(t-s)^{\frac{n}{2}},$
$\displaystyle s-\varepsilon^{2}(t-s)\in I,$

where $\varepsilon$ is any positive constant. If so, then the constants in the
conclusions also depend on $\varepsilon$.

  2. 2.

Theorem 1.7 and Theorem 1.8 do not imply each other. The reason is because
unless the Nash entropy is known to be bounded, one does not know whether an
$\ell$-center is close to an $H_{n}$-center. In fact, the statement of Theorem
1.8 is much stronger than Theorem 1.7. This is because of Bamler’s good
gradient estimates for heat kernels.

  3. 3.

Comparing Theorem 1.8 with [Bam20a, Theorem 6.2], we have that, if
$(z,t-r^{2})$ is an $H_{n}$-center of $(x,t)$, then
$r^{-n}\operatorname{Vol}_{g_{t-r^{2}}}\left(B_{t-r^{2}}(z,\sqrt{2H_{n}}r)\right)$
and $\mathcal{N}_{x,t}(r^{2})$ mutually bound each other.

Theorem 1.7 and Theorem 1.8 show an interesting behavior of the Nash entropy:
it is the geometry around the $H_{n}$-centers or the $\ell$-centers that
determines (a lower bound of) the Nash entropy. This is somewhat in contrast
to our first impression, since the Nash entropy is defined as a global
integral. But the $H_{n}$-concentration property provides a likely
explanation: as we know, the $H_{n}$-centers are points around which the
conjugate heat flow (regarded as a probability measure) accumulates its
measure ([Bam20a, Proposition 3.13]), and hence the region far away from an
$H_{n}$-center should make little contribution to the Nash entropy.
Furthermore, this “locality” behavior is also in time. In other words,
$\mathcal{N}_{x,t}(t-s)$ is bounded so long as the geometry near $(z,s)$, an
$H_{n}$-center of $(x,t)$, is bounded, regardless of what is happening to the
Ricci flow on $M\times[s,t]$. We hope these results to be useful in future
studies.

This paper is organized as follows. In section 2 we review some basic
definitions and results. In section 3 we prove Bamler’s conjugate heat kernel
estimates for noncompact Ricci flows. In section 4 we prove Theorem 1.7 and
Theorem 1.8. In section 5 we prove some basic properties for ancient solutions
satisfying Assumption B, including the boundedness of the Nash entropy;
Theorem 1.1(1) is proved in this section. In section 6 we explore the relation
between smooth convergence and $\mathbb{F}$-convergence. In section 7 we prove
Theorem 1.1(2)(3). In section 8 we prove Corollary 1.2. In section 9 we prove
Theorem 1.4. In section 10 we consider two classical cases in which the
asymptotic shrinker is known to exist, and show that they are special cases of
Theorem 1.1. In section 11 we prove Theorem 1.3.

_Acknowledgement:_ The authors are much indebted to Professor Bennett Chow for
many insightful discussions. The first author would also like to thank
Professor Richard Bamler for many good suggestions.

##  2 Preliminaries

###  2.1 Conjugate heat kernel

Let $(M^{n},g(t))_{t\in I}$ be a complete Ricci flow. We denote by
$K(x,t\,|\,y,s)$ the unique minimal fundamental solution to the heat equation
coupled with $g(t)$, that is,

$\displaystyle\Box_{x,t}K(x,t\,|\,y,s)=0,$
$\displaystyle\quad\lim_{t\rightarrow s+}K(x,t\,|\,y,s)=\delta_{y}(x),$
$\displaystyle\Box^{*}_{y,s}K(x,t\,|\,y,s)=0,$
$\displaystyle\quad\lim_{s\rightarrow t-}K(x,t\,|\,y,s)=\delta_{x}(y),$

where

$\Box_{x,t}:=\partial_{t}-\Delta_{g_{t},x},\quad\Box^{*}_{y,s}:=-\partial_{s}-\Delta_{g_{s},y}+R(y,s).$

Throughout this paper, the laplacians and the covariant derivatives are all
time-dependent, computed using the evolving metric of the Ricci flow. We shall
suppress the subindices in the notations such as $\Delta_{g_{t},x}$ when the
metric and the variables are understood. It is well known that, whenever the
integration by parts at infinity is valid, the conjugate heat equation
preserves the integral. Hence, at least for Ricci flows with bounded curvature
within each compact time interval, the measure $\nu_{x,t\,|\,s}$ defined as

$\displaystyle\nu_{x,t\,|\,s}(A):=\int_{A}K(x,t\,|\,\cdot,s)dg_{s},\quad
A\subset M$ (2.1)

is always a probability measure, where $x\in M$, $s,t\in I$, and $s\leq t.$ In
this article, both the fundamental solution $K(x,t\,|\,\cdot,\cdot)$ and the
evolving probability measure $(\nu_{x,t\,|\,s})_{s\in I\cap(-\infty,t]}$ will
be referred to as the _conjugate heat kernel_ , whenever there is no
ambiguity. One can also use other positive solutions to the conjugate heat
equation instead of a fundamental solution to construct time-dependent
probability measures as (2.1), and such evolving probability measures are
usually referred to as _conjugate heat flows_. For the conjugate heat kernel,
there are the following logarithmic Sobolev and Poincaré inequalities proved
in [HN14].

######  Proposition 2.1 (Hein-Naber’s logrithmic Sobolev and Poincaré
inequalities [HN14]).

Suppose that $(M^{n},g(t))_{t\in[-T,0]}$ is a complete Ricci flow with bounded
curvature. Let $x_{0}\in M$ be a fixed point and
$\nu_{s}:=\nu_{x_{0},0\,|\,s}$ for $s\in[-T,0).$ Then, for any function $u\geq
0,\sqrt{u}\in C_{0}^{0,1}(M)$, we have

$\int_{M}u\log
u\,d\nu_{s}-\left(\int_{M}u\,d\nu_{s}\right)\log\left(\int_{M}u\,d\nu_{s}\right)\leq|s|\int_{M}\frac{|\nabla
u|^{2}}{u}d\nu_{s}.$ (2.2)

For any $u\in C_{0}^{0,1}(M),$ we have

$\int_{M}u^{2}d\nu_{s}-\left(\int_{M}ud\nu_{s}\right)^{2}\leq
2|s|\int_{M}|\nabla u|^{2}d\nu_{s}.$ (2.3)

In the original statement of the above results, Hein-Naber [HN14] assumed
bounded geometry for the Ricci flow. However, given all the estimates for the
conjugate heat kernel (c.f. [CCGGIIKLLN10, Corollary 26.26, Theorem 26.31],
[EKNT08, Theorem 10], and [Zhq06, BCP10]), it is not difficult to weaken the
assumption to bounded curvature alone. We shall include the proof in Appendix
A.

Hein-Naber also proved the following Gaussian concentration theorem using
Davies’ technique and their logarithmic Sobolev inequalities.

######  Proposition 2.2 (Hein-Naber’s Gaussian concentration [HN14]).

Under the same assumption as the above proposition, we have

$\nu_{s}(A)\nu_{s}(B)\leq\exp\left\\{-\frac{{\rm
dist}^{2}_{s}(A,B)}{8|s|}\right\\},$ (2.4)

for any measurable subsets $A$, $B\subset M$. Here ${\rm dist}_{s}$ means the
classical distance between two sets computed using the metric $g(s)$, not the
Hausdorff distance.

###  2.2 Perelman’s entropy functionals

Let us recall Perelman’s definition of the $\mathcal{W}$-functional:

$\displaystyle\mathcal{W}(g,f,\tau):=\int_{M}\Big{(}\tau\big{(}|\nabla
f|^{2}+R\big{)}+f-n\Big{)}(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg,$ (2.5)

where $(M^{n},g)$ is a Riemannian manifold, $f$ is a smooth function, and
$\tau>0$ is a positive scale. If we let $u:=(4\pi\tau)^{-\frac{n}{2}}e^{-f}$,
then we may rewrite

$\displaystyle\overline{\mathcal{W}}(g,u,\tau)$ $\displaystyle:=$
$\displaystyle\int_{M}\left(\tau\left(\frac{|\nabla
u|^{2}}{u^{2}}+R\right)-\log u-\frac{n}{2}\log(4\pi\tau)-n\right)u\,dg.$ (2.6)

In fact, this form of the $\mathcal{W}$ functional is what we will apply in
the discussion of section 9. Note that for any $c>0$, we have

$\displaystyle\mathcal{W}(cg,f,c\tau)=\mathcal{W}(g,f,\tau),\quad\overline{\mathcal{W}}(cg,c^{-\frac{n}{2}}u,c\tau)=\overline{\mathcal{W}}(g,u,\tau).$

Perelman’s $\mu$-functional and $\nu$-functional are defined as follows.

$\displaystyle\mu(g,\tau)$ $\displaystyle:=$
$\displaystyle\inf\left\\{\overline{\mathcal{W}}(g,u,\tau)\ \bigg{|}\
\sqrt{u}\in C^{\infty}_{0}(M),\ u\geq 0,\ \text{ and }\int_{M}udg=1\right\\},$
$\displaystyle\nu(g)$ $\displaystyle:=$
$\displaystyle\inf_{\tau>0}\mu(g,\tau).$

Here for the $\mu$ functional we are adopting the definition in
[CCGGIIKLLN07]; see the formula above Lemma 6.28 therein. It is well
understood that $\mu(g,\tau)$ is the logarithmic Sobolev constant at scale
$\tau$, and that $\nu$ is the Sobolev constant.

Let us fix a point $(p_{0},t_{0})$ in the space-time of a Ricci flow
$(M^{n},g(t))_{t\in I}$, and denote the conjugate heat kernel based at
$(p_{0},t_{0})$ as

$\displaystyle
K(p_{0},t_{0}\,|\,x,t):=(4\pi(t_{0}-t))^{-\frac{n}{2}}e^{-f(x,t)},\quad t\in
I\cap(-\infty,t_{0}).$

Then, _Perelman’s entropy_ and the _Nash entropy_ are respectively defined as

$\displaystyle\mathcal{W}_{p_{0},t_{0}}(\tau)$ $\displaystyle:=$
$\displaystyle\mathcal{W}\big{(}g(t_{0}-\tau),f(\cdot,t_{0}-\tau),\tau\big{)}$
$\displaystyle=$ $\displaystyle\int_{M}\Big{(}\tau\left(|\nabla
f|^{2}+R\right)+f-n\Big{)}(\cdot,t_{0}-\tau)\,d\nu_{p_{0},t_{0}\,|\,t_{0}-\tau},$
$\displaystyle\mathcal{N}_{p_{0},t_{0}}(\tau)$ $\displaystyle:=$
$\displaystyle\int_{M}f(\cdot,t_{0}-\tau)\,d\nu_{p_{0},t_{0}\,|\,t_{0}-\tau}-\frac{n}{2},$
(2.8)

for all $\tau>0$ and $t_{0}-\tau\in I$. The point $(p_{0},t_{0})$ is called
the _base point_. It is well known from Perelman [Per02] that both Perelman’s
entropy and the Nash entropy are increasing in time (and hence decreasing in
$\tau$). Although the monotonocity of $\mathcal{W}_{p_{0},t_{0}}$ or
$\mathcal{N}_{p_{0},t_{0}}$ requires the integration by parts at infinity, yet
this is valid at least for Ricci flows with bounded curvature.

###  2.3 Perelman’s reduced distance

We briefly review Perelman’s reduced distance and reduced volume. Let
$(M,g(t))_{t\in[-T,0]}$ be a Ricci flow with bounded curvature. Let
$(p_{0},t_{0})\in M\times(-T,0]$ be a fixed point in space-time. Then,
Perelman’s _reduced distance_ is defined as

$\displaystyle\ell_{p_{0},t_{0}}(x,\tau):=\frac{1}{2\sqrt{\tau}}\inf_{\gamma}\int_{0}^{\tau}\sqrt{s}\left(|\dot{\gamma}(s)|_{g(t_{0}-s)}^{2}+R(\gamma(s),t_{0}-s)\right)ds,$
(2.9)

where $x\in M$, $\tau\in(0,T-|t_{0}|]$, and the infimum is taken over all
piecewise smooth curves $\gamma:[0,\tau]\rightarrow M$ satisfying
$\gamma(0)=p_{0}$ and $\gamma(\tau)=x$. The minimizer of (2.9) is usually
called a minimal $\mathcal{L}$-geodesic from $(p_{0},t_{0})$ to
$(x,t_{0}-\tau)$. $(p_{0},t_{0})$ is called the base point of $\ell$, and
whenever the base point is understood, we shall suppress the subindex in the
notaion $\ell_{p_{0},t_{0}}(\cdot,\cdot)$. The _reduced volume based at
$(p_{0},t_{0})$_ is defined as

$\displaystyle\mathcal{V}_{p_{0},t_{0}}(\tau):=\int_{M}(4\pi\tau)^{-\frac{n}{2}}e^{-\ell_{p_{0},t_{0}}(\cdot,\tau)}dg_{t_{0}-\tau}.$
(2.10)

Perelman’s reduced distance and reduced volume satisfy many nice equations and
inequalities. Among them the most important one is the monotonicity of the
reduced volume.

######  Proposition 2.3.

Perelman’s reduced volume $\mathcal{V}(\tau)$ is an increasing function in
time (and hence a decreasing function in $\tau$).

The underlying reason for the monotonicity of the reduced volume is the fact
that its integrand is a “sub”-conjugate heat kernel.

######  Proposition 2.4.

Let $\ell_{p,0}(x,\tau)$ be the reduced distance based at $(p,0)$. Then,
$\displaystyle u(x,t):=(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_{p,0}(x,|t|)}$ is a
subsolution to the conjugate heat equation $-\partial_{t}u-\Delta u+Ru=0$
which also converges to the Dirac delta measure based at $p$ as
$\tau\rightarrow 0+$. Precisely, this means

$\displaystyle\frac{\partial\ell}{\partial\tau}-\Delta_{g_{-\tau}}\ell+\left|\,\nabla_{g_{-\tau}}\ell\,\right|_{g_{-\tau}}^{2}-R_{g_{-\tau}}+\frac{n}{2\tau}\geq
0,$ $\displaystyle\lim_{\tau\rightarrow
0+}(4\pi\tau)^{-\frac{n}{2}}e^{-\ell_{p,0}(\cdot,\tau)}=\delta_{p}.$

Both of the formulas above are understood in the sense of distribution. In
consequence, we have

$\displaystyle(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_{p,0}(x,|t|)}\leq
K(p,0\,|\,x,t)\quad\text{ for all }(x,t)\in M\times[-T,0),$ (2.11)

where $K(p,0\,|\,\cdot,\cdot)$ is the conjugate heat kernel based at $(p,0)$.

By an elementary application of the maximum principle, Perelman [Per02] proved
that $\ell(\cdot,\tau)$ always attains its minimum. This minimum point should
be viewed as the “center” of the reduced distance.

######  Proposition 2.5.

Let $\ell_{p,0}$ be the reduced distance based at $(p,0)$. Then we have

$\displaystyle\min_{M}\ell(\cdot,\tau)\leq\frac{n}{2}\quad\text{ for all
}\quad\tau\in(0,T].$ (2.12)

The point(s) where the minimum in formula (2.12) is attained plays an
important role in our arguments. In most of the cases, it turns out that such
a minimum point is not far from Bamler’s $H_{n}$-centers. Hence, we would like
to assign a special term to these points.

######  Definition 2.6.

Let $(M^{n},g(t))_{t\in I}$ be a Ricci flow, and let $(x,t)\in M\times I$ be a
point in space time. Let $s\in I\cap(-\infty,t)$. Then, $(z,s)$ is called an
_$\ell$ -center_ of $(x,t)$ if

$\displaystyle\ell_{x,t}(z,t-s)\leq\frac{n}{2}.$

Remark: Similar to the case of the $H_{n}$-center, the $\ell$-center is not
necessarily unique at a fixed time $s$ for a fixed base point $(x,t)$.
Furthermore, in practice (especially when considering the base points for the
blow-down sequence from which we obtain an asymptotic shrinker), a sequence of
space-time points along which $\ell$ is uniformly bounded serves equally well
as a sequence of $\ell$-centers; see Definition 2.8.

###  2.4 Shrinking gradient Ricci soliton and its entropy

A shrinking gradient Ricci soliton (or Ricci shrinker for short) is a tuple
$(M^{n},g,f)$, where $(M^{n},g)$ is a smooth Riemannian manifold, and $f$ is a
smooth function on $M$ called a _shrinker potential_ , satisfying

$\operatorname{Ric}+\nabla^{2}f=\tfrac{1}{2}g.$

Apparently, the shrinker potential is not unique, for one may always add a
constant to it. If we normalize $f$ such that

$(4\pi)^{-n/2}\int_{M}e^{-f}dg=1,$

then

$\mu:=f-(|\nabla f|^{2}+R)$

is a constant called the _shrinker entropy_.

Let $\Phi_{t}$ be the 1-parameter group of self-diffeomorphisms generated by
$\nabla f$ with $\Phi_{0}:M\rightarrow M$ being the identity map. Define
$\phi_{t}:=\Phi_{-\log|t|}$, $g(t):=|t|\phi^{*}_{t}g$, and
$f_{t}=f\circ\phi_{t}$. Then, $(M,g(t))_{t\in(-\infty,0)}$ is an ancient
solution to the Ricci flow, and $(4\pi|t|)^{-\frac{n}{2}}e^{-f_{t}}$ is a
solution to the conjugate heat equation. The evolving tuple
$(M,g(t),f_{t})_{t\in(-\infty,0)}$ is called the _canonical form_ of the Ricci
shrinker. The canonical form satisfies

$\displaystyle\operatorname{Ric}_{g_{t}}+\nabla^{2}f_{t}=\frac{1}{2|t|}g(t),$
(2.13) $\displaystyle-|t|\big{(}|\nabla
f_{t}|^{2}_{g_{t}}+R_{g_{t}}\big{)}+f_{t}\equiv\mu,$

where $\mu$ is the shrinker entropy. It turns out that the entropy of a fixed
shrinker is unique (even if the normalized potential is not necessarily
unique). In fact, Li-Wang [LW20] proved the following strong statement.

######  Proposition 2.7 ([LW20]).

Let $(M^{n},g,f)$ be a Ricci shrinker, where $f$ is normalized so that
$\int_{M}(4\pi)^{-\frac{n}{2}}dg=1$. Let $\mu$ be the shrinker entropy. Then
we have

$\displaystyle\mu=\mathcal{W}(g,f,1)=\mu(g,1)=\nu(g).$

Furthermore, if we let $(M,g(t),f_{t})_{t\in(-\infty,0)}$ be the canonical
form, then we also have

$\displaystyle\mu=\mathcal{W}(g(t),f_{t},|t|)=\mu(g(t),|t|)=\nu(g(t))\quad\text{
for all }\quad t\in(-\infty,0).$

###  2.5 Locally uniformly Type I ancient solution

Cheng-Zhang [CZ20] studied a geometric condition for ancient Ricci flows,
called the locally uniformly Type I condition. Precisely, it is defined as
follows.

######  Definition 2.8 (Locally uniformly Type I ancient solutions).

Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient Ricci flow. Fix $p_{0}\in M$
and let $\\{(p_{i},-\tau_{i})\\}_{i=1}^{\infty}\subset M\times(-\infty,0)$ be
a sequence of space-time points with $\tau_{i}\nearrow\infty$. Then,
$(M,g(t))_{t\in(-\infty,0]}$ is called _locally uniformly Type I_ along the
sequence $\\{(p_{i},-\tau_{i})\\}_{i=1}^{\infty}$, if the following hold.

  1. (1)

$\ell_{p_{0},0}$ is bounded along $\\{(p_{i},-\tau_{i})\\}_{i=1}^{\infty}$,
that is,

$\displaystyle\limsup_{i\rightarrow\infty}\ell(p_{i},\tau_{i})<\infty.$

  2. (2)

Around the space-time points $(p_{i},-\tau_{i})$, the curvature has a locally
uniformly Type I bound. More precisely, there exists a positive function
$C:(0,\infty)\rightarrow(0,\infty)$ with the following property: for all
$A>0$, there exists $i_{0}\in\mathbb{N}$, depending on $A$, such that

$\displaystyle\sup_{B_{-\tau_{i}}(p_{i},r\sqrt{\tau_{i}})\times[-2\tau_{i},-\tau_{i}]}|{\operatorname{Rm}}|\leq\frac{C(r)}{\tau_{i}}\quad\text{for
all}\quad i\geq i_{0}\quad\text{and for all}\quad r\leq A.$

  3. (3)

There is a time-wise Ricci curvature lower bound for $g(t)$. In other words,
there exists a continuous positive function
$K:(-\infty,0]\rightarrow(0,\infty)$, such that

$\displaystyle\operatorname{Ric}_{g_{t}}\geq-K(t)g(t)\quad\text{ for all
}\quad t\in(-\infty,0].$

  4. (4)

$g(t)$ is noncollapsed along $(p_{i},-\tau_{i})$. In other words,

$\displaystyle\liminf_{i\rightarrow\infty}\Big{(}(\tau_{i})^{-\frac{1}{2}}\text{inj}\big{(}g(-\tau_{i}),p_{i}\big{)}\Big{)}>0,$

where $\text{inj}(g,x)$ stands for the injectivity radius of the Riemannian
metric $g$ at $x$.

With the locally uniformly Type I condition, Cheng-Zhang [CZ20] obtained
locally uniformly $C^{0}$ and $C^{1}$ estimates for the reduced distance
$\ell$, and consequently proved the existence of an asymptotic shrinker. We
shall include these results below.

######  Proposition 2.9 (Proposition 5.1 in [CZ20]).

Let $(M,g(t))_{t\in(-\infty,0]}$ be a locally uniformly Type I ancient
solution as described in Definition 2.8. Let
$\ell_{i}(\cdot,\tau):=\ell_{p_{0},0}(\cdot,\tau_{i}\tau)$ and
$g_{i}(t):=\tau_{i}^{-1}g(\tau_{i}t)$. Then, for any
$\varepsilon\in(0,\frac{1}{4})$, there is a positive function
$C(\cdot,\varepsilon):(0,\infty)\rightarrow(0,\infty)$ with the following
property: for any $r>0$, it holds that $0\leq\ell_{i}(x,|t|)\leq
C(r,\varepsilon)$ for all $(x,t)\in
B_{g_{i,-1}}(p_{i},r)\times[-2,-1-\varepsilon]$ whenever $i$ is large enough.

######  Proposition 2.10 (Asymptotic Shrinker [CZ20]).

Let $g_{i}(t)$ and $\ell_{i}$ be as described in the above proposition, then,
after possibly passing to a subsequence, the sequence of tuples
$\displaystyle\Big{\\{}\big{(}M,g_{i}(t),\ell_{i}(\cdot,|t|)\big{)}_{t\in[-2,-1]}\Big{\\}}_{i=1}^{\infty}$
converges to (the canonical form of) a shrinking gradient Ricci soliton
$\big{(}M_{\infty},g_{\infty}(t),\ell_{\infty}(\cdot,|t|)\big{)}_{t\in(-2,-1)}$,
satisfying

$\displaystyle\operatorname{Ric}_{g_{\infty}}(t)+\nabla^{2}\ell_{\infty}(\cdot,|t|)=\frac{1}{2|t|}g_{\infty}(t).$

The convergence of the Ricci flows is in the pointed smooth Cheeger-Gromov-
Hamilton sense, and $\ell_{i}\rightarrow\ell_{\infty}$ in the
$C^{0,\alpha}_{\operatorname{loc}}$ sense or in the weak
$*W^{1,2}_{\operatorname{loc}}$ sense on $M_{\infty}\times(1,2)$, where
$\ell_{i}$ should be understood to be pulled back by the defining
diffeomorphisms of the Cheeger-Gromov-Hamilton convergence, and $\alpha$ is
any number in $(0,1)$.

######  Corollary 2.11 (Noncollapsing [CZ20]).

A locally uniformly Type I ancient solution is (weakly) $\kappa$-noncollapsed
on all scales, where $\kappa>0$ depends only on the entropy of the asymptotic
shrinker.

###  2.6 Metric flow and $H_{n}$-center

In this subsection we briefly review the notion of the metric flow introduced
by Bamler [Bam20b]. Since here (and in the next subsection) we are not
attempting to be comprehensive about all the nuances, the reader should always
resort to [Bam20b] for more details. We shall use the notation
$\mathcal{P}(X)$ to represent the space of probability measures on a metric
space $X.$ The metric flow is introduced as a natural generalization of the
Ricci flow space-time. A metric flow over $I\subset\mathbb{R}$ is a tuple

$\big{(}\mathcal{X},\mathfrak{t},({\rm dist}_{t})_{t\in
I},(\nu_{x\,|\,s})_{x\in\mathcal{X},s\in I,s\leq\mathfrak{t}(x)}\big{)},$

where $\mathfrak{t}:\mathcal{X}\to I$ is the time function and
$\mathcal{X}_{t}:=\mathfrak{t}^{-1}(t)$ is called the time slice at $t$, ${\rm
dist}_{t}$ is a metric on $\mathcal{X}_{t}$,
$\nu_{x\,|\,s}\in\mathcal{P}(\mathcal{X}_{s})$ is a family of probability
measures called the _conjugate heat kernel_ based at $x\in\mathcal{X}$, and it
satisfies the usual reproduction formula: for any $t_{1}\leq t_{2}\leq t_{3}$
in $I$ and for any $x\in\mathcal{X}_{t_{3}}$, we have

$\nu_{x\,|\,t_{1}}=\int_{\mathcal{X}_{t_{2}}}\nu_{\cdot\,|\,t_{1}}\,d\nu_{x\,|\,t_{2}}.$

The sharp gradient estimate of Bamler [Bam20b, Theorem 4.1] is also axiomized
into the definition of the metric flow. More details could be found in
[Bam20b, Definition 3.2]. The last two authors generalized [Bam20b, Theorem
4.1] to noncompact Ricci flows with bounded curvature within each compact time
interval (c.f. [MZ21]). Hence, the following observation is straightforward.

######  Theorem 2.12 (Bamler).

Let $(M^{n},g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature
within each time interval compact in $I$. Then it induces a canonical metric
flow in the sense of [Bam20b, Definition 3.2].

###### Proof.

By [CCGGIIKLLN10, Lemma 26.16], the reproduction formula holds for the minimal
heat kernel $K(x,t\,|\,y,s)$ coupled with the flow $(M,g(t))$. So [Bam20b,
Definition 3.2 (7)] is satisfied if we choose
$d\nu_{x,t\,|\,s}:=K(x,t\,|\,\cdot,s)\,dg_{s}.$ As mentioned in the above
paragraph, [Bam20b, Definition 3.2 (1)-(6)] are satisfied. ∎

We shall then introduce some definitions and results for the metric flow. In
particular, they can be applied to smooth Ricci flows satisfying the condition
of the above theorem. Before we proceed to define the notion of
$H$-concentration, let us recall the _variance_ of two probability measures.
Let $\mu,\nu\in\mathcal{P}(X)$, where $X$ is a metric space, then their
variance is defined as

${\rm Var}(\mu,\nu):=\int_{X\times X}{\rm
dist}^{2}(y_{1},y_{2})\,d\mu(y_{1})d\nu(y_{2}),$

where ${\rm dist}$ is the metric on $X$. If $\mu$ is the same as $\nu$, then
we also denote ${\rm Var}(\mu):={\rm Var}(\mu,\mu)$. $H$-concentration is
defined as follows.

######  Definition 2.13 (Definition 3.30 in [Bam20b]).

A metric flow $\mathcal{X}$ over $I\subset\mathbb{R}$ is said to be
$H$-concentrated, where $H$ is a positive number, if for any $s,t\in I$,
$s\leq t$, and $x_{1},x_{2}$ $\in\mathcal{X}_{t}$, we have,

$\text{Var}(\nu_{x_{1}\,|\,s},\nu_{x_{2}\,|\,s})\leq\text{dist}_{t}^{2}(x_{1},x_{2})+H(t-s).$
(2.14)

Combining (2.14) with the reproduction formula, we have that, on an
$H$-concentrated metric flow $\mathcal{X}$, for any
$x_{1},x_{2}\in\mathcal{X}_{t},$

${\rm Var}(\nu_{x_{1}\,|\,s},\nu_{x_{2}\,|\,s})+Hs$

is non-decreasing in $s\in I\cap(-\infty,t]$, and is bounded from above by
${\rm dist}_{t}^{2}(x_{1},x_{2})+Ht$ (c.f. [Bam20b, Proposition 3.34]). This
fact guarantees the existence of $H$-centers (c.f. [Bam20b, Proposition
3.36]), which are defined as follows.

######  Definition 2.14 (Definition 3.35 in [Bam20b]).

Let $s,t\in I$ and $s\leq t$. A point $z$ $\in\mathcal{X}_{s}$ is called an
$H$-center of $x$ $\in\mathcal{X}_{t}$ if

$\text{Var}(\delta_{z},\nu_{x\,|\,s})\leq H(t-s).$

The $H$-center adopted its name partially because the conjugate heat kernel
accumulates its measure around it. Precisely, we have ([Bam20a, Proposition
3.13] and [Bam20b, Lemma 3.37]):

######  Proposition 2.15 (Bamler).

Let $\mathcal{X}$ be an $H$-concentrated metric flow over
$I\subset\mathbb{R}$. Let $x\in\mathcal{X}_{t}$ be a fixed point, and let
$z\in\mathcal{X}_{s}$ be an $H$-center of $x$, where $s<t$ and $s,t\in I$.
Then, for any $A>1$, we have

$\displaystyle\nu_{x\,|\,s}\big{(}B_{s}(z,\sqrt{AH(t-s)})\big{)}\geq
1-\frac{1}{A}.$

Remark: Bamler [Bam20a, Proposition 3.2] proved that if
$\mathcal{X}=M^{n}\times I$ is a Ricci flow space-time on a closed manifold
$M^{n}$ over an interval $I$, then $\mathcal{X}$ must be $H_{n}$-concentrated,
where

$H_{n}:=\frac{(n-1)\pi^{2}}{2}+4.$

The same argument also works when the Ricci flow is complete and has bounded
curvature within compact time intervals. _This is a fact which shall be used
throughout this article._ In general, given $x$ $\in\mathcal{X}_{t}$, and
$s\leq t$, $H$-centers of $x$ may not be unique in $\mathcal{X}_{s}$.

###  2.7 Metric flow pair and $\mathbb{F}$-convergence

Suppose $\mathcal{X}$ is a metric flow over $I\subset\mathbb{R}$. A family of
probability measures $\mu_{s}\in\mathcal{P}(\mathcal{X}_{s})$, where $s\in
I^{\prime}\subset I$, is called a _conjugate heat flow_ if it satisfies the
reproduction formula: for any $s,t\in I^{\prime},s\leq t,$ we have

$\mu_{s}=\int_{\mathcal{X}_{t}}\nu_{x\,|\,s}\,d\mu_{t}(x).$

A metric flow pair is then defined to be a metric flow coupled with a
conjugate heat flow.

######  Definition 2.16 (Definition 5.1 in [Bam20b]).

A pair $\left(\mathcal{X},(\mu_{t})_{t\in I^{\prime}}\right)$ is called a
_metric flow pair_ over $I\subset\mathbb{R}$ if the following conditions are
satisfied:

  * •

$I^{\prime}\subset I$, and $|I\setminus I^{\prime}|=0$, where $|\,\cdot\,|$ is
the Lebesgue measure;

  * •

$\mathcal{X}$ is a metric flow over $I^{\prime}$;

  * •

$(\mu_{t})_{t\in I^{\prime}}$ is a conjugate heat flow on $\mathcal{X}$ with
$\operatorname{spt}\mu_{t}=\mathcal{X}_{t}$ for all $t\in I^{\prime}$.

The definition of $\mathbb{F}$-convergence requires the notions of coupling
and $1$-Wasserstein distance between probability measures. Let $X,Y$ be metric
spaces. 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
$X.$

It is to be noted that for any metric flow $\mathcal{X}$, the $1$-Wassernstein
distance between two conjugate heat flows satisfies a monotonicity property
([Bam20b, Proposition 3.24(2)]), namely, for any conjugate heat flows
$(\mu^{1}_{s})_{s\in I^{\prime}}$ and $(\mu^{2}_{s})_{s\in I^{\prime\prime}}$,
we have

$\displaystyle{\rm
dist}_{W_{1}}^{\mathcal{X}_{s}}(\mu^{1}_{2},\mu^{2}_{s})\quad\text{ is non-
decreasing in }\quad s\in I^{\prime}\cap I^{\prime\prime}.$ (2.15)

Consequently, for any $x_{1}$ and $x_{2}\in\mathcal{X}_{t}$, we have

$\displaystyle{\rm
dist}_{W_{1}}^{\mathcal{X}_{s}}(\nu_{x_{1}\,|\,s},\nu_{x_{2}\,|\,s})\leq{\rm
dist}_{t}(x_{1},x_{2})\quad\text{ for all }\quad s<t.$ (2.16)

In fact, this monotonicity property is a consequence of the prescribed
gradient estimate in the definition of the metric flow ([Bam20b, Definition
3.2(6)]).

We now introduce the definition of $\mathbb{F}$-convergence within a
correspondence, because this is the only version we shall use, and because it
is essentially equivalent to the definition of $\mathbb{F}$-convergence
itself. See more details in [Bam20b, Section 6]. Let
$\displaystyle\big{\\{}(\mathcal{X}^{i},(\mu_{t}^{i})_{t\in
I^{\prime,i}})\big{\\}}_{i\in\mathbb{N}\cup\\{\infty\\}}$ be a sequence of
metric flow pairs over a finite interval $I\subset\mathbb{R}$. A
correspondence $\mathfrak{C}$ is a collection of complete and separable metric
spaces $(Z_{t},{\rm dist}^{Z_{t}})_{t\in I}$ together with isometric
embeddings $\phi_{t}^{i}:(\mathcal{X}_{t}^{i},{\rm
dist}_{t}^{i})\to(Z_{t},{\rm dist}^{Z_{t}})$ for $t\in I^{\prime,i}.$ Then,
$(\mathcal{X}^{i},(\mu_{t}^{i})_{t\in I^{\prime,i}})$ $\mathbb{F}$-converges
to $(\mathcal{X}^{\infty},(\mu_{t}^{\infty})_{t\in I^{\prime,\infty}})$ within
the correspondence $\mathfrak{C}$ uniformly on $J\subset I$, denoted as

$\left(\mathcal{X}^{i},(\mu_{t}^{i})_{t\in
I^{\prime,i}}\right)\xrightarrow{\makebox[42.67912pt]{$\mathbb{F},\mathfrak{C},J$}}\left(\mathcal{X}^{\infty},(\mu_{t}^{\infty})_{t\in
I^{\prime,\infty}}\right),$

if for any $\varepsilon>0,$ there is an $\bar{i}\in\mathbb{N},$ such that if
$i\geq\bar{i},$ there is a measurable subset $E_{i}\subset I$ with

$J\subset I\setminus E_{i}\subset I^{\prime,i}\cap I^{\prime,\infty},$

and there are couplings $q_{t}^{i}\in\Pi(\mu_{t}^{i},\mu_{t}^{\infty})$ for
$t\in I\setminus E_{i}$ with the following properties:

  * •

$|E_{i}|\leq\varepsilon^{2}$.

  * •

For any $s,t\in I\setminus E_{i},s\leq t,$ it holds that

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

If $J$ above can be taken as any compact sub-interval of $I$, we say that the
convergence is uniform over any compact sub-intervals.

The tangent flow at infinity of an ancient solution is then defined to be the
$\mathbb{F}$-limit of a sequence of Ricci flows obtained by a Type I scaling
process for the ancient solution.

######  Definition 2.17 (Tangent flow at infinity [Bam20b]).

Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution. A metric flow pair
$\big{(}\mathcal{X},(\mu_{t})_{t\in(-\infty,0)}\big{)}$, where $\mathcal{X}$
is a metric flow over $(-\infty,0]$, is called a _tangent flow at infinity_ of
$(M,g(t))_{t\in(-\infty,0]}$, if there exist a fixed point $(p_{0},t_{0})\in
M\times(-\infty,0]$ and a sequence $\tau_{i}\nearrow\infty$, such that

$\displaystyle\left((M,g_{i}(t))_{t\in(-\infty,0]},(\nu^{i}_{t})_{t\in(-\infty,0]}\right)\xrightarrow{\makebox[28.45274pt]{$\mathbb{F}$}}\left(\mathcal{X},(\mu_{t})_{t\in(-\infty,0)}\right),$

where

$\displaystyle g_{i}(t):=\tau_{i}^{-1}g(t_{0}+\tau_{i}t),$
$\displaystyle\nu^{i}_{t}:=\nu_{p_{0},t_{0}\,|\,t_{0}+\tau_{i}t},$

for all $t\in(-\infty,0]$. Since the interval $(-\infty,0]$ is infinite, the
$\mathbb{F}$-convergence here should be understood to be the
$\mathbb{F}$-convergence on each finite subinterval of $(-\infty,0]$.

If we assume that the ancient solution in question has bounded curvature
within each compact time intervals, then it must be $H_{n}$-concentrated. By
[Bam20b, Theorem 7.6], a tangent flow always exists for any fixed base point
$(p_{0},t_{0})$ and for any sequence of scaling factors
$\\{\tau_{i}\\}_{i=1}^{\infty}$. However, without a uniform lower bound on the
Nash entropy, the tangent flow could be collapsed.

##  3 Conjugate heat kernel estimates on noncompact manifolds

In this section, we shall verify that [Bam20a, Theorem 7.2] and [Bam20a,
Proposition 5.13] are still valid for Ricci flows with bounded curvature
within each compact time interval. Bamler’s original proof was for Ricci flows
on closed manifolds. Given the assumptions which we make, these
generalizations are but straightforward arguments, yet they are very important
to the proofs of our main theorems.

###  3.1 The Gaussian upper bound

######  Theorem 3.1 (Theorem 7.1 in [Bam20a]).

Let $(M^{n},g(t))_{t\in I}$, where $I$ is a compact interval, be a complete
Ricci flow with bounded curvature. Suppose $[s,t]\subset I$ and $R\geq
R_{\operatorname{min}}$ on $M\times[s,t]$. Then for any $x,y\in M$, we have

$\displaystyle
K(x,t\,|\,y,s)\leq\frac{C}{(t-s)^{\frac{n}{2}}}\exp(-\mathcal{N}_{x,t}(t-s)),$

where $C:=C(R_{\operatorname{min}}(t-s))$ is a constant depending on
$R_{\operatorname{min}}(t-s)$.

######  Theorem 3.2 (Theorem 7.2 in [Bam20a]).

Let $(M^{n},g(t))_{t\in I}$, where $I$ is a compact interval, be a complete
Ricci flow with bounded curvature. Suppose $[s,t]\subset I$ and $R\geq
R_{\operatorname{min}}$ on $M\times[s,t]$. Let $(z,s)\in M\times I$ be an
$H_{n}$-center of a point $(x,t)\in M\times I$. Then for any $\varepsilon>0$
and $y\in M$, we have

$\displaystyle
K(x,t\,|\,y,s)\leq\frac{C\exp(-\mathcal{N}_{x,t}(t-s))}{(t-s)^{\frac{n}{2}}}\exp\left(-\frac{{\rm
dist}_{s}^{2}(z,y)}{(8+\varepsilon)(t-s)}\right),$

where $C:=C(R_{\operatorname{min}}(t-s),\varepsilon)$ is a constant depending
on $R_{\operatorname{min}}(t-s)$ and $\varepsilon$.

###### Proof of Theorem 3.1.

The proof is not essentially different from [Bam20a, Theorem 7.1], since the
latter does not rely heavily on the fact that the background manifold is
closed. The only potential issue is the application of the maximum principle.
Hence, we shall verify [Bam20a, (7.11)] below. By parabolic rescaling, we may
assume that $s=0$ and $t=1$. It suffices to show

$v(x,t)\leq\int_{M}K(x,t\,|\,y,\tfrac{1}{2})v(y,\tfrac{1}{2})\,dg_{\frac{1}{2}}(y),$
(3.1)

for all $(x,t)\in M\times[\frac{1}{2},1]$, where

$v:=(t-\tfrac{1}{2})|\nabla u|^{2}+u^{2},\quad u(x,t):=K(x,t\,|\,y,0),$ (3.2)

and $y\in M$ is a fixed point.

In fact, by elementary computations, we have that $v$ is a subsolution to the
heat equation, that is,

$\Box v=|\nabla u|^{2}-2(t-\tfrac{1}{2})|\nabla^{2}u|^{2}-2|\nabla u|^{2}\leq
0.$

Hence, to prove (3.1), we need only to verify that the maximum principle can
be applied to $v$ on $M\times[\frac{1}{2},1]$.

By [CCGGIIKLLN10, Lemma 26.17], we can find a positive constant $J>0$,
depending on the curvature bound on $M\times[0,1]$ and
$\operatorname{Vol}_{g_{0}}\big{(}B_{g_{0}}(y,1)\big{)}>0$, such that
$0<u(x,t)\leq J$ for all $(x,t)\in M\times[\frac{1}{4},1]$. We may then apply
[Zhq06, Theorem 3.2] (for the proof in our noncompact setting, see [Zh21,
Lemma 2.4]) to obtain

$\displaystyle\frac{|\nabla
u|^{2}}{u^{2}}\leq\frac{1}{t-\tfrac{1}{4}}\log\frac{J}{u}\quad\text{ on }\quad
M\times(\tfrac{1}{4},1].$

Hence, we have the following estimate on $M\times[\frac{1}{2},1]$.

$\displaystyle|\nabla u|^{2}\leq 4\Big{(}u^{2}\log J-u^{2}\log u\Big{)}\leq
C(J),$

where $C(J)$ is a constant depending only on $J$. Here we have applied the
fact that $0<u\leq J$ on $M\times[\frac{1}{2},1]$. It then follows that the
function $v$ as defined in (3.2) is bounded on $M\times[\frac{1}{2},1]$, and
the standard maximum principle (c.f. [CCGGIIKLLN08, Theorem 12.10]) implies
that

$(t-\tfrac{1}{2})|\nabla u|^{2}(x,t)\leq
v(x,t)\leq\int_{M}K(x,t\,|\,y,\tfrac{1}{2})u^{2}(y,\tfrac{1}{2})dg_{\tfrac{1}{2}}.$

This is exactly [Bam20a, (7.11)].

For the rest of the proof, the reader may follow [Bam20a] line by line, and we
shall not include all the details therein. Besides the argument above,
Bamler’s original proof also applied [Bam20a, Theorem 5.9], [Bam20a, Corollary
5.11], and [Bam20a, Theorem 6.2]; the first two of them are already proved in
[MZ21, Theorem 4.4, Corollary 4.5] for our case, and the third result can be
easily generalized to our case using (3.4) below. ∎

The proof of Theorem 3.2 also follows closely after Bamler’s argument, for it
does not depend on anything holding exclusively on closed manifolds. In the
course of the proof, all the auxiliary results have already been established
in our case, as described in the last paragraph above. We shall not exhibit
all these details here, since this could be a distraction from our main
purpose, and since this type of generalizations are usually understood to be
straightforward.

###  3.2 An integral inequality involved in the Nash entropy

Next, we prove [Bam20a, Proposition 5.13] on a complete Ricci flow with
bounded curvature on compact time intervals. The reader will see later that
formula (3.4) is particularly useful for the local analysis of the Nash
entropy, that is, the estimation of the Nash entropy using local geometry (see
section 4 below). Interestingly, we shall apply this formula in a somewhat
reverse way as Bamler did in the proof of [Bam20a, Theorem 6.2].

######  Proposition 3.3 (Proposition 5.13 in [Bam20a]).

Let $(M^{n},g(t))_{t\in I}$, where $I$ is a compact interval, be a Ricci flow
with bounded curvature. Let $t_{0}-\tau$, $t_{0}\in I$, where $\tau>0$, and
let $d\nu:=(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg$ be the conjugate heat kernel
based at $(x_{0},t_{0})$. Furthermore, assume $R(\cdot,t_{0}-\tau)\geq
R_{\operatorname{min}}$, then we have

$\displaystyle\int_{M}\tau(|\nabla f|^{2}+R)d\nu_{t_{0}-\tau}$
$\displaystyle\leq$ $\displaystyle\frac{n}{2},$ (3.3)
$\displaystyle\int_{M}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)^{2}d\nu_{t_{0}-\tau}$
$\displaystyle\leq$ $\displaystyle n-2R_{\operatorname{min}}\tau.$ (3.4)

###### Proof.

(3.3) follows from the fact that the Nash entropy is always no smaller than
Perelman’s entropy, which is always true when the Ricci flow has bounded
curvature. (3.4) follows from (3.3) together with Hein-Naber’s Poincaré
inequality (2.3). To apply the Poincaré inequality to the function
$f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}$, let us first of all recall
the coarse heat kernel Gaussian upper and lower bounds (c.f. Theorem 26.25 and
Theorem 26.31 in [CCGGIIKLLN10]), that is,

$\displaystyle C^{-1}{\rm dist}^{2}_{t_{0}-\tau}(x,x_{0})-C\leq
f(x,t_{0}-\tau)\leq C{\rm dist}^{2}_{t_{0}-\tau}(x_{0},x)+C\quad\text{ for all
}\quad x\in M,$

where $C$ is a constant depending only on the curvature bounds on
$M\times[t_{0}-\tau,t_{0}]$, the value of $\tau$, and
$\operatorname{Vol}_{g_{t_{0}}}\big{(}B_{g_{t_{0}}}(x_{0},1)\big{)}$. It then
follows from some straightforward computations that

$\displaystyle\big{|}\,\mathcal{N}_{x_{0},t_{0}}(\tau)\,\big{|}=\left|\,\int_{M}fd\nu_{t_{0}-\tau}-\frac{n}{2}\,\right|<\infty,$
(3.5)
$\displaystyle\int_{M}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)^{2}d\nu_{t_{0}-\tau}<\infty.$

Let us then fix an arbitrary number $A>0$, and let
$\phi^{A}:M\rightarrow[0,1]$ be a smooth cut-off function such that
$\phi^{A}\equiv 1$ on $B_{t_{0}-\tau}(x_{0},A)$, $\phi^{A}\equiv 0$ on
$M\setminus B_{t_{0}-\tau}(x_{0},2A)$, and
$|\nabla\phi^{A}|_{g_{t_{0}-\tau}}\leq 2A^{-1}$ everywhere on $M$. We may then
apply the PoincaRé inequality (2.3) to
$u=\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)$ and
obtain

$\displaystyle\int_{M}\left(\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)\right)^{2}d\nu_{t_{0}-\tau}-\left(\int_{M}\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)d\nu_{t_{0}-\tau}\right)^{2}$
(3.6) $\displaystyle\leq
2\tau\int_{M}\left|\nabla\left(\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)\right)\right|^{2}d\nu_{t_{0}-\tau}.$

Taking $A\rightarrow\infty$ and using (3.5), we have the following
computations for all terms in formula (3.6).

$\displaystyle\int_{M}\left(\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)\right)^{2}d\nu_{t_{0}-\tau}\rightarrow\int_{M}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)^{2}d\nu_{t_{0}-\tau},$
$\displaystyle\int_{M}\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)d\nu_{t_{0}-\tau}\rightarrow\int_{M}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)d\nu_{t_{0}-\tau}=0,$
$\displaystyle\int_{M}\left|\nabla\left(\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)\right)\right|^{2}d\nu_{t_{0}-\tau}=\int_{M}|\nabla\phi^{A}|^{2}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)^{2}d\nu_{t_{0}-\tau}$
$\displaystyle\quad+\int_{M}(\phi^{A})^{2}|\nabla
f|^{2}d\nu_{t_{0}-\tau}+2\int_{M}\langle\nabla\phi^{A},\nabla
f\rangle\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)d\nu_{t_{0}-\tau},$

and for the last three terms, we have

$\displaystyle\int_{M}(\phi^{A})^{2}|\nabla
f|^{2}d\nu_{t_{0}-\tau}\rightarrow\int_{M}|\nabla f|^{2}d\nu_{t_{0}-\tau},$
$\displaystyle\int_{M}|\nabla\phi^{A}|^{2}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)^{2}d\nu_{t_{0}-\tau}\leq\frac{4}{A^{2}}\int_{M}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)^{2}d\nu_{t_{0}-\tau}\rightarrow
0$ $\displaystyle\left|\int_{M}\langle\nabla\phi^{A},\nabla
f\rangle\phi^{A}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)d\nu_{t_{0}-\tau}\right|$
$\displaystyle\quad\quad\leq\frac{2}{A}\left(\int_{M}|\nabla
f|^{2}d\nu_{t_{0}-\tau}\right)^{\frac{1}{2}}\left(\int_{M}\left(f-\mathcal{N}_{x_{0},t_{0}}(\tau)-\frac{n}{2}\right)^{2}d\nu_{t_{0}-\tau}\right)^{\frac{1}{2}}\rightarrow
0.$

Hence, (3.4) follows from (3.6). ∎

##  4 A pseudolocality for the Nash entropy

In this section we prove Theorem 1.7 and Theorem 1.8. These results shall be
important tools in the proof of Theorem 1.1. We emphasize again that though
the statements of Theorem 1.7 and Theorem 1.8 are closely related, yet without
a Nash entropy bound, there is no Gaussian upper bound for the conjugate heat
kernel, and it is not clear whether the $H_{n}$-centers are close to the
$\ell$-centers (compare with Proposition 5.6). Therefore, Theorem 1.7 and
Theorem 1.8 do not imply each other.

###### Proof of Theorem 1.7.

In the proof, we shall use the lower case letter $c$ to represent a positive
estimate constant which is intuitively small, and the capital letter $C$ to
represent a positive estimate constant which is intuitively large. These
constants may vary from line to line, we nevertheless use the same letters in
order not to cause notational complexity. Throughout the proof, we define

$\displaystyle r:=\sqrt{t-s},\quad s^{\prime}:=s-\tfrac{r^{2}}{9}.$

First of all, we state a consequence of the Bishop-Gromov comparison theorem,
which will be used in the estimation of the local upper and lower bounds for
the conjugate heat kernel.

Claim 1. There are positive constants $c$ and $C$, depending only on $\alpha$
and $C_{0}$, such that the following hold for all $\displaystyle y\in
B_{s}(z,\tfrac{r}{3})$.

$\displaystyle|{\operatorname{Ric}}|\leq C_{0}r^{-2}\quad\text{ on }\quad
B_{s}(y,\tfrac{r}{3})\times[s^{\prime},s],$ (4.1)
$\displaystyle\operatorname{Vol}_{g_{s}}\left(B_{s}(y,\tfrac{r}{3})\right)\geq
c\left(\tfrac{r}{3}\right)^{n}.$ (4.2)

###### Proof of Claim 1.

(4.1) is almost a restatement of an assumption of the theorem. We will prove
(4.2) by the Bishop-Gromov comparison theorem. Fixing any point $y\in
B_{s}(z,\frac{r}{3})$, since $\operatorname{Ric}\geq-C_{0}r^{-2}$ on
$B_{s}(z,r)$, we may argue as follows. Note that all the volumes are computed
using $g(s)$.

$\displaystyle\frac{\operatorname{Vol}\big{(}B_{s}(y,\tfrac{r}{3})\big{)}}{(\tfrac{r}{3})^{n}}$
$\displaystyle\geq$ $\displaystyle
c\frac{\operatorname{Vol}\big{(}B_{s}(y,\tfrac{2r}{3})\big{)}}{(\tfrac{2r}{3})^{n}}\geq
c\frac{\operatorname{Vol}\big{(}B_{s}(z,\tfrac{r}{3})\big{)}}{(\tfrac{2r}{3})^{n}}$
$\displaystyle=$ $\displaystyle
2^{-n}c\frac{\operatorname{Vol}\big{(}B_{s}(z,\tfrac{r}{3})\big{)}}{(\tfrac{r}{3})^{n}}\geq
c\frac{\operatorname{Vol}\big{(}B_{s}(z,r)\big{)}}{r^{n}}\geq c,$

where the constant $c$ depends only on $\alpha$ and $C_{0}$. ∎

Next, we shall obtain some estimates for the conjugate heat kernel
$K(x,t\,|\,\cdot,\cdot)$. The local upper bound follows straightforwardly from
the standard mean value inequality, and the local lower bound is a consequence
of (2.11).

Claim 2. There is a constant $C$, depending only on $\alpha$ and $C_{0}$, such
that

$\displaystyle K\left(x,t\,\left|\,y,s^{\prime}\right.\right)\leq
Cr^{-n}\leq\frac{C}{(t-s^{\prime})^{\tfrac{n}{2}}}\quad\text{ for all }\quad
y\in B_{s}(z,\tfrac{r}{3}).$ (4.3)

###### Proof of Claim 2.

Let us fix an arbitrary $y\in B_{s}(z,\frac{r}{3})$. Since
$K(x,t\,|\,\cdot,\cdot)$ is a conjugate heat kernel, we have

$\displaystyle\int_{s-\frac{1}{9}r^{2}}^{s}\int_{B_{s}(y,\frac{r}{3})}K(x,t\,|\,\cdot,\eta)dg_{\eta}d\eta\leq\int_{s-\frac{1}{9}r^{2}}^{s}\int_{M}K(x,t\,|\,\cdot,\eta)dg_{\eta}d\eta=\int_{s-\frac{1}{9}r^{2}}^{s}1d\eta=\frac{1}{9}r^{2}.$

On the other hand, the bounds provided by (4.1) and (4.2) are sufficient for
the implementation of the standard mean value inequality (c.f. [CCGGIIKLLN10,
Theorem 25.2]) on the parabolic disk
$B_{s}(y,\frac{r}{3})\times[s-\frac{1}{9}r^{2},s]$. We then obtain that

$\displaystyle K\left(x,t\,\left|\,y,s^{\prime}\right.\right)$
$\displaystyle\leq$
$\displaystyle\frac{C}{\frac{1}{9}r^{2}\operatorname{Vol}_{g_{s}}\big{(}B_{s}(y,\frac{r}{3})\big{)}}\int_{s-\frac{1}{9}r^{2}}^{s}\int_{B_{s}(y,\frac{r}{3})}K(x,t|\cdot,\eta)dg_{\eta}d\eta$
$\displaystyle\leq$ $\displaystyle\frac{C}{\frac{1}{9}r^{2}\cdot
c(\frac{r}{3})^{n}}\cdot\tfrac{1}{9}r^{2}\leq Cr^{-n}.$

∎

Claim 3: There is a constant $c$, depending only on $\alpha$ and $C_{0}$, such
that

$\displaystyle K(x,t\,|\,y,s^{\prime})\geq
cr^{-n}\geq\frac{c}{(t-s^{\prime})^{\frac{n}{2}}}\quad\text{ for all }\quad
y\in B_{s}(z,\tfrac{r}{3}).$

###### Proof of Claim 3.

Let us fix an arbitrary $y\in B_{s}(z,\frac{1}{3}r)$, and let
$\gamma_{2}:[0,\frac{1}{9}r^{2}]\rightarrow B_{s}(z,\frac{1}{3}r)$ be the
minimal constant-speed $g(s)$-geodesic, such that $\gamma_{2}(0)=z$ and
$\gamma_{2}(\frac{1}{9}r^{2})=y$. Then, obviously we have

$\displaystyle|\gamma_{2}^{\prime}(\tau)|^{2}_{g_{s}}\leq\left(\frac{r/3}{r^{2}/9}\right)^{2}=9r^{-2}\quad\text{
for all }\quad\tau\in[0,\tfrac{1}{9}r^{2}].$

Therefore, by the Ricci curvature bound in (4.1), we have

$\displaystyle|\gamma^{\prime}_{2}(\tau)|^{2}_{g_{s-\tau}}\leq|\gamma^{\prime}_{2}(\tau)|^{2}_{g_{s}}\exp\left(\int_{0}^{\tfrac{1}{9}r^{2}}\left(\sup_{B_{s}\left(z,\tfrac{1}{3}r\right)\times[s^{\prime},s]}|\operatorname{Ric}|\right)d\eta\right)\leq
Cr^{-2},$ (4.4)

for all $\tau\in[0,\tfrac{1}{9}r^{2}]$. Let $\gamma_{1}:[0,r^{2}]\rightarrow
M$ be a minimal $\mathcal{L}$-geodesic from $(x,t)$ to $(z,s)$. Then, we may
use

$\displaystyle\gamma(\tau):=\left\\{\begin{array}[]{lcr}\gamma_{1}(\tau)&\text{if}&\tau\in[0,r^{2}],\\\
\gamma_{2}(\tau-r^{2})&\text{if}&\tau\in[r^{2},\tfrac{10}{9}r^{2}].\end{array}\right.$

as a test curve in (2.9) and estimate

$\displaystyle\ell_{x,t}(y^{\prime},t-s^{\prime})$ $\displaystyle=$
$\displaystyle\ell_{x,t}\left(y^{\prime},\tfrac{10}{9}r^{2}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\sqrt{\tfrac{10}{9}r^{2}}}\bigg{\\{}\int_{0}^{r^{2}}\sqrt{\tau}\left(|\gamma_{1}^{\prime}(\tau)|^{2}_{g_{t-\tau}}+R_{g_{t-\tau}}(\gamma_{1}(\tau))\right)d\tau$
$\displaystyle\quad\quad\quad+\int_{0}^{\tfrac{1}{9}r^{2}}\sqrt{\tau+r^{2}}\left(|\gamma_{2}^{\prime}(\tau)|^{2}_{g_{s-\tau}}+R_{g_{s-\tau}}(\gamma_{2}(\tau))\right)d\tau\bigg{\\}}$
$\displaystyle\leq$
$\displaystyle\frac{3}{2\sqrt{10}r}\left(2\sqrt{r^{2}}\ell_{x,t}(z,r^{2})+\int_{0}^{\frac{1}{9}r^{2}}\sqrt{\tau+r^{2}}Cr^{-2}d\tau\right)$
$\displaystyle\leq$ $\displaystyle\frac{3}{2\sqrt{10}r}\left(nr+Cr\right)$
$\displaystyle\leq$ $\displaystyle C.$

Here we have used the facts that $\ell_{x,t}(z,t-s)\leq\frac{n}{2}$ and that
$\displaystyle\sup_{\tau\in[0,\frac{1}{9}r^{2}]}\left|R_{g_{s-\tau}}(\gamma_{2}(\tau))\right|\leq
C_{0}r^{-2}$, where the latter is because $\gamma_{2}\subset
B_{s}(z,\frac{r}{3})$. Claim 3 then follows from (2.11). ∎

With all the preparations above, we are ready to estimate the Nash entropy. By
(4.1), we have

$\displaystyle\operatorname{Vol}_{g_{s^{\prime}}}\big{(}B_{s}(z,\tfrac{r}{3})\big{)}$
$\displaystyle\geq$
$\displaystyle\operatorname{Vol}_{g_{s}}\big{(}B_{s}(z,\tfrac{r}{3})\big{)}\exp\left(-\int_{0}^{\tfrac{1}{9}r^{2}}\left(\sup_{B_{s}(z,\tfrac{r}{3})\times[s^{\prime},s]}|R|\right)d\eta\right)$
$\displaystyle\geq$ $\displaystyle cr^{n}\geq c(t-s^{\prime})^{\tfrac{n}{2}}.$

Combining this with Claim 3, we have

$\displaystyle\nu_{x,t\,|\,s^{\prime}}\big{(}B_{s}(z,\tfrac{r}{3})\big{)}=\int_{B_{s}(z,\tfrac{r}{3})}K(x,t\,|\,\cdot,s^{\prime})dg_{s^{\prime}}\geq
c.$ (4.6)

Since $[s-r^{2},s]\subset I$, we have, by the maximum principle,

$\displaystyle R(\cdot,s^{\prime})\geq-\frac{n}{2(s^{\prime}-(s-r^{2}))}\geq-
Cr^{-2}\geq-\frac{C}{t-s^{\prime}}.$ (4.7)

Let us define function $f$ by

$\displaystyle
K(x,t\,|\,\cdot,s^{\prime}):=\frac{1}{(4\pi(t-s^{\prime}))^{\frac{n}{2}}}e^{-f(\cdot,s^{\prime})}.$

Then, Claim 2 and Claim 3 imply

$\displaystyle-C\leq f(\cdot,s^{\prime})\leq C\quad\text{ on }\quad
B_{s}(z,\tfrac{r}{3}).$ (4.8)

With the estimates (4.6)—(4.8), we may compute by using (3.4)

$\displaystyle\quad\quad n+2(t-s^{\prime})\cdot\frac{C}{t-s^{\prime}}$
$\displaystyle\geq\int_{M}\left(f(\cdot,s^{\prime})-\mathcal{N}_{x,t}(t-s^{\prime})-\tfrac{n}{2}\right)^{2}d\nu_{x,t\,|\,s^{\prime}}$
$\displaystyle\geq\int_{B_{s}(z,\tfrac{r}{3})}\left(f(\cdot,s^{\prime})-\mathcal{N}_{x,t}(t-s^{\prime})-\tfrac{n}{2}\right)^{2}d\nu_{x,t\,|\,s^{\prime}}$
$\displaystyle=\int_{B_{s}(z,\tfrac{r}{3})}\bigg{(}f^{2}(\cdot,s^{\prime})-2f\left(\mathcal{N}_{x,t}(t-s^{\prime})+\tfrac{n}{2}\right)+\left(\mathcal{N}_{x,t}(t-s^{\prime})+\tfrac{n}{2}\right)^{2}\bigg{)}d\nu_{x,t\,|\,s^{\prime}}$
$\displaystyle\geq\int_{B_{s}(z,\tfrac{r}{3})}\left(-f^{2}(\cdot,s^{\prime})+\tfrac{1}{2}\left(\mathcal{N}_{x,t}(t-s^{\prime})+\tfrac{n}{2}\right)^{2}\right)d\nu_{x,t\,|\,s^{\prime}}$
$\displaystyle\geq\tfrac{1}{2}\left(\mathcal{N}_{x,t}(t-s^{\prime})+\tfrac{n}{2}\right)^{2}\nu_{x,t\,|\,s^{\prime}}\big{(}B_{s}(z,\tfrac{r}{3})\big{)}-\nu_{x,t\,|\,s^{\prime}}\big{(}B_{s}(z,\tfrac{r}{3})\big{)}\sup_{B_{s}(z,\tfrac{r}{3})}f^{2}$
$\displaystyle\geq
c\left(\mathcal{N}_{x,t}(t-s^{\prime})+\tfrac{n}{2}\right)^{2}-C.$

It then follows that

$\displaystyle\mathcal{N}_{x,t}(t-s)\geq\mathcal{N}_{x,t}(t-s^{\prime})\geq-\beta.$

This finishes the proof of the theorem. ∎

Before we proceed to prove Theorem 1.8, a few preparatory results are needed.
These results are all proved in [Bam20a] for Ricci flows on closed manifolds,
but they can be easily generalized to our case without much effort.

######  Proposition 4.1 (Corollary 5.11 in [Bam20a]; c.f. Corollary 4.5 in
[MZ21]).

Let $(M^{n},g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature
within each time interval compact in $I$. Assume $s$, $t^{*}$, $t_{1}$, and
$t_{2}\in I$ with $s<t^{*}\leq t_{1},t_{2}$. Moreover, assume
$R(\cdot,t^{*})\geq R_{\operatorname{min}}$. Then, for any $x_{1}$ and
$x_{2}\in M$, we have

$\displaystyle\mathcal{N}^{*}_{s}(x_{1},t_{1})-\mathcal{N}_{s}^{*}(x_{2},t_{2})\leq\left(\frac{n}{2(t^{*}-s)}-R_{\operatorname{min}}\right)^{\frac{1}{2}}{\rm
dist}_{W_{1}}^{g_{t^{*}}}(\nu_{x_{1},t_{1}\,|\,t^{*}},\nu_{x_{2},t_{2}\,|\,t^{*}})+\frac{n}{2}\log\left(\frac{t_{2}-s}{t^{*}-s}\right),$
(4.9)

where we have defined $\mathcal{N}_{s}^{*}(x,t):=\mathcal{N}_{x,t}(t-s)$.

######  Theorem 4.2 (Theorem 8.1 in [Bam20a]).

Let $(M^{n},g(t))_{t\in I}$ be a complete Ricci flow with bounded curvature
within each time interval compact in $I$. Assume $[t-r^{2},t]\subset I$ and
$R\geq R_{\operatorname{min}}$ on $M\times[t-r^{2},t]$. Then for all
$A\in[1,\infty)$, we have

$\operatorname{Vol}_{g_{t}}\left(B_{g_{t}}(x,Ar)\right)\leq
C(R_{\operatorname{min}}r^{2})\exp(\mathcal{N}_{x,t}(r^{2}))\exp(C_{0}A^{2})r^{n},$

where $C_{0}$ is a constant depending only on the dimension.

###### Proof.

In the original proof of [Bam20a, Theorem 8.1], the results applied are
[Bam20a, Theorem 5.9] and [Bam20a, Theorem 7.5], and the former is already
established in [MZ21, Theorem 4.4] under the assumption of bounded curvature
within each compact time interval. For [Bam20a, Theorem 7.5], the results
applied in the original proof are [Bam20a, Theorem 5.9] (c.f. [MZ21, Theorem
4.4]), [Bam20a, Corollary 5.11] (c.f. [MZ21, Corollary 4.5]), the Gaussian
concentration theorem (c.f. Proposition 2.2 above), [Bam20a, Theorem 7.1]
(c.f. Theorem 3.1 above), and Proposition 4.2 (c.f. [MZ21, Proposition 3.4]).
In conclusion, the proof in our case follows step by step after [Bam20a,
Theorem 8.1]; all these results hold under our current assumption. Since the
proofs of both [Bam20a, Theorem 7.5] and [Bam20a, Theorem 8.1] are short, we
shall leave the verification to the readers. ∎

###### Proof of Theorem 1.8.

Under the assumption of the theorem, we denote

$r:=\sqrt{t-s}.$

Since $s-r^{2}\in I$, by the maximum principle, we have
$R(\cdot,s)\geq-\frac{n}{2r^{2}}$. Then, applying Theorem 4.2 to the point
$(z,s)$ and $A=1$, we have

$\alpha r^{n}\leq\operatorname{Vol}_{g_{s}}\big{(}B_{s}(z,r)\big{)}\leq
C(-\tfrac{n}{2})\exp(\mathcal{N}_{z,s}(r^{2}))\exp(C_{0})r^{n},$

that is,

$\displaystyle\mathcal{N}^{*}_{s-r^{2}}(z,s)=\mathcal{N}_{z,s}(r^{2})\geq-
C_{1},$ (4.10)

where $C_{1}$ depends only on $\alpha$ and $n$.

Next, applying Proposition 4.1 with $(z,s)\to(x_{1},t_{1})$,
$(x,t)\to(x_{2},t_{2})$, $s\to t^{*}$, $s-r^{2}\to s$, $-\frac{n}{2r^{2}}\to
R_{\operatorname{min}}$, we have

$\displaystyle\mathcal{N}_{s-r^{2}}^{*}(z,s)-\mathcal{N}_{s-r^{2}}^{*}(x,t)$
$\displaystyle\leq$
$\displaystyle\left(\frac{n}{r^{2}}\right)^{\frac{1}{2}}{\rm
dist}_{W_{1}}^{g_{s}}(\nu_{x,t\,|\,s},\delta_{z})+\frac{n}{2}\log\left(\frac{t-s+r^{2}}{s-s+r^{2}}\right)$
$\displaystyle\leq$
$\displaystyle\left(\frac{n}{r^{2}}\right)^{\frac{1}{2}}\sqrt{H_{n}(t-s)}+\frac{n}{2}\log\frac{2r^{2}}{r^{2}}$
$\displaystyle\leq$ $\displaystyle C(n),$

where we have applied the definition of the $H_{n}$-center and the fact ${\rm
dist}_{W_{1}}^{g_{s}}(\nu_{x,t\,|\,s},\,\delta_{z})\leq\sqrt{\operatorname{Var}(\nu_{x,t\,|\,s},\,\delta_{z})}\leq\sqrt{H_{n}(t-s)}$
(c.f. [Bam20a, Lemma 3.2]). In combination with (4.10), we have

$\mathcal{N}_{x,t}(2r^{2})=\mathcal{N}_{s-r^{2}}^{*}(x,t)\geq\mathcal{N}_{s-r^{2}}^{*}(z,s)-C(n)\geq-
C_{1}-C(n).$

This finishes the proof of the theorem. ∎

##  5 Properties of ancient solutions satisfying Assumption B

In this section, we prove several properties for an ancient solution
satisfying (1.1) and Assumption B. We first of all verify that such an ancient
solution must be locally uniformly Type I, which implies a shrinker structure
on the blow-down limit in (1.2). This fact together with Theorem 1.7 implies
that the ancient solution has bounded Nash entropy. Finally, by using Theorem
3.1, we show that an $H_{n}$-center is always not far from an $\ell$-center.

Throughout this section, we let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient
solution satisfying (1.1) and Assumption B with respect to
$\big{(}p_{0},0,\\{\tau_{i}\\}_{i=1}^{\infty},\\{p_{i}\\}_{i=1}^{\infty}\big{)}$.
Let $(M_{\infty},g_{\infty}(t))_{t\in[-2,-1]}$ be the limit Ricci flow in
(1.2). From the smooth convergence (1.2), we may easily check that
$(M,g(t))_{t\in(-\infty,0]}$ satisfies the locally uniformly Type I condition.

######  Proposition 5.1.

$(M,g(t))_{t\in(-\infty,0]}$ is locally uniformly Type I along the sequence
$\\{(p_{i},-\tau_{i})\\}_{i=1}^{\infty}$.

###### Proof.

By the following Cheeger-Gromov-Hamilton convergence in the statement of
Assumption B

$\displaystyle\big{(}M,g_{i}(t),p_{i}\big{)}_{t\in[-2,-1]}\xrightarrow{\makebox[28.45274pt]{}}\big{(}M_{\infty},g_{\infty}(t),p_{\infty}\big{)}_{t\in[-2,-1]},$

where $g_{i}(t)=\tau_{i}^{-1}g(\tau_{i}t)$, we may find
$\varepsilon_{i}\searrow 0$, open sets $U_{i}\subset M_{\infty}$, and
diffeomorphisms

$\displaystyle\Psi_{i}:M_{\infty}\supset U_{i}\rightarrow V_{i}\subset M,$
(5.1)

such that the following hold

$\displaystyle U_{i}\supset
B_{g_{\infty,-1}}(p_{\infty},\varepsilon_{i}^{-1})\quad\text{ and }\quad
V_{i}=\Psi_{i}(U_{i})\supset B_{g_{i,-1}}(p_{i},\varepsilon_{i}^{-1}),$ (5.2)
$\displaystyle\Psi_{i}(p_{\infty})=p_{i},$ (5.3)
$\displaystyle\big{\|}\Psi_{i}^{*}g_{i}-g_{\infty}\big{\|}_{C^{[\varepsilon_{i}^{-1}]}(U_{i}\times[-2,-1])}\leq\varepsilon_{i}.$
(5.4)

As a consequence of (5.3) and (5.4), we can find a positive function
$C:(0,\infty)\rightarrow(0,\infty)$, such that

$\displaystyle\lim_{i\rightarrow\infty}\operatorname{inj}_{g_{i,-1}}(p_{i})=\operatorname{inj}_{g_{\infty,-1}}(p_{\infty})>0,$
(5.5)
$\displaystyle\lim_{i\rightarrow\infty}\sup_{B_{g_{i,-1}}(p_{i},r)\times[-2,-1]}|\operatorname{Rm}_{g_{i}}|=\sup_{B_{g_{\infty,-1}}(p_{\infty},r)\times[-2,-1]}|\operatorname{Rm}_{g_{\infty,-1}}|\leq
C(r)<\infty,$ (5.6)

where the latter holds for all $r>0$. Scaling (5.5) and (5.6) with factor
$\tau_{i}$, we have verified Definition 2.8(2)(4) for $(M,g(t))$. Furthermore,
by (1.1), we have a time-wise Ricci curvature lower bound for $g(t)$. Lastly,
since $(p_{i},-\tau_{i})$’s are $\ell$-centers of $(p_{0},0)$, we also have
$\ell_{p_{0},0}(p_{i},\tau_{i})\leq\frac{n}{2}$ for all $i$. Therefore,
$(M,g(t))$ is locally uniformly Type I along the sequence
$\\{(p_{i},-\tau_{i})\\}_{i=1}^{\infty}$. ∎

The following proposition and corollary are but restatements of Proposition
2.10 and Corollary 2.11, respectively. Theorem 1.1(1) follows from Proposition
5.2.

######  Proposition 5.2.

The limit $(M_{\infty},g_{\infty}(t))_{t\in[-2,-1]}$ has a shrinker structure.
More precisely, $\ell_{i}\rightarrow\ell_{\infty}$ in the
$C^{\alpha}_{\operatorname{loc}}$ sense or in the weak
$*W^{1,2}_{\operatorname{loc}}$ sense on $M_{\infty}\times(1,2)$, where
$\ell_{i}(\cdot,\tau)=\ell(\cdot,\tau_{i}\tau):M\rightarrow\mathbb{R}$ for
$\tau\in[1,2]$, $\ell$ is the reduced distance based at $(p_{0},0)$ with
respect to the Ricci flow $(M,g(t))$, and $\ell_{\infty}$ is a smooth shrinker
potential satisfying

$\displaystyle\operatorname{Ric}_{g_{\infty}}(-\tau)+\nabla^{2}\ell_{\infty}(\cdot,\tau)=\frac{1}{2\tau}g_{\infty}(-\tau)\quad\text{
for all }\quad\tau\in(1,2).$

######  Corollary 5.3.

$(M,g(t))_{t\in(-\infty,0)}$ is weakly $\kappa$-noncollapsed on all scales. As
a consequence, $(M,g(t))_{t\in(-\infty,0]}$ has bounded geometry in each
compact time-interval.

Remark: From this point on, any ancient solution satisfying (1.1) and
Assumption B is understood to have bounded geometry within each compact time
interval.

######  Theorem 5.4.

The Nash entropy of $(M,g(t))_{t\in(-\infty,0]}$ based at $(p_{0},0)$ is
bounded independent of time, that is,

$\displaystyle\mathcal{N}_{p_{0},0}(\tau)\geq-Y\quad\text{ for all
}\quad\tau>0,$

where $Y\in(0,\infty)$ depends on the local geometry bounds around the
$\ell$-centers $(p_{i},-\tau_{i})$.

###### Proof.

By the assumption (1.1), we have that each $(M,g_{i}(t))_{t\in[-2,0]}$ is a
complete Ricci flow with bounded curvature, where
$g_{i}(t):=\tau_{i}^{-1}g(\tau_{i}t)$. Since $(p_{i},-1)$ is an $\ell$-center
of $(p_{0},0)$ with respect to the Ricci flow $(M,g_{i}(t))$, and since (5.5)
and (5.6) imply that

$\displaystyle\big{|}\operatorname{Rm}_{g_{i}}\big{|}\leq C\quad\text{ on
}\quad B_{g_{i,-1}}(p_{i},1)\times[-2,-1],$
$\displaystyle\operatorname{Vol}_{g_{i,-1}}(p_{i},1)\geq c,$

where $c$ and $C$ are positive constants independent of $i$, we have, by
Theorem 1.7,

$\displaystyle\mathcal{N}^{i}_{p_{0},0}(1)\geq-Y\quad\text{ for all }\quad
i\in\mathbb{N},$

where $\mathcal{N}^{i}$ is the Nash entropy of the Ricci flow $(M,g_{i}(t))$
and $Y$ depends only on $c$ and $C$. Therefore, by the scaling property of the
Nash entropy, we have

$\mathcal{N}_{p_{0},0}(\tau_{i})\geq-Y\quad\text{ for all }\quad i,$

and the theorem follows from the monotonicity of $\mathcal{N}$. ∎

Next, we show that the $H_{n}$-centers and the $\ell$-centers are not far from
each other, and consequently the base points in (1.2) can be replaced by
$H_{n}$-centers. First of all, the following result is merely a consequence of
Theorem 3.2 and Theorem 5.4.

######  Proposition 5.5.

There exists a positive number $C$ depending only on the number $Y$ in the
conclusion of Theorem 5.4, such that

$\displaystyle
K(p_{0},0\,|\,x,t)\leq\frac{C}{|t|^{\frac{n}{2}}}\exp\left(-\frac{{\rm
dist}_{t}^{2}(x,z)}{C|t|}\right)\quad\text{ for all }\quad x\in M\quad\text{
and }\quad t\in(-\infty,0),$ (5.7)

where $(z,t)$ is an $H_{n}$-center of $(p_{0},0)$.

Let us find another sequence of points $\\{z_{i}\\}_{i=0}^{\infty}\subset M$,
such that each $(z_{i},-\tau_{i})$ is an $H_{n}$-centers of $(p_{0},0)$. For
$H_{n}$-concentrated Ricci flows (this category obviously contains all Ricci
flows with bounded curvature in compact time intervals; see the remark below
Proposition 2.15), one may always find an $H_{n}$-center at each time for any
base point. We then have the following estimate.

######  Proposition 5.6.

There is a constant $C$ depending on the number Y in the conclusion of Theorem
5.4, such that the following holds.

$\displaystyle{\rm dist}_{-\tau_{i}}(p_{i},z_{i})\leq
C\sqrt{\tau_{i}}\quad\text{ for all }\quad i.$ (5.8)

Remark: Indeed, the same argument of the proof below shows that, for any
$t<0$, we have

${\rm dist}_{t}(p,z)\leq C\sqrt{|t|},$

where $(p,t)$ and $(z,t)$ are an $\ell$-center and an $H_{n}$-center of
$(p_{0},0)$, respectively.

###### Proof.

Combining (2.11) and (5.7), we have

$\displaystyle\frac{C}{\tau_{i}^{\frac{n}{2}}}\exp\left(-\frac{{\rm
dist}_{-\tau_{i}}^{2}(p_{i},z_{i})}{C\tau_{i}}\right)\geq
K(p_{0},0\,|\,p_{i},-\tau_{i})\geq\frac{1}{(4\pi\tau_{i})^{\frac{n}{2}}}e^{-\ell_{p_{0},0}(p_{i},\tau_{i})}\geq\frac{1}{(4\pi\tau_{i})^{\frac{n}{2}}}e^{-\frac{n}{2}},$

where we have used the fact that
$\ell_{p_{0},0}(p_{i},\tau_{i})\leq\frac{n}{2}$. It then follows that

$\displaystyle{\rm dist}_{-\tau_{i}}^{2}(p_{i},z_{i})\leq C\tau_{i}\quad\text{
for all }\quad i.$

∎

######  Corollary 5.7.

In the convergence in (1.2), possibly after passing to a subsequence, one may
replace $p_{i}$ by $z_{i}$. If so, then the limit Ricci flow is still the
same, whose base point is correspondingly replace by some $z_{\infty}$
satisfying ${\rm dist}_{g_{\infty,-1}}(p_{\infty},z_{\infty})\leq C$, where
$C$ is the constant in (5.8).

###### Proof.

After the parabolic scaling $g_{i}(t)=\tau_{i}^{-1}g(\tau_{i}t)$, (5.8)
becomes

$\displaystyle{\rm dist}_{g_{i,-1}}(p_{i},z_{i})\leq C\quad\text{ for all
}\quad i.$

The corollary then follows from the definition of Cheeger-Gromov-Hamilton
convergence. ∎

##  6 Smooth convergence and $\mathbb{F}$-convergence

For the purpose of proving Theorem 1.1(3), we shall prove a more general
statement in this section, namely, that from a sequence of complete smooth
Ricci flows, if, using a sequence of $H_{n}$-centers as base points, one
obtains a smooth Cheeger-Gromov-Hamilton limit, then, this smooth limit is the
same as the $\mathbb{F}$-limit.

Let $\\{(M_{i},g_{i}(t))_{t\in(-T_{i},0]}\\}_{i=1}^{\infty}$ be a sequence of
complete Ricci flows, each one with bounded curvature within each compact time
interval, where $\infty\geq T_{i}>c>0$ for some constant $c$. For each $i$,
let $p_{i}\in M_{i}$ be a fixed point and $(\nu^{i}_{t})_{t\in(-T_{i},0]}$ the
conjugate heat kernel on $(M_{i},g_{i})$ based at $(p_{i},0)$. According to
[Bam20b, Theorem 7.4, Corollary 7.5],
$\displaystyle\left\\{\big{(}M_{i},g_{i}(t))_{t\in(-T_{i},0]},(\nu^{i}_{t})_{t\in(-T_{i},0]}\big{)}\right\\}_{i=1}^{\infty}$
has an $\mathbb{F}$-convergent subsequence.

######  Theorem 6.1.

Assume that there exist a compact interval $I=[a,b]\subset(-T_{\infty},0)$,
where $T_{\infty}:=\limsup_{i\rightarrow\infty}T_{i}$, and a smooth Ricci flow
$(M_{\infty},g_{\infty}(t),z_{\infty})_{t\in I}$, such that

$\displaystyle\left(M_{i},g_{i}(t),z_{i}\right)_{t\in
I}\xrightarrow{\makebox[28.45274pt]{}}\left(M_{\infty},g_{\infty}(t),z_{\infty}\right)_{t\in
I}$ (6.1)

in the smooth Cheeger-Gromov-Hamilton sense, where $(z_{i},b)$ is an
$H_{n}$-center of $(p_{i},0)$ for each $i\in\mathbb{N}$. Then
$(M_{\infty},g_{\infty}(t))_{t\in[a,b]}$ induces an $H_{n}$-concentrated
continuous metric flow $\mathcal{X}^{\infty}$ and there is a conjugate heat
flow $(\nu_{t}^{\infty})_{t\in[a,b)}$ on $\mathcal{X}^{\infty}$, such that, by
passing to a subsequence, we have

$\displaystyle\left((M_{i},g_{i}(t))_{t\in[a,b]},(\nu^{i}_{t})_{t\in[a,b]}\right)\xrightarrow{\makebox[28.45274pt]{$\mathbb{F}$}}\left(\mathcal{X}^{\infty},(\nu_{t}^{\infty})_{t\in[a,b)}\right),$
(6.2)

where the convergence is uniform over any compact sub-interval of $[a,b).$

Remarks:

  1. 1.

This theorem implies that, for any subsequence of
$\left\\{\left((M_{i},g_{i}(t))_{t\in[a,b]},(\nu^{i}_{t})_{t\in[a,b]}\right)\right\\}_{i=1}^{\infty}$,
there is a further subsequence that $\mathbb{F}$-converges to
$\left(\mathcal{X}^{\infty},(\nu_{t}^{\infty})_{t\in[a,b)}\right)$, where
$\nu_{t}^{\infty}$ is a conjugate heat flow on $\mathcal{X}^{\infty}$. So any
continuous metric flow pair $(\mathcal{Y}^{\infty},(\mu_{t})_{t\in[a,b)})$
that arises as an $\mathbb{F}$-limit given by the compactness theorem [Bam20b,
Theorem 7.6] should be of the form
$\left(\mathcal{X}^{\infty},(\nu_{t}^{\infty})_{t\in[a,b)}\right)$. However,
this does not imply that the $\mathbb{F}$-convergence in
$(\ref{theequivalentFconvergence})$ holds without passing to a subsequence,
since we are not able to show that $\nu_{t}^{\infty}$ is independent of the
subsequence.

  2. 2.

We may replace the $H_{n}$-centers $(z_{i},b)$ in the assumptions with
$\ell$-centers and the conclusions still hold. This is because $H_{n}$-centers
are not far away from $\ell$-centers by the arguments in Proposition 5.6. This
point will become clear from the proof.

Throughout this section, we assume that the conditions of Theorem 6.1 hold.
Just as we have done in (5.1)—(5.4), by the definition of smooth convergence,
we may find an increasing sequence of pre-compact open sets $U_{i}\subset
M_{\infty}$ with $\cup_{i=1}^{\infty}U_{i}=M_{\infty}$ and a sequence of
diffeomorphisms $\Psi_{i}:U_{i}\to V_{i}\subset M_{i}$, such that
$\Psi_{i}(z_{\infty})=z_{i}$ and

$\displaystyle\|\Psi_{i}^{*}g_{i}-g_{\infty}\|_{C^{[\varepsilon_{i}^{-1}]}(U_{i}\times[a,b])}<\varepsilon_{i},$
(6.3)

for some $\varepsilon_{i}\searrow 0.$

The proof of Theorem 6.1 is divided into several components. We shall first of
all show that the smooth limit flow is indeed a metric flow. This is not as
obvious as it appears to be, since we do not make any geometric assumption for
$(M_{\infty},g_{\infty}(t))_{t\in[a,b]}$ except for its smoothness and
completeness. The idea is to construct a metric flow structure using the
converging sequence.

######  Lemma 6.2.

$(M_{\infty},g_{\infty}(t))_{t\in[a,b]}$ induces an $H_{n}$-concentrated
continuous metric flow $\mathcal{X}^{\infty}$.

###### Proof.

As mentioned in Theorem 2.12, each $(M_{i},g_{i}(t))$ induces an
$H_{n}$-concentrated metric flow (see also the remark below Proposition 2.15).
Let us then verify [Bam20b, Definition 3.2] and the $H_{n}$-concentrarion
condition for $(M_{\infty},g_{\infty}(t))_{t\in I}$.

The conjugate heat kernel $K_{\infty}(\cdot,\cdot\,|\,\cdot,\cdot)$ on
$(M_{\infty},g_{\infty}(t))_{t\in I}$ is given by Theorem B.1. Furthermore, by
Theorem B.2, if we define
$d\nu^{\infty}_{x,t\,|\,s}=K(x,t\,|\,\cdot,s)dg_{\infty,s}$, then we have that
$\nu^{\infty}_{x,t\,|\,s}\in\mathcal{P}(M_{\infty})$ and [Bam20b, Definition
3.2(1)—(5),(7)] holds.

To verify that
$\left(M_{\infty},g_{\infty}(t),\nu^{\infty}_{x,t\,|\,s}\right)$ satisfies
[Bam20b, Definition 3.2(6)], we shall apply [Bam20b, Lemma 3.8]. Let us fix
$s,t\in I$ with $s<t$, a positive number $T$, and a measurable function
$u:M_{\infty}\rightarrow(0,1)$ such that $f:=\Phi^{-1}\circ u$ is
$T^{-\frac{1}{2}}$-Lipschitz with respect to the metric $g_{\infty}(s)$, where
$\Phi:\mathbb{R}\rightarrow(0,1)$ is defined as in [Bam20b, (3.1)]. Let
$\eta:[0,\infty)\rightarrow[0,1]$ be the standard cut-off function such that
$\eta\equiv 1$ on $[0,1]$, $\eta\equiv 0$ on $[2,\infty)$, and
$0\geq\eta^{\prime}\geq-2$ everywhere. Let $\zeta:\mathbb{R}\rightarrow[-1,1]$
be the function satisfying $\zeta(t)=t$ for $|t|<1$,
$\zeta|_{[1,\infty)}\equiv 1$, and $\zeta|_{(-\infty,-1]}\equiv-1.$ For any
$A>0$, we define

$\displaystyle f^{A}:=\eta\left(\tfrac{1}{A^{2}}{\rm
dist}_{g_{\infty,s}}(z_{\infty},\cdot)\right)\cdot
A\zeta\left(\tfrac{f}{A}\right),$ $\displaystyle u^{A}:=\Phi\circ f^{A}.$

Then we have

$\displaystyle u^{A}\in[\Phi(-A),\Phi(A)]\subset(0,1)\quad\text{ everywhere on
}\quad M_{\infty},$ (6.4) $\displaystyle\big{|}\,u-u^{A}\,\big{|}\leq
1-\Phi(A)=\Phi(-A)\quad\text{ on }\quad B_{g_{\infty,s}}(z_{\infty},A^{2}),$
(6.5) $\displaystyle u^{A}\equiv 1/2\quad\text{ on }\quad M_{\infty}\setminus
B_{g_{\infty,s}}(z_{\infty},2A^{2}),$ (6.6)
$\displaystyle\operatorname{Lip}f^{A}\leq\tfrac{2}{A^{2}}\sup\left|A\zeta\left(\tfrac{f}{A}\right)\right|+\sup|\eta|\cdot
A\operatorname{Lip}\zeta\cdot\operatorname{Lip}\tfrac{f}{A}\leq
T^{-\frac{1}{2}}+\tfrac{2}{A}.$ (6.7)

Let us fix two points $x,y\in M_{\infty}$. By (6.4), (6.5), and the bounded
convergence theorem, we have

$\displaystyle\lim_{A\rightarrow\infty}u^{A}_{t}(x),\
\lim_{A\rightarrow\infty}u^{A}_{t}(y)=u_{t}(x),\ u_{t}(y),$ (6.8)

where

$u_{t}^{A}(\cdot):=\int_{M_{\infty}}u^{A}\,d\nu^{\infty}_{\cdot,t\,|\,s},\quad
u_{t}(\cdot):=\int_{M_{\infty}}u\,d\nu^{\infty}_{\cdot,t\,|\,s}.$

In view of (6.6), for all $i$ large enough we may define

$\displaystyle f_{i}^{A}$ $\displaystyle=$ $\displaystyle
f^{A}\circ\Psi_{i}^{-1}\quad\text{ on }\quad V_{i}=\Psi_{i}(U_{i})$
$\displaystyle f_{i}^{A}$ $\displaystyle\equiv$ $\displaystyle 0\quad\text{ on
}\quad M_{i}\setminus V_{i}.$ $\displaystyle u_{i}^{A}$ $\displaystyle:=$
$\displaystyle\Phi\circ f_{i}^{A}.$

Then, by the smooth convergence (6.3), we have

$\displaystyle\operatorname{Lip}f_{i}^{A}\leq
T^{-\frac{1}{2}}+\frac{2}{A}+\varepsilon_{i},$

and consequently (c.f. [MZ21, Theorem 3.1]),

$\displaystyle\Big{|}\,\Phi^{-1}\circ u^{A}_{i,t}(\Psi_{i}(x))-\Phi^{-1}\circ
u^{A}_{i,t}(\Psi_{i}(y))\,\Big{|}$ (6.9)
$\displaystyle\quad\quad\leq\left(\left(T^{-\frac{1}{2}}+\tfrac{2}{A}+\varepsilon_{i}\right)^{2}+t-s\right)^{-\frac{1}{2}}{\rm
dist}_{g_{i,t}}\big{(}\Psi_{i}(x),\Psi_{i}(y)\big{)},$

for all $i$ large enough, where
$u_{i,t}^{A}:=\int_{M_{i}}u_{i}^{A}d\nu^{i}_{\cdot,t\,|\,s}$. In view of (6.4)
and Proposition 2.15, the integral of
$\int_{M_{i}}u_{i}^{A}d\nu^{i}_{\cdot,t\,|\,s}$ is uniformly negligible
outside a large disk. Then, by the locally smoothly convergence of the
conjugate heat kernels, we have (c.f. the proof of (B.9))

$\lim_{i\rightarrow\infty}u^{A}_{i,t}(\Psi_{i}(x)),\
\lim_{i\rightarrow\infty}u^{A}_{i,t}(\Psi_{i}(y))=u_{t}^{A}(x),\
u_{t}^{A}(y).$

Hence, (6.9) implies that

$\displaystyle\big{|}\Phi^{-1}\circ u_{t}^{A}(x)-\Phi^{-1}\circ
u_{t}^{A}(y)\big{|}\leq\left(\left(T^{-\frac{1}{2}}+\tfrac{2}{A}\right)^{2}+t-s\right)^{-\frac{1}{2}}{\rm
dist}_{g_{\infty,t}}(x,y).$

Therefore, by (6.8) and taking $A\rightarrow\infty$, we have

$\big{|}\Phi^{-1}\circ u_{t}(x)-\Phi^{-1}\circ
u_{t}(y)\big{|}\leq\left(T+t-s\right)^{-\frac{1}{2}}{\rm
dist}_{g_{\infty,t}}(x,y).$

Since $x,y\in M_{\infty}$ are arbitrary, we have verified [Bam20b, Definition
3.2(6)].

Finally, it is also easy to verify the $H_{n}$-concentration property for
$\mathcal{X}^{\infty}$, since it is but a consequence of Fatou’s lemma. ∎

######  Lemma 6.3.

There is a positive solution to the conjugate heat equation
$v:M_{\infty}\times[a,b)\rightarrow\mathbb{R}$ coupled with
$(M_{\infty},g_{\infty}(t))$, satisfying
$d\nu^{\infty}_{s}:=v_{s}\,dg_{\infty,s}\in\mathcal{P}(M_{\infty})$ and

$\Psi_{i}^{*}K_{i}(p_{i},0\,|\,\cdot,\cdot)\to v$

locally smoothly on $M_{\infty}\times[a,b).$

###### Proof.

Arguing in the same way as the proof of [Lu12, Theorem 2.1], we can find a
nonnegative solution $v:M_{\infty}\times[a,b)\rightarrow\mathbb{R}$ to the
conjugate heat equation, such that

$\displaystyle\Psi_{i}^{*}K_{i}(p_{i},0\,|\,\cdot,\cdot)\to v$ (6.10)

locally smoothly on $M_{\infty}\times[a,b)$.

In this case, Theorem B.1 does not imply that $\int v_{s}\,dg_{\infty,s}=1$ as
it did in the proof of formula (B.5). This is because the base point
$(p_{i},0)$ of the conjugate heat kernel $K_{i}(p_{i},0\,|\,\cdot,\cdot)$ is
not in the region of the Cheeger-Gromov-Hamilton convergence. We first of all
observe from (6.3) that, there is a positive function
$C:(0,\infty)\rightarrow(0,\infty)$ with the following property: for any
$r>0$, we have

$\displaystyle\left|\operatorname{Rm}_{g_{i}}\right|\leq C(r)\quad\text{ on
}\quad B_{g_{i,b}}(z_{i},r)\times[a,b]$ (6.11)

whenever $i$ is large enough. Since we also have

$\displaystyle\liminf_{i\rightarrow\infty}\operatorname{Vol}_{g_{i,b}}\big{(}B_{g_{i,b}}(z_{i},1)\big{)}=\operatorname{Vol}_{g_{\infty,b}}\big{(}B_{g_{\infty,b}}(z_{\infty},1)\big{)}>0,$

we may apply Theorem 1.8 and the remark below it to the $H_{n}$-center
$(z_{i},b)$ of $(p_{i},0)$, and consequently, we can find a positive number
$Y$ independent of $i$, such that

$\displaystyle\mathcal{N}^{i}_{p_{i},0}(|b|)\geq-Y\quad\text{ for all }\quad
i\in\mathbb{N},$

where $\mathcal{N}^{i}$ is the Nash entropy of the Ricci flow
$(M_{i},g_{i}(t))$. The argument in Proposition 5.6 then implies that

$\displaystyle{\rm dist}_{g_{i,b}}(z_{i},p^{\prime}_{i})\leq C\quad\text{ for
all }\quad i\in\mathbb{N},$ (6.12)

where $(p^{\prime}_{i},b)$ is an $\ell$-center of $(p_{i},0)$, and $C$ is a
constant depending only on $Y$.

Now we will use (2.11) to obtain a uniform lower bound for
$K_{i}(p_{i},0\,|\,p^{\prime}_{i},b-\varepsilon)$, where $\varepsilon$ is an
arbitrarily fixed number in $(0,b-a]$. Let us fix such an $\varepsilon$.
(6.11) and (6.12) imply

$\sup_{t\in[a,b]}\left|R_{g_{i}}(p^{\prime}_{i},t)\right|\leq C\quad\text{ for
all }i\in\mathbb{N},$

where $C$ is a constant independent of $i$. We may concatenate a minimal
$\mathcal{L}$-geodesic from $(p_{i},0)$ to $(p^{\prime}_{i},b)$ and the static
curve from $(p^{\prime}_{i},b)$ to $(p^{\prime}_{i},b-\varepsilon)$ as a test
curve in (2.9). This yields

$\displaystyle\ell_{p_{i},0}(p^{\prime}_{i},|b|+\varepsilon)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\sqrt{|b|+\varepsilon}}\left(2\sqrt{|b|}\ell_{p_{i},0}(p^{\prime}_{i},|b|)+\int_{|b|}^{|b|+\varepsilon}\sqrt{\tau}R_{g_{i}}(p^{\prime}_{i},-\tau)\,d\tau\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\sqrt{|b|+\varepsilon}}\left(2\sqrt{|b|}\cdot\frac{n}{2}+C(|b|+\varepsilon)^{3/2}\right)$
$\displaystyle\leq$ $\displaystyle C\quad\text{ for all }\quad
i\in\mathbb{N}.$

Hence, by (2.11), we have

$K_{i}(p_{i},0\,|\,p^{\prime}_{i},b-\varepsilon)\geq\frac{1}{4\pi(|b|+\varepsilon)^{\frac{n}{2}}}e^{-\ell_{p_{i},0}(p^{\prime}_{i},|b|+\varepsilon)}\geq
c(\varepsilon)>0\quad\text{ for all }\quad i\in\mathbb{N}.$

By (6.12) again, we may find a point $p^{\prime}_{\infty}\in M_{\infty}$, such
that $\Psi_{i}^{-1}(p^{\prime}_{i})\rightarrow p^{\prime}_{\infty}$ after
possibly passing to a subsequence. Consequently

$v(p^{\prime}_{\infty},b-\varepsilon)=\lim_{i\rightarrow\infty}K_{i}(p_{i},0\,|\,p^{\prime}_{i},b-\varepsilon)\geq
c(\varepsilon)>0.$

Since $\varepsilon\in(0,b-a]$ is arbitrary, it then follows from the strong
maximum principle that $v>0$ everywhere on $M_{\infty}\times[a,b)$.

Once we know that $v$ is positive, by exactly the same argument as in the
proof of (B.5), we have

$\int_{M_{\infty}}v_{s}\,dg_{\infty,s}=1,$

for any $s\in[a,b).$ Indeed, here one may use the fact that

$\lim_{i\rightarrow\infty}\int_{B_{g_{i,s}}(z_{i},1)}K_{i}(p_{i},0\,|\,\cdot,s)dg_{i,s}=\int_{B_{g_{\infty,s}}(z_{\infty},1)}v(\cdot,s)dg_{\infty,s}>0$

to show that $(z_{i},s)$ is not far from an $H_{n}$-center of $(p_{i},0)$, for
any $s\in[a,b)$, and the rest of the argument is the same as the proof of
(B.5). ∎

######  Lemma 6.4.

Let $(M^{n},g(t))_{t\in I}$ be a complete Ricci flow over a compact interval
$I$ and $o\in M$. Then for any $A>0,\Omega:=\cup_{t\in I}\bar{B}_{t}(o,A)$ is
compact.

###### Proof.

Let $x_{j}\in\bar{B}_{s_{j}}(o,A)\subset\Omega$ be an arbitrary sequence.
Assume that $s_{j}\to\bar{s}\in I.$ We shall prove that $\\{x_{j}\\}$ has a
convergent subsequence. Suppose that

$\sup_{B_{\bar{s}}(o,10A)\times I}|{\operatorname{Ric}}|\leq\Lambda.$

We claim that for any $\epsilon\in(0,1),$ there is $\bar{j},$ such that if
$j\geq\bar{j},$ $x_{j}\in B_{\bar{s}}(o,(1+\epsilon)A)$. Suppose not. Then by
passing to a subsequence, we may assume that there are $g(s_{j})$-minimal
geodesics $\gamma_{j}:[0,\sigma_{j}]\to M$ with
$\gamma_{j}(0)=o,\gamma_{j}(\sigma_{j})=x_{j},\sigma_{j}<A$ but there is a
first time $\lambda_{j}<\sigma_{j}$ such that
$\gamma_{j}(\lambda_{j})\in\partial B_{\bar{s}}(o,(1+\epsilon/2)A).$ Pick
$\delta>0$ such that $e^{-\Lambda\delta}>1-\epsilon/4.$ When
$|s_{j}-\bar{s}|<\delta$, we have

$\displaystyle A\geq L_{s_{j}}(\gamma_{j}|_{[0,\lambda_{j}]})$
$\displaystyle\geq
e^{-\Lambda|s_{j}-\bar{s}|}L_{\bar{s}}(\gamma_{j}|_{[0,\lambda_{j}]})\geq(1-\epsilon/4){\rm
dist}_{\bar{s}}(o,\gamma_{j}(\lambda_{j}))$
$\displaystyle=(1-\epsilon/4)(1+\epsilon/2)A>(1+\epsilon/8)A,$

which is a contradiction. Hence, after passing to a subsequence, we have
$x_{j}\to\bar{x}$ for some $\bar{x}\in\bar{B}_{\bar{s}}(o,A).$ ∎

We are now ready to prove Theorem 6.1.

###### Proof of Theorem 6.1.

We divide the proof into several steps.

Step 1: Construction of the correspondence. For $t\in I$, set
$Z^{i}_{t}:=M_{i}\sqcup M_{\infty}$ and we shall extend the metrics on
$(M_{i},g_{i}(t))$ and $(M_{\infty},g_{\infty}(t))$ to $Z^{i}_{t}.$ For any
$y_{i}\in M_{i},y\in M_{\infty},$ define

${\rm dist}^{Z^{i}_{t}}(y,y_{i})={\rm dist}^{Z^{i}_{t}}(y_{i},y):=\inf_{w\in
U_{i}}{\rm dist}_{g_{\infty,t}}(y,w)+{\rm
dist}_{g_{i,t}}(\Psi_{i}(w),y_{i})+\varepsilon_{i}.$

It is routine to verify that this is indeed a metric, and
$M_{i},M_{\infty}\rightarrow M_{i}\sqcup M_{\infty}$ are isometric embeddings.
By [Bam20b, Lemma 2.13], we may assume that $Z_{t}^{i}$ are isometrically
embedded into a common metric space $Z_{t}$ that is complete and separable.
Let $\phi^{i}_{t}:(M_{i},g_{i}(t))\to Z_{t}$ be the isometric embedding for
$i\in\mathbb{N}\cup\\{\infty\\}.$ Note that for any $x\in U_{i},$

${\rm
dist}^{Z_{t}}(\phi_{t}^{\infty}(x),\phi_{t}^{i}(\Psi_{i}(x)))=\varepsilon_{i}.$

Step 2: Construction of the couplings. Henceforth until the end of the proof
of the theorem, we shall fix an arbitrarily small $\varepsilon>0$ and denote
$E=(b-\varepsilon^{2},b].$ By Lemma 6.3, there is a conjugate heat flow

$d\nu^{\infty}_{s}:=v_{s}\,dg_{\infty,s},$

where $\nu^{\infty}_{s}\in\mathcal{P}(M_{\infty})$ for $s\in[a,b)$, and
$\Psi_{i}^{*}\nu_{s}^{i}\to\nu_{s}^{\infty}$ on $M_{\infty}\times[a,b)$ in the
$C_{c}^{\infty}$-topology as smooth $n$-forms.

Claim 1: For $i=\infty$ or for all $i$ sufficiently large, if $s\in I\setminus
E$ and $r\geq 10\sqrt{|a|}$, then

$\nu^{i}_{s}(M_{i}\setminus B_{g_{i,s}}(z_{i},r))\leq Ce^{-cr^{2}},$ (6.13)

where $c$ and $C$ are constants depending only on the geometry of
$(M_{\infty},g_{\infty}(t))_{t\in I\setminus E})$.

###### Proof of Claim 1..

By the smooth convergence of $\Psi_{i}^{*}K_{i}(p_{i},0\,|\,\cdot,\cdot)$ and
the fact that $v>0$, there is $c_{0}>0,$ such that for any $s\in I\setminus
E$, we have

$\displaystyle\nu_{s}^{i}\left(B_{g_{i,s}}(z_{i},\sqrt{|s|})\right)\geq\tfrac{1}{2}\nu_{s}^{\infty}\left(B_{g_{\infty,s}}(z_{\infty},\sqrt{|s|})\right)\geq
c_{0},$ (6.14)

if $i\geq\bar{i}$ is sufficiently large. For $i\geq\bar{i}$ and $r\geq
10\sqrt{|a|},$ by the Gaussian concentration (2.4) and (6.14), we have

$\displaystyle c_{0}\nu_{s}^{i}\left(M_{i}\setminus
B_{g_{i,s}}(z_{i},r)\right)\leq\nu_{s}^{i}\left(B_{g_{i,s}}(z_{i},\sqrt{|s|})\right)\nu_{s}^{i}\left(M_{i}\setminus
B_{g_{i,s}}(z_{i},r)\right)\leq\exp\left\\{-\frac{(r-\sqrt{|s|})^{2}}{8|s|}\right\\}\leq
e^{-cr^{2}},$

for some $c$ depending on $a$ and $b$. Note that the Gaussian concentration is
also true for $\nu^{\infty}_{s}$ by Fatou’s lemma. Thus, (6.13) also holds for
$i=\infty.$ ∎

For all $s\in I\setminus E$, set
$\Omega_{s}=\bar{B}_{g_{\infty,s}}(z_{\infty},A)$, where $A$ is some large
number to be determined. Let us also denote $\Omega:=\cup_{s\in I\setminus
E}\Omega_{s}$, which is compact by Lemma 6.4. For all $s\in I\setminus E$ and
for $i$ large enough or $i=\infty$, define

$\mu_{s}^{\infty}:=\nu_{s}^{\infty}|_{\Omega_{s}}+\eta_{s}\delta_{z_{\infty}},\quad\mu_{s}^{i}:=\Psi_{i*}(\mu_{s}^{\infty}),$

where the push-forward by $\Psi_{i}$ makes sense because
$\operatorname{spt}\mu_{s}^{\infty}\subset U_{i}$ when $i$ is large enough.

Claim 2: We can fix $A$ large enough, such that for $i=\infty$ or for
$i\in\mathbb{N}$ sufficiently large, we have

$\sup_{s\in I\setminus E}{\rm
dist}_{W_{1}}^{(M_{i},g_{i}(s))}(\nu_{s}^{i},\mu_{s}^{i})<\varepsilon.$

###### Proof of Claim 2..

For any $s\in I\setminus E$ and any $1$-Lipschitz function $\phi$ on
$(M_{\infty},g_{\infty}(s)),$ by (6.13), we have

$\displaystyle\int\phi\,d(\nu_{s}^{\infty}-\mu_{s}^{\infty})$
$\displaystyle=\int(\phi-\phi(z_{\infty}))\,d(\nu_{s}^{\infty}-\mu_{s}^{\infty})\leq\int_{M_{\infty}\setminus\Omega_{s}}{\rm
dist}_{g_{\infty,s}}(z_{\infty},x)\,d\nu_{s}^{\infty}(x)$ $\displaystyle\leq
ACe^{-cA^{2}}+C\int_{A}^{\infty}re^{-cr^{2}}dr<\varepsilon,$

if $A$ is large enough. Here we have used a standard real analysis result
(c.f. [MZ21, Lemma 3.3]). Let $s\in I\setminus E$ and $\phi$ be any
$1$-Lipschitz function on $(M_{i},g_{i}(s)).$ By (6.13), if $A$ is fixed large
enough, then whenever $i$ is sufficiently large (depending on $A$), we have

$\displaystyle\int\phi\,d(\nu_{s}^{i}-\mu_{s}^{i})=\int[\phi-\phi(z_{i})]\,d(\nu_{s}^{i}-\mu_{s}^{i})$
$\displaystyle\leq$ $\displaystyle
A\int_{\Omega_{s}}d(\Psi_{i}^{*}\nu_{s}^{i}-\nu_{s}^{\infty})+\int_{M_{i}\setminus\Psi_{i}(\Omega_{s})}{\rm
dist}_{g_{i,s}}(z_{i},x)\,d\nu^{i}_{s}(x)$ $\displaystyle\leq$ $\displaystyle
2A|\Omega|_{g_{\infty,s}}\Big{\|}K_{i}(p_{i},0\,|\,\Psi_{i}(\cdot),s)-v_{s}^{\infty}\Big{\|}_{C^{0}(\Omega)}+ACe^{-cA^{2}}+C\int_{A}^{\infty}re^{-cr^{2}}dr<\varepsilon.$

Here we have used the smooth convergence (6.10) and the fact
$\Psi_{i}(z_{\infty})=z_{i}$. This finishes the proof of the claim.

∎

Next, we define a sequence of coupling by

$\tilde{q}_{s}^{i}:=({\rm
id},\Psi_{i})_{*}(\mu_{s}^{\infty})\in\Pi(\mu_{s}^{\infty},\mu_{s}^{i}).$

Then, for any $s,t\in I$ with $s<t,$ we have

$\displaystyle\int_{M_{\infty}\times
M_{i}}d^{Z_{s}}_{W_{1}}\left(\phi^{\infty}_{s*}\nu^{\infty}_{x,t\,|\,s},\phi^{i}_{s*}\nu^{i}_{y,t\,|\,s}\right)\,d\tilde{q}_{t}^{i}(x,y)$
$\displaystyle=$
$\displaystyle\int_{\Omega_{t}}d^{Z_{s}}_{W_{1}}\left(\phi^{\infty}_{s*}\nu^{\infty}_{x,t\,|\,s},\phi^{i}_{s*}\nu^{i}_{\Psi_{i}(x),t\,|\,s}\right)\,d\nu^{\infty}_{t}(x)+\eta_{t}\cdot
d^{Z_{s}}_{W_{1}}\left(\phi^{\infty}_{s*}\nu^{\infty}_{z_{\infty},t\,|\,s},\phi^{i}_{s*}\nu^{i}_{z_{i},t\,|\,s}\right).$

Claim 3: There is a large $\bar{i}\in\mathbb{N}$, such that if $i\geq\bar{i}$,
then, for any $s,t\in I\setminus E=[a,b-\varepsilon^{2}]$ with $s<t$ and for
any $x\in\Omega_{t}$, we have

${\rm
dist}^{Z_{s}}_{W_{1}}\left(\phi^{\infty}_{s*}\nu^{\infty}_{x,t\,|\,s},\phi^{i}_{s*}\nu^{i}_{\Psi_{i}(x),t\,|\,s}\right)<\varepsilon.$

###### Proof of Claim 3.

Suppose not. By passing to a subsequence, we may assume that there are
$s_{i},t_{i}\in I\setminus E,s_{i}<t_{i},x_{i}\in\Omega_{t_{i}},$ such that

${\rm
dist}^{Z_{s_{i}}}_{W_{1}}\left(\phi^{\infty}_{s_{i}*}\nu^{\infty}_{x_{i},t_{i}\,|\,s_{i}},\phi^{i}_{s_{i}*}\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}\right)\geq\varepsilon.$
(6.15)

By passing to a further subsequence, we may assume that
$t_{i}\to\bar{t},s_{i}\to\bar{s}\leq\bar{t},x_{i}\to\bar{x}\in\Omega_{\bar{t}}.$

Case A: $\bar{s}=\bar{t}.$ Write $\bar{x}_{i}=\Psi_{i}(\bar{x}).$

$\displaystyle{\rm
dist}^{Z_{s_{i}}}_{W_{1}}\left(\phi^{\infty}_{s_{i}*}\nu^{\infty}_{x_{i},t_{i}\,|\,s_{i}},\phi^{i}_{s_{i}*}\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}\right)$
$\displaystyle\leq$ $\displaystyle\ {\rm
dist}^{Z_{s_{i}}}_{W_{1}}\left(\phi^{\infty}_{s_{i}*}\nu^{\infty}_{x_{i},t_{i}\,|\,s_{i}},\phi^{\infty}_{s_{i}*}\delta_{\bar{x}}\right)+{\rm
dist}^{Z_{s_{i}}}_{W_{1}}\left(\phi^{\infty}_{s_{i}*}\delta_{\bar{x}},\phi^{i}_{s_{i}*}\delta_{\Psi_{i}(\bar{x})}\right)+{\rm
dist}^{Z_{s_{i}}}_{W_{1}}\left(\phi^{i}_{s_{i}*}\delta_{\Psi_{i}(\bar{x})},\phi^{i}_{s_{i}*}\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}\right)$
$\displaystyle\leq$ $\displaystyle\ {\rm
dist}^{g_{\infty,s_{i}}}_{W_{1}}\left(\nu^{\infty}_{x_{i},t_{i}\,|\,s_{i}},\delta_{\bar{x}}\right)+\varepsilon_{i}+{\rm
dist}^{g_{i,s_{i}}}_{W_{1}}\left(\delta_{\bar{x}_{i}},\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}\right)$
$\displaystyle\leq$ $\displaystyle\ {\rm
dist}^{g_{\infty,s_{i}}}_{W_{1}}\left(\nu^{\infty}_{x_{i},t_{i}\,|\,s_{i}},\nu^{\infty}_{\bar{x},t_{i}\,|\,s_{i}}\right)+{\rm
dist}^{g_{\infty,s_{i}}}_{W_{1}}\left(\nu^{\infty}_{\bar{x},t_{i}\,|\,s_{i}},\delta_{\bar{x}}\right)+\varepsilon_{i}$
$\displaystyle\quad\quad+{\rm
dist}^{g_{i,s_{i}}}_{W_{1}}\left(\nu^{i}_{\bar{x}_{i},t_{i}\,|\,s_{i}},\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}\right)+{\rm
dist}^{g_{i,s_{i}}}_{W_{1}}\left(\nu^{\infty}_{\bar{x}_{i},t_{i}\,|\,s_{i}},\delta_{\bar{x}_{i}}\right)$
$\displaystyle\leq$ $\displaystyle\ 2{\rm
dist}_{g_{\infty,t_{i}}}(x_{i},\bar{x})+2\varepsilon_{i}+{\rm
dist}^{g_{\infty,s_{i}}}_{W_{1}}\left(\nu^{\infty}_{\bar{x},t_{i}\,|\,s_{i}},\delta_{\bar{x}}\right)+{\rm
dist}^{g_{i,s_{i}}}_{W_{1}}\left(\nu^{\infty}_{\bar{x}_{i},t_{i}\,|\,s_{i}},\delta_{\bar{x}_{i}}\right)\to
0,$

as $i\to\infty$, which is a contradiction to (6.15). Here we have also applied
(2.15). The last convergence above is due to [Bam20a, Proposition 9.5] which
is but a consequence of Proposition 2.15.

Case B: $\bar{s}<\bar{t}.$ By Theorem B.1, after possibly passing to a
subsequence, we have

$\Psi_{i}^{*}K_{i}(\Psi_{i}(x_{i}),t_{i}\,|\,\cdot,\cdot)\to
K_{\infty}(\bar{x},\bar{t}\,|\,\cdot,\cdot)$

in the $C_{c}^{\infty}$-topology and the convergence is uniform on compact
subsets of $M_{\infty}\times[a,\bar{t}).$ In particular,

$\left\|\Psi_{i}^{*}\nu_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}-\nu^{\infty}_{\bar{x},\bar{t}\,|\,s_{i}}\right\|_{C^{0}(\mathcal{K})}\to
0,$

as $n$-forms for any compact subset $\mathcal{K}\subset M_{\infty}.$ Let
$(z,\bar{s})$ be an $H_{n}$-center of $(\bar{x},\bar{t})$ and let
$B:=B_{g_{\infty,\bar{s}}}(z,10D)$ for some large constant $D$ to be
determined. First choose $D$ large enough so that
$\nu^{\infty}_{\bar{x},\bar{t}\,|\,\bar{s}}(M_{\infty}\setminus
B)<\frac{\varepsilon}{10D}.$ This is possible because of Proposition 2.15.
Then we have

$\displaystyle{\rm
dist}^{Z_{s_{i}}}_{W_{1}}\left(\phi^{\infty}_{s_{i}*}\nu^{\infty}_{x_{i},t_{i}|s_{i}},\phi^{i}_{s*}\nu^{i}_{\Psi_{i}(x_{i}),t_{i}|s_{i}}\right)$
$\displaystyle\leq$ $\displaystyle\ {\rm
dist}^{g_{\infty,s_{i}}}_{W_{1}}\left(\nu^{\infty}_{x_{i},t_{i}|s_{i}},\nu^{\infty}_{\bar{x},\bar{t}|s_{i}}\right)+{\rm
dist}^{Z_{s_{i}}}_{W_{1}}\left(\phi^{\infty}_{s_{i}*}\nu^{\infty}_{\bar{x},\bar{t}|s_{i}},\phi^{i}_{s*}\nu^{i}_{\Psi_{i}(x_{i}),t_{i}|s_{i}}\right).$

The first term above clearly converges to 0. For the second term, we argue in
the same way as Claim 2 above. By the local distance distortion estimates, we
may assume that $B_{i}=B_{g_{\infty,s_{i}}}(z,D)\subset B.$ Consider any
bounded $1$-Lipschitz function $\phi$ defined on $Z_{s_{i}}.$ We may assume
that $\phi(\phi_{s_{i}}^{\infty}(z))=0,$ for otherwise we may replace it with
$\phi-\phi(\phi_{s_{i}}^{\infty}(z))$. Then, we compute

$\displaystyle\int_{M_{\infty}}\phi\circ\phi^{\infty}_{s_{i}}\,d\nu^{\infty}_{\bar{x},\bar{t}\,|\,s_{i}}-\int_{M_{i}}\phi\circ\phi^{i}_{s_{i}}\,d\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}$
$\displaystyle\leq$ $\displaystyle\
\int_{B_{i}}\phi\circ\phi^{\infty}_{s_{i}}\,d\nu^{\infty}_{\bar{x},\bar{t}\,|\,s_{i}}-\int_{\Psi_{i}(B_{i})}\phi\circ\phi^{i}_{s_{i}}\,d\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}$
$\displaystyle\ +CDe^{-cD^{2}}+C\int_{D}^{\infty}se^{-cs^{2}}\,ds,$

where we used the Gaussian concentration as in Claim 1. We can fix $D$ so that
the last line above is less than $\varepsilon/4$ for all $i$ large. Then, we
have

$\displaystyle\int_{B_{i}}\phi\circ\phi^{\infty}_{s_{i}}\,d\nu^{\infty}_{\bar{x},\bar{t}\,|\,s_{i}}-\int_{\Psi_{i}(B_{i})}\phi\circ\phi^{i}_{s_{i}}\,d\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}$
$\displaystyle\leq$ $\displaystyle\
\int_{B_{i}}\left\\{\phi\circ\phi^{\infty}_{s_{i}}-\phi\circ\phi^{i}_{s_{i}}\circ\Psi_{i}\right\\}\,d\nu^{\infty}_{\bar{x},\bar{t}\,|\,s_{i}}+\int_{B_{i}}\phi\circ\phi^{i}_{s_{i}}\circ\Psi_{i}\left\\{d\nu^{\infty}_{\bar{x},\bar{t}\,|\,s_{i}}-\Psi_{i}^{*}d\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}\right\\}$
$\displaystyle\leq$ $\displaystyle\
\varepsilon_{i}+2D\left\|\Psi_{i}^{*}\nu_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}-\nu^{\infty}_{\bar{x},\bar{t}\,|\,s_{i}}\right\|_{C^{0}(B)}.$

Note that the last line does not depend on $\phi$ and converges to $0$ since
$B$ is compact. Hence, we have

${\rm
dist}^{Z_{s_{i}}}_{W_{1}}\left(\phi^{\infty}_{s_{i}*}\nu^{\infty}_{x_{i},t_{i}\,|\,s_{i}},\phi^{i}_{s_{i}*}\nu^{i}_{\Psi_{i}(x_{i}),t_{i}\,|\,s_{i}}\right)<\varepsilon/2,$

when $i$ is large enough; this is a contradiction agains (6.15). ∎

By the definition of the $1$-Wassernstein distance, there are couplings
$\theta^{i}_{s}\in\Pi(\mu_{s}^{i},\nu_{s}^{i})$ such that

$\int_{M_{i}\times M_{i}}{\rm dist}_{g_{i,s}}(x,y)\,d\theta^{i}_{s}(x,y)<{\rm
dist}_{W_{1}}^{(M_{i},g_{i,s})}(\nu_{s}^{i},\mu_{s}^{i})+\varepsilon<2\varepsilon,$

if $i\geq\bar{i}.$ Applying [Bam20b, Lemma 2.2] for three times, there is
$Q_{s}^{i}\in\mathcal{P}(M_{i}\times M_{i}\times M_{\infty}\times M_{\infty})$
such that the marginal into the first and second factors equals
$\theta^{i}_{s},$ the marginal into the third and first factors equals
$\tilde{q}^{i}_{s},$ and the marginal into the third and fourth factors equals
$\theta^{\infty}_{s}.$ Define $q_{s}^{i}$ to be the marginal of $Q_{s}^{i}$
into the second and fourth factors. Then
$q_{s}^{i}\in\Pi(\nu_{s}^{i},\nu_{s}^{\infty}).$

Step 4: Final verification. For any $s,t\in I\setminus
E=[a,b-\varepsilon^{2}],s\leq t,$ we have

$\displaystyle\int_{M_{\infty}\times
M_{i}}d^{Z_{s}}_{W_{1}}\left(\phi^{\infty}_{s*}\nu^{\infty}_{x,t\,|\,s},\phi^{i}_{s*}\nu^{i}_{y,t\,|\,s}\right)\,dq_{t}^{i}(x,y)$
$\displaystyle=$ $\displaystyle\ \int_{M_{i}\times M_{i}\times
M_{\infty}\times
M_{\infty}}d^{Z_{s}}_{W_{1}}\left(\phi^{\infty}_{s*}\nu^{\infty}_{x,t\,|\,s},\phi^{i}_{s*}\nu^{i}_{y,t\,|\,s}\right)\,dQ_{t}^{i}(y,y_{1},x,x_{1})$
$\displaystyle\leq$ $\displaystyle\ \int\left\\{{\rm
dist}_{g_{\infty,t}}(x,x_{1})+{\rm
dist}_{g_{i,t}}(y,y_{1})+d^{Z_{s}}_{W_{1}}\left(\phi^{\infty}_{s*}\nu^{\infty}_{x_{1},t\,|\,s},\phi^{i}_{s*}\nu^{i}_{y_{1},t\,|\,s}\right)\right\\}\,dQ_{t}^{i}(y,y_{1},x,x_{1})$
$\displaystyle=$ $\displaystyle\ \int_{M_{\infty}\times M_{\infty}}{\rm
dist}_{g_{\infty,t}}\,d\theta^{\infty}_{t}+\int_{M_{i}\times M_{i}}{\rm
dist}_{g_{i,t}}\,d\theta^{i}_{t}+\int_{M_{\infty}\times
M_{i}}d^{Z_{s}}_{W_{1}}\left(\phi^{\infty}_{s*}\nu^{\infty}_{x,t\,|\,s},\phi^{i}_{s*}\nu^{i}_{y,t\,|\,s}\right)\,d\tilde{q}_{t}^{i}(x,y)$
$\displaystyle<$ $\displaystyle\ 10\varepsilon,$

if $i\geq\bar{i},$ where we have also used the monotonicity formula (2.15).

∎

As a special case of this Theorem, under the assumptions of Theorem 1.1, we
have, after passing to a subsequence,

$\displaystyle\Big{(}(M,g_{i}(t))_{t\in[-2,-1]},(\nu^{i}_{s})_{s\in[-2,-1]}\Big{)}\xrightarrow{\makebox[28.45274pt]{$\mathbb{F}$}}\Big{(}(M_{\infty},g_{\infty}(t))_{t\in[-2,-1]},(\nu_{s}^{\infty})_{s\in[-2,-1)}\Big{)}.$

To finish the proof of Theorem 1.1, we only need to show that any
$\nu_{s}^{\infty}$ in the proof of Theorem 6.1 is induced by the shrinker
potential function. We defer the proof of this fact to Proposition 7.6 in the
next section.

##  7 Conergence of the Nash entropy and Smooth tangent flow at infinity

In this section, we prove Theorem 1.1(2)(3). To this end, we shall show that
the conjugate heat kernel based at $(p_{0},0)$, after scaling, converges to
the shrinker potential of the asymptotic shrinker. Throughout this section, we
still consider an ancient solution $(M,g(t))_{t\in(-\infty,0]}$ satisfying
Assumption B with respect to
$\big{(}p_{0},0,\\{\tau_{i}\\}_{i=1}^{\infty},\\{p_{i}\\}_{i=1}^{\infty}\big{)}$.
Let $(M_{\infty},g_{\infty}(t))_{t\in[-2,-1]}$ be the limit in (1.2), which is
known to be Perelman’s asymptotic shrinker by Prposition 5.2. Recall that at
this moment we only have a Gaussian upper bound for
$K(p_{0},0\,|\,\cdot,\cdot)$. The lack of a Gaussian lower bound is because of
the fact that we currently do not have a good estimate for the reduced
distance, which, however, is not needed now. For the sake of convenience, we
define the following notations for the scaled conjugate heat kernel

$\displaystyle
u_{i}(x,t):=\tau_{i}^{\frac{n}{2}}K(p_{0},0\,|\,x,\tau_{i}t)=\frac{1}{(4\pi|t|)^{\frac{n}{2}}}e^{-f_{i}(x,t)}$
(7.1) $\displaystyle
d\nu^{i}_{t}:=K(p_{0},0\,|\,\cdot,\tau_{i}t)dg_{\tau_{i}t}=u_{i}(\cdot,t)dg_{i,t}.$

######  Lemma 7.1.

There is a smooth function
$u_{\infty}:M_{\infty}\times[-2,-1)\rightarrow\mathbb{R}$, satisfying

  1. (1)

$u_{i}\rightarrow u_{\infty}$ locally smoothly on $M_{\infty}\times[-2,-1)$,
where $u_{i}$ should be understood to be pulled back by the defining
diffeomorphisms in the Cheeger-Gromov-Hamilton convergence.

  2. (2)

$u_{\infty}$ is a solution to the conjugate heat equation coupled with
$g_{\infty}(t)$.

  3. (3)

$u_{\infty}>0$ everywhere on $M_{\infty}\times[-2,-1)$.

  4. (4)

$\displaystyle\int_{M_{\infty}}u_{\infty}(\cdot,t)dg_{\infty,t}\equiv
1\quad\text{ for all }\quad t\in[-2,-1).$

###### Proof.

In view of Corollary 5.7, this lemma is but a restatement of Lemma 6.3. ∎

Since $u_{\infty}>0$ everywhere, we may define the function $f_{\infty}$ by

$\displaystyle
u_{\infty}(x,t):=(4\pi|t|)^{-\frac{n}{2}}e^{-f_{\infty}(x,t)}\quad\text{ for
all }\quad(x,t)\in M_{\infty}\times[-2,-1).$

Then, we obviously have

$f_{i}\rightarrow f_{\infty}\quad\text{ locally smoothly on }\quad
M_{\infty}\times[-2,-1).$

######  Lemma 7.2.

For any $\varepsilon\in(0,1)$, there exists a constant $C$ independent of $i$,
such that the following holds for each $i\in\mathbb{N}\cup\\{\infty\\}$.

$\displaystyle f_{i}(x,t)\geq\frac{1}{C}{\rm
dist}_{g_{i,t}}^{2}(x,z_{i})-C\quad\text{ for all }\quad(x,t)\in
M\times[-2,-(1+\varepsilon)].$

###### Proof.

Fixing an $\varepsilon\in(0,1)$, we have that
$\cup_{t\in[-2,-1-\varepsilon]}B_{g_{\infty,t}}(z_{\infty},1)\times\\{t\\}$ is
a precompact set in $M_{\infty}\times[-2,-1)$, and hence $u_{i}\rightarrow
u_{\infty}$ uniformly on this set. Consequently we have

$\displaystyle\liminf_{i\rightarrow\infty}\int_{B_{g_{i,t}}(z_{i},1)}u_{i}(\cdot,t)dg_{i,t}$
$\displaystyle=$
$\displaystyle\int_{B_{g_{\infty,t}}(z_{\infty},1)}u_{\infty}(\cdot,t)dg_{\infty,t}$
$\displaystyle\geq$
$\displaystyle\inf_{s\in[-2,-1-\varepsilon]}\int_{B_{g_{\infty,s}}(z_{\infty},1)}u_{\infty}(\cdot,s)dg_{\infty,s}>0,$

for all $t\in[-2,-1-\varepsilon]$. Hence, we can find a positive number
$c_{0}>0$, such that

$\displaystyle\int_{B_{g_{i,t}}(z_{i},1)}u_{i}(\cdot,t)dg_{i,t}\geq
c_{0}\quad\text{ for all }\quad t\in[-2,-1-\varepsilon]\quad\text{ and for all
}\quad i\in\mathbb{N}.$ (7.2)

Let $t\in[-2,-1-\varepsilon]$ and let $(z^{\prime}_{i},t)$ be an
$H_{n}$-center of $(p_{0},0)$ with respect to the scaled Ricci flow
$(M,g_{i}(t))$. By Proposition 2.15 and (7.2), we have that

$\displaystyle{\rm
dist}_{g_{i,t}}(z_{i},z^{\prime}_{i})\leq\sqrt{\frac{2}{c_{0}}H_{n}|t|}+1\leq
C\quad\text{ for all }\quad t\in[-2,-1-\varepsilon]\quad\text{ and for all
}\quad i\in\mathbb{N}.$ (7.3)

The case when $i\in\mathbb{N}$ follows from Proposition 5.5 and (7.3). If
$i=\infty$, then this case follows from the fact that $f_{\infty}$ is the
local smooth limit of $\\{f_{i}\\}_{i=1}^{\infty}$. ∎

######  Lemma 7.3.

There is a constant $C$ independent of $i$, such that the following hold for
all $t\in[-2,-1)$ and $i\in\mathbb{N}\cup\\{\infty\\}$.

$\displaystyle\int_{M_{i}}f_{i}^{2}(\cdot,t)u_{i}(\cdot,t)dg_{i,t}\leq C.$
(7.4)

Here we have let $M_{i}\equiv M$ for all $i\in\mathbb{N}$.

###### Proof.

By (3.4) and the Nash entropy bound in Theorem 5.4, we have that (7.4) hold
for $i\in\mathbb{N}$. The case when $i=\infty$ then follows from Fatou’s lemma
and the locally smooth convergences of $f_{i}$ and $u_{i}$. ∎

######  Proposition 7.4.

The following statements are true.

  1. (1)

$\displaystyle\lim_{i\rightarrow\infty}\mathcal{N}_{i}(|t|)=\int_{M_{\infty}}f_{\infty}(\cdot,t)u_{\infty}(\cdot,t)dg_{\infty,t}-\frac{n}{2}$
for all $t\in[-2,-1)$.

  2. (2)

$N_{\infty}:=\displaystyle\int_{M_{\infty}}f_{\infty}(\cdot,t)u_{\infty}(\cdot,t)dg_{\infty,t}-\frac{n}{2}$
is a constant in $t\in[-2,-1)$.

Here $\mathcal{N}_{i}(\tau):=\mathcal{N}_{p_{0},0}(\tau_{i}\tau)$ stands for
the Nash entropy of the scaled Ricci flow $(M,g_{i}(t))$ based at $(p_{0},0)$.

###### Proof.

We shall only prove part (1), since part (2) obviously follows from part (1)
and the monotonicity of the Nash entropy. Let us fix an arbitrary
$t\in[-2,-1)$. Applying Lemma 7.2 and (7.4), we may find a positive number
$C$, such that the following holds for all $A>C$.

$\displaystyle C\geq\int_{M}f_{i}^{2}u_{i}dg_{i,t}\geq\int_{M\setminus
B_{g_{i,t}}(z_{i},A)}f_{i}^{2}u_{i}dg_{i,t}\geq\frac{A^{2}}{2C}\int_{M\setminus
B_{g_{i,t}}(z_{i},A)}f_{i}u_{i}dg_{i,t}.$

Hence

$\displaystyle 0<\int_{M\setminus
B_{g_{i,t}}(z_{i},A)}f_{i}u_{i}\,dg_{i,t}\leq\frac{C}{A^{2}}\quad\text{ for
all }\quad A>C.$ (7.5)

Since $f_{i}\rightarrow f_{\infty}$ locally smoothly, the following holds for
all $A>0$.

$\displaystyle\int_{B_{g_{\infty,t}}(z_{i},A)}f_{\infty}u_{\infty}dg_{\infty,t}$
$\displaystyle=$
$\displaystyle\lim_{i\rightarrow\infty}\int_{B_{g_{i,t}}(z_{i},A)}f_{i}u_{i}dg_{i,t}$
$\displaystyle=$
$\displaystyle\lim_{i\rightarrow\infty}\left(\int_{M}f_{i}u_{i}dg_{i,t}-\int_{M\setminus
B_{g_{i,t}}(z_{i},A)}f_{i}u_{i}dg_{i,t}\right)$ $\displaystyle=$
$\displaystyle\lim_{i\rightarrow\infty}\left(\mathcal{N}_{i}(|t|)+\tfrac{n}{2}+O(A^{-2})\right).$

Here we have applied (7.5). Note that the constant $C$ in formula (7.5) is
independent of both $i$ and $A$. Finally, taking $i\rightarrow\infty$ first
and then $A\rightarrow\infty$, the conclusion follows. ∎

Therefore, if we use $u_{\infty}$ to construct a Nash entropy, then what we
obtain is a constant—the critical point of Perelman’s monotonicity formula.
This should imply that $f_{\infty}$ is a shrinker potential. However, since we
do not have any geometric condition on $(M_{\infty},g_{\infty}(t))$ except for
the fact that it is a shrinker, and since Perelman’s monotonicity formula
depends heavily on the integration by parts at infinity, we cannot so easily
conclude that $f_{\infty}$ is a shrinker potential. One way to resolve this is
to apply the cut-off function constructed by [LW20, Lemma 3]. We shall use an
alternative method. Recall the following result of Bamler.

######  Proposition 7.5 (Proposition 6.1 in [Bam20c]).

For any $Y<\infty$ and $\varepsilon>0$, there is a
$\bar{\delta}(Y,\varepsilon)>0$, such that the following holds whenever
$\delta\in(0,\bar{\delta})$. Let $(M,g(t))_{t\in I}$ be a complete Ricci flow
with bounded curvature within each compact time interval. Let
$(x_{0},t_{0})\in M\times I$ and $r>0$ be such that
$[t_{0}-\delta^{-1}r^{2},t_{0}-\delta r^{2}]\subset I$. Suppose
$\mathcal{N}_{x_{0},t_{0}}(r^{2})\geq-Y$. If

$\mathcal{N}_{x_{0},t_{0}}(\delta^{-1}r^{2})\geq\mathcal{N}_{x_{0},t_{0}}(\delta
r^{2})-\delta,$

then we have

$\int_{t_{0}-\varepsilon^{-1}r^{2}}^{t_{0}-\varepsilon
r^{2}}\int_{M}\tau\left|\,\operatorname{Ric}+\nabla^{2}f-\frac{1}{2\tau}g\,\right|^{2}d\nu_{x_{0},t_{0}\,|\,t}dg_{t}dt\leq\varepsilon,$

where $\tau(t):=t_{0}-t$ and
$d\nu_{x_{0},t_{0}\,|\,t}:=(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg_{t}$ is the
conjugate heat kernel based at $(x_{0},t_{0})$.

###### Proof.

In fact, in the proof of [Bam20c, Proposition 6.1], the conclusion stated in
the current proposition relies only on Perelman’s monotonicity formulas for
entropies, which are obviously true for Ricci flows with bounded curvature.
Hence, Bamler’s proof can be directly applied to our case. ∎

We are then ready to show that $f_{\infty}$ is a shrinker potential.

######  Proposition 7.6.

$f_{\infty}$ is a shrinker potential function, satisfying

$\displaystyle\operatorname{Ric}_{\infty}+\nabla^{2}f_{\infty}=\frac{1}{2|t|}g_{\infty}\quad\text{
for all }\quad t\in[-2,-1).$

###### Proof.

By Theorem 5.4, we have that, for any $\delta>0$, it holds that

$\displaystyle\lim_{i\rightarrow\infty}\mathcal{N}_{p_{0},0}(\delta^{-1}\tau_{i})-\mathcal{N}_{p_{0},0}(\delta\tau_{i})=0.$

Hence, Proposition 7.5 implies the existence of a sequence
$\varepsilon_{i}\searrow 0$, such that

$\int_{-\varepsilon_{i}^{-1}\tau_{i}}^{-\varepsilon_{i}\tau_{i}}\int_{M}|t|\left|\,\operatorname{Ric}+\nabla^{2}f-\frac{1}{2|t|}g\,\right|^{2}d\nu_{p_{0},0\,|\,t}dg_{t}dt\leq\varepsilon_{i},$

where $d\nu_{p_{0},0\,|\,t}=(4\pi|t|)^{-\frac{n}{2}}e^{-f}dg_{t}$ is the
conjugate heat kernel based at $(p_{0},0)$. Scaling by $\tau_{i}$, we
immediately have

$\displaystyle\quad\int_{-2}^{-1}\int_{M}|t|\left|\,\operatorname{Ric}_{g_{i}}+\nabla^{2}f_{i}-\frac{1}{2|t|}g_{i}\,\right|^{2}u_{i}dg_{i,t}dt$
(7.6)
$\displaystyle\leq\int_{-\varepsilon_{i}^{-1}}^{-\varepsilon_{i}}\int_{M}|t|\left|\,\operatorname{Ric}_{g_{i}}+\nabla^{2}f_{i}-\frac{1}{2|t|}g_{i}\,\right|^{2}u_{i}dg_{i,t}dt$
$\displaystyle\leq\varepsilon_{i}\rightarrow 0,$

where $g_{i}(t):=\tau_{i}^{-1}g(\tau_{i}t)$, $f_{i}$ and $u_{i}$ are defined
in (7.1). Because of the local smooth convergence of $g_{i}$, $f_{i}$, and
$u_{i}$, taking a limit for (7.6) and applying Fatou’s lemma, we have

$\int_{-2}^{-1}\int_{M_{\infty}}|t|\left|\,\operatorname{Ric}_{g_{\infty}}+\nabla^{2}f_{\infty}-\frac{1}{2|t|}g_{\infty}\,\right|^{2}u_{\infty}dg_{\infty,t}dt=0.$

Since $u_{\infty}>0$ everywhere, the Proposition follows immediately. ∎

###### Proof of Theorem 1.1(2).

To prove Theorem 1.1(2), we need only to show $N_{\infty}=\mu_{\infty}$, where
$N_{\infty}$ is defined in Proposition 7.4(2) and $\mu_{\infty}$ is the
entropy of the asymptotic shrinker. Because of Lemma 7.1(4), by (2.13) we have

$\displaystyle N_{\infty}$ $\displaystyle=$
$\displaystyle\int_{M_{\infty}}f_{\infty}u_{\infty}dg_{\infty,t}-\frac{n}{2}$
$\displaystyle=$ $\displaystyle\int_{M_{\infty}}|t|(|\nabla
f_{\infty}|^{2}+R_{\infty})u_{\infty}dg_{\infty,t}-\frac{n}{2}+\mu_{\infty}$
$\displaystyle=$ $\displaystyle\int_{M_{\infty}}|t|(\Delta
f_{\infty}+R_{\infty})u_{\infty}dg_{\infty,t}-\frac{n}{2}+\mu_{\infty}$
$\displaystyle=$
$\displaystyle\int_{M_{\infty}}|t|\frac{n}{2|t|}u_{\infty}dg_{\infty,t}-\frac{n}{2}+\mu_{\infty}$
$\displaystyle=$ $\displaystyle\mu_{\infty}.$

Here we have applied integration by parts at infinity. This is valid, since
both $f_{\infty}$ and $|\nabla f_{\infty}|^{2}$ have quadratic growth bounds
([CZ10]), and since a Ricci shrinker has at most Euclidean volume growth
([CZ10, Mun09]); a standard cut-off argument easily verifies the integration
by parts at infinity. ∎

###### Proof of Theorem 1.1(3).

Since any Ricci flow with bounded curvature within each compact time interval
is $H_{n}$-concentrated, we have

$\displaystyle\big{(}(M,g_{i}(t))_{t\in(-\infty,0]},(\nu^{i}_{s})_{s\in(-\infty,0]}\big{)}\xrightarrow{\makebox[28.45274pt]{$\mathbb{F}$}}\mathcal{X},$

where $d\nu_{s}^{i}=u_{i}(\cdot,s)dg_{i,s}$ and $\mathcal{X}$ is a metric flow
pair over $(-\infty,0]$ (c.f. [Bam20b, Theorem 7.8]). Here the
$\mathbb{F}$-convergence should be understood to be the
$\mathbb{F}$-convergence over each finite subinterval of $(-\infty,0]$.

By Corollary 5.7, we have

$\displaystyle\big{(}M,g_{i}(t),z_{i}\big{)}_{t\in[-2,-1]}\xrightarrow{\makebox[28.45274pt]{}}\big{(}M_{\infty},g_{\infty}(t),z_{\infty}\big{)}_{t\in[-2,-1]}.$

Therefore, Theorem 6.1 implies that

$\displaystyle\mathcal{X}_{[-2,-1)}=\big{(}(M_{\infty},g_{\infty}(t))_{t\in[-2,-1)},(\nu^{\infty}_{s})_{s\in[-2,-1)}\big{)},$

where $\nu_{s}^{\infty}:=u_{\infty}(\cdot,s)dg_{s}$ is a conjugate heat flow
on $(M_{\infty},g_{\infty}(t))$.

Finally, the proof of Theorem 6.1 indicates that $u_{i}\rightarrow u_{\infty}$
locally smoothly, where $u_{i}$ is defined in (7.1). Hence, by Proposition
7.6, $u_{\infty}=(4\pi|t|)^{-\frac{n}{2}}e^{-f_{\infty}}$ is indeed a
conjugate heat flow made of a shrinker potential function; this proves Theorem
1.1(3). ∎

##  8 Independence of the base point

In this section, we prove Corollary 1.2. Recall that in the statement of
Assumption B, the choices of the base point $(p_{0},0)$ and the sequence of
$\ell$-centers $\\{(p_{i},-\tau_{i})\\}_{i=1}^{\infty}$ are involved. If an
ancient solution satisfies Assumption B with respect to some certain base
point $(p_{0},0)$ and some certain sequence of scales
$\\{\tau_{i}\\}_{i=1}^{\infty}$, it is not immediately clear whether
Assumption B is still valid if we alter the base point and the sequence of
scales. And even if it were, it is not yet clear whether the limit is the same
as before. Although the choices of the $\ell$-centers $p_{i}$ are not unique
either, yet, according to what we have developed by far, this fact does not
affect the validity of Assumption B or the limit flow in (1.2). What we shall
prove next is that for an ancient solution satisfying Assumption B, we may
freely alter the base point, while the asymptotic shrinker is still the same.
(But we may not change the sequence of scales $\\{\tau_{i}\\}_{i=1}^{\infty}$
at will.) This leads to the proof of Corollary 1.2.

######  Theorem 8.1.

Let $(M,g(t))_{t\in(-\infty,0]}$ be an ancient solution satisfying (1.1) and
Assumption B with respect to
$\big{(}p_{0},0,\\{\tau_{i}\\}_{i=1}^{\infty},\\{p_{i}\\}_{i=1}^{\infty}\big{)}$.
Then, for any $(p^{\prime}_{0},t^{\prime}_{0})\in M\times(-\infty,0]$, the
following holds.

$\displaystyle\big{(}M,g^{\prime}_{i}(t),p^{\prime}_{i}\big{)}_{t\in[-2,-1]}\xrightarrow{\makebox[28.45274pt]{}}\big{(}M_{\infty},g_{\infty}(t),p^{\prime}_{\infty}\big{)}_{t\in[-2,-1]},$

where $(p^{\prime}_{i},-\tau_{i})$ is an $\ell$-center of
$(p^{\prime}_{0},t^{\prime}_{0})$ with respect to the Ricci flow $(M,g(t))$,
$g^{\prime}_{i}(t):=\tau^{\prime-1}_{i}g(t^{\prime}_{0}+\tau^{\prime}_{i}t)$,
$\tau^{\prime}_{i}:=\tau_{i}+t^{\prime}_{0}$ for all $i$ large, and
$(M_{\infty},g_{\infty}(t))$ is the same limit as in (1.2).

Remark: In fact, if the existing interval of $(M,g(t))$ extends beyond $t=0$,
then $t_{0}^{\prime}$ can be taken to be greater than $0$, so long as the
curvature remains to be bounded up to $t^{\prime}_{0}$. This can be easily
observed from the proof of this theorem.

###### Proof.

Since $\displaystyle
g^{\prime}_{i}(t)=\frac{\tau_{i}}{\tau^{\prime}_{i}}g_{i}\left(\frac{\tau^{\prime}_{i}}{\tau_{i}}t+\tau_{i}^{-1}t^{\prime}_{0}\right)$,
where $g_{i}(t):=\tau_{i}^{-1}g_{i}(\tau_{i}t)$,

$\tau_{i}^{-1}t^{\prime}_{0}\rightarrow 0\quad\text{ and
}\quad\frac{\tau_{i}}{\tau^{\prime}_{i}}\rightarrow 1$

as $i\rightarrow\infty$. It is then clear that we need only to show the
equivalence of the base points, that is

$\displaystyle\limsup_{i\rightarrow\infty}{\rm
dist}_{g^{\prime}_{i,-1}}(p_{i},p^{\prime}_{i})<\infty.$ (8.1)

Obviously, our concern is only in the case when $i$ is large. Hence,
throughout this proof, we assume

$\displaystyle\tau_{i}\gg|t^{\prime}_{0}|\quad\text{ and
}\quad\left|\frac{\tau_{i}}{\tau^{\prime}_{i}}-1\right|\ll 1.$

Let $(z^{\prime}_{i},-\tau_{i})$ be an $H_{n}$-center of
$(p^{\prime}_{0},t^{\prime}_{0})$, and $(z_{i},-\tau_{i})$ and $H_{n}$-center
of $(p_{0},0)$. By [MZ21, Proposition 4.6] and Theorem 5.4, we have

$\displaystyle\lim_{\tau\rightarrow\infty}\mathcal{N}_{p^{\prime}_{0},t^{\prime}_{0}}(\tau)=\lim_{\tau\rightarrow\infty}\mathcal{N}_{p_{0},0}(\tau)\geq-Y>-\infty.$

It then follows from Proposition 5.6 that

$\displaystyle{\rm dist}_{g_{i,-1}}(p_{i},z_{i}),\ {\rm
dist}_{g^{\prime}_{i,-1}}(p^{\prime}_{i},z^{\prime}_{i})\leq C,$

where $C$ is independent of $i$. Hence, to prove (8.1), it suffices to show
that

$\displaystyle{\rm dist}_{g_{i,-1}}(z_{i},z^{\prime}_{i}),\ {\rm
dist}_{g^{\prime}_{i,-1}}(z_{i},z^{\prime}_{i})\leq C,$

or equivalently,

$\displaystyle{\rm dist}_{g_{-\tau_{i}}}(z_{i},z^{\prime}_{i})\leq
C\sqrt{\tau_{i}},$ (8.2)

for some constant $C$ independent of $i$.

Let us prove (8.2) by contradiction. Assume that there are positive numbers
$A_{i}\nearrow\infty$ such that

$\displaystyle{\rm dist}_{g_{-\tau_{i}}}(z_{i},z_{i}^{\prime})\geq
10\sqrt{A_{i}H_{n}\tau_{i}}.$

Then, whenever $i$ is large enough, we may find a nonnegative $1$-Lipschitz
function $\varphi_{i}$, satisfying the following properties:

  1. (1)

$\varphi_{i}$ is compactly supported on
$B_{g_{-\tau_{i}}}(z_{i},2\sqrt{A_{i}H_{n}\tau_{i}})$,

  2. (2)

$\varphi_{i}\geq\sqrt{A_{i}H_{i}\tau_{i}}$ on
$B_{g_{-\tau_{i}}}(z_{i},\sqrt{A_{i}H_{n}\tau_{i}})$,

  3. (3)

$0\leq\varphi_{i}\leq 2\sqrt{A_{i}H_{n}\tau_{i}}$ everywhere,

  4. (4)

the support of $\varphi_{i}$ is disjoint from
$B_{g_{-\tau_{i}}}(z^{\prime}_{i},\sqrt{A_{i}H_{n}\tau^{\prime}_{i}})$.

Using $\varphi_{i}$ as a test function, we may compute

$\displaystyle\quad\quad{\rm
dist}_{W_{1}}^{g_{-\tau_{i}}}(\nu_{p_{0},0\,|\,-\tau_{i}},\nu_{p^{\prime}_{0},t^{\prime}_{0}\,|\,-\tau_{i}})$
$\displaystyle\geq\int_{M}\varphi_{i}d\nu_{p_{0},0\,|\,-\tau_{i}}-\int_{M}\varphi_{i}d\nu_{p^{\prime}_{0},t^{\prime}_{0}\,|\,-\tau_{i}}$
$\displaystyle\geq\int_{B_{g_{-\tau_{i}}}(z_{i},\sqrt{A_{i}H_{n}\tau_{i}})}\varphi_{i}d\nu_{p_{0},0\,|\,-\tau_{i}}-\int_{M\setminus
B_{g_{-\tau_{i}}}(z^{\prime}_{i},\sqrt{A_{i}H_{n}\tau^{\prime}_{i}})}\varphi_{i}d\nu_{p^{\prime}_{0},t^{\prime}_{0}\,|\,-\tau_{i}}$
$\displaystyle\geq\left(1-\frac{1}{A_{i}}\right)\sqrt{A_{i}H_{n}\tau_{i}}-\frac{1}{A_{i}}2\sqrt{A_{i}H_{n}\tau_{i}}$
$\displaystyle=\left(1-\frac{3}{A_{i}}\right)\sqrt{A_{i}H_{n}\tau_{i}}\rightarrow\infty,$

as $i\rightarrow\infty$; here we have applied Proposition 2.15. On the other
hand, by (2.15), we have

$\displaystyle
d_{W_{1}}^{g_{-\tau_{i}}}(\nu_{p_{0},0\,|\,-\tau_{i}},\nu_{p^{\prime}_{0},t^{\prime}_{0}\,|\,-\tau_{i}})\leq
d_{W_{1}}^{g_{t^{\prime}_{0}}}(\nu_{p_{0},0\,|\,t^{\prime}_{0}},\delta_{p^{\prime}_{0}})<\infty.$

This is a contradiction, and (8.2) follows; we have finished the proof of the
theorem. ∎

We are now ready to prove Corollary 1.2. As before, we still consider an
ancient solution $(M,g(t))_{t\in(-\infty,0]}$ satisfying (1.1) and Assumption
B with respect to
$\big{(}p_{0},0,\\{\tau_{i}\\}_{i=1}^{\infty},\\{p_{i}\\}_{i=1}^{\infty}\big{)}$.
We first of all need to deal with the case when the base point is fixed at
$(p_{0},0)$.

######  Lemma 8.2.

Let $\ell_{\infty}:M_{\infty}\times(1,2]\rightarrow\mathbb{R}$ be the shrinker
potential of $(M_{\infty},g_{\infty}(t))_{t\in[-2,-1)}$ as given by
Proposition 5.2. Then we have

$\displaystyle
V_{\infty}:=\int_{M_{\infty}}(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_{\infty}(\cdot,|t|)}dg_{\infty,t}\equiv\lim_{\tau\rightarrow\infty}\mathcal{V}_{p_{0},0}(\tau)$

for all $t\in[-2,-1)$.

###### Proof.

Let $t\in[-2,-1)$ be arbitrarily fixed. We need only to show

$\displaystyle\int_{M_{\infty}}(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_{\infty}(\cdot,|t|)}dg_{\infty,t}=\lim_{i\rightarrow\infty}\int_{M}(4\pi|t|)^{-\frac{n}{2}}e^{-\ell_{i}(\cdot,|t|)}dg_{i,t},$

where, as before, $g_{i}(t)=\tau_{i}^{-1}g(\tau_{i}t)$ and
$\ell_{i}(\cdot,\tau)=\ell_{p_{0},0}(\cdot,\tau_{i}\tau)$.