# On noncollapsed $\mathbb{F}$-limit metric solitons

Pak-Yeung Chan111Pak-Yeung Chan’s research is supported by EPSRC grant
EP/T019824/1., Zilu Ma, Yongjia Zhang222Yongjia Zhang’s research is supported
by National Natural Science Foundation of China NSFC12301076 and Shanghai
Sailing Program 23YF1420400.

###### Abstract

A noncollapsed $\mathbb{F}$-limit metric soliton is a self-similar singularity
model that inevitably arises when studying the Ricci flow with the tool of
$\mathbb{F}$-convergence [Bam20a, Bam20b, Bam20c]. In this article, we shall
present a systematic study of the noncollapsed $\mathbb{F}$-limit metric
soliton, and show that, apart from the known results in [Bam20c], it satisfies
many properties of smooth Ricci shrinkers. In particular, we show a quadratic
lower bound for the scalar curvature, a local gap theorem, a global Sobolev
inequality, and an optimal volume growth lower bound.

###### Contents

  1. 1 Introduction
    1. 1.1 Basic properties of the potential function and the volume
    2. 1.2 A lower bound for the scalar curvature
    3. 1.3 A local gap theorem
    4. 1.4 Global $\nu$-functional and Sobolev inequality
    5. 1.5 Volume growth lower estimate
  2. 2 Metric soliton as an $\mathbb{F}$-limit
  3. 3 Metric soliton as an ancient $\mathbb{F}$-limit
  4. 4 Center and the potential function
    1. 4.1 Center of metric soliton
    2. 4.2 Estimates of the potential function
    3. 4.3 Scalar curvature near a center
  5. 5 A lower bound for the scalar curvature
    1. 5.1 Local lower bound of the scalar curvature near $H_{n}$-centers
    2. 5.2 Proof of the scalar curvature lower bound
  6. 6 A local gap theorem
    1. 6.1 Perelman’s functionals
    2. 6.2 A preliminary soliton entropy gap
    3. 6.3 Construction of the cut-off functions
    4. 6.4 Proof of the local gap theorem
  7. 7 Global $\nu$-functional and Sobolev inequality
  8. 8 Volume growth lower estimate
    1. 8.1 Preliminary volume decay lower bound
    2. 8.2 Differential equation of the volume and the scalar curvature
    3. 8.3 Noncompact metric soliton has infinite volume
    4. 8.4 Noncompact metric soliton has at least linear volume growth

##  1 Introduction

In the development of the Ricci flow, especially in the proof of the
geometrization conjecture via the Hamilton-Perelman program [Ham93, Per02,
Per03a, Per03b], singularity analysis is a critical problem. The efficiency of
the singularity analysis, to certain degree, determines the potential of the
Ricci flow in its geometric application. In Hamilton’s classical singularity
analysis [Ham93], strong local a priori estimates (especially the “bounded
curvature at bounded distance condition”) are essential to the existence of
singularity models. In other words, one cannot freely detect the local
geometry at arbitrary regions; only the geometric information at the highest-
curvature region is available. Although Perelman improved Hamilton’s method
and overcame this difficulty in the three-dimensional case, the singularity
analysis in general higher dimensional cases remains a major issue in the
field of Ricci flow.

Bamler’s breakthrough [Bam20a, Bam20b, Bam20c] provides some remedy to the
aforementioned problem. By developing sophisticated estimates for the heat
kernel and the entropy on the Ricci flow [Bam20a], Bamler invented proper
objects for compactness theory, called metric flows [Bam20b]. A metric flow is
a family of time-dependent metric spaces, with evolving probability measures
that behave like conjugate heat kernels. In particular, a Ricci flow can be
regarded as a metric flow. The convergence of metric flows is called the
$\mathbb{F}$-convergence. Just like the pointed-Gromov-Hausdorff convergences,
one needs also to pick a sequence of “base points” for the
$\mathbb{F}$-convergence, which is, in reality, a sequence of conjugate heat
kernels.

_Any_ sequence of $n$-dimensional Ricci flows convergences, possibly after
passing to a subsequence, to a metric flow. This compactness result, if
applied to the singularity analysis, naturally leads to a more general notion
of singularity model—the $\mathbb{F}$-limit of smooth Ricci flows. Bamler
[Bam20c] showed many geometric properties for an $\mathbb{F}$-limit of a
noncollapsed sequence of $n$-dimensional Ricci flows; here by noncollapsed we
mean that the sequence of Ricci flows has bounded Nash entropy at some fixed
time. Among many others, the most important properties of such an
$\mathbb{F}$-limit are: (1) it is almost everywhere smooth, and the singular
set has space-time codimension at least four; (2) the convergence is a smooth
convergence on the regular part of the limit, and consequently, the regular
part of the $\mathbb{F}$-limit is a smooth Ricci flow space-time.

If, in addition to the noncollapsing assumption, the $\mathbb{F}$-limit has
constant entropy, then it is a self-similar metric flow called a metric
soliton (see Section 2 for more details). Metric solitons are not only special
$\mathbb{F}$-limits, but arise in many scenarios, and therefore are important
objects of study. In fact, so long as the monotonicity of the Nash entropy
comes into play, one expects the $\mathbb{F}$-limit to be a metric soliton.
For instance, the tangent flow at one point, or the tangent flow of an ancient
solution at infinity, are both metric solitons.

Not surprisingly, the metric soliton satisfies the same differential equation
as the smooth Ricci shrinker on the regular part. Furthermore, it is known
that a four-dimensional $\mathbb{F}$-limit metric soliton has only isolated
orbifold singularities. But little is known beyond these facts. For classical
smooth Ricci shrinkers, many nice estimates and analyses are obtained, among
which quite a few promoted the study of the Ricci flow. One naturally wonders
whether the same or similar results are true for metric solitons, at least on
its regular part. The goal of the current paper is to show some of these
results. Hopefully our work will be helpful to the singularity analysis in
higher dimensions. In order to present our project, we shall review some
classical results of the Ricci shrinker, and our plan will flush out
naturally.

Let $(M^{n},g,f)$ be a complete and smooth Ricci shrinker, satisfying the
following equation and normalization conditions.

$\displaystyle\operatorname{Ric}+\nabla^{2}f=\frac{1}{2}g,$
$\displaystyle\int_{M}(4\pi)^{-\frac{n}{2}}e^{-f}\,dg=1,$
$\displaystyle\mu_{g}:\equiv 2\Delta f-|\nabla f|^{2}+R+f-n<0.$

We summarize some classical results below.

  1. (1)

([CZ10]) The potential function has quadratic growth:

$\frac{1}{4}\left(d(x,x_{0})-5n\right)^{2}_{+}\leq
f(x)-\mu_{g}\leq\frac{1}{4}\left(d(x,x_{0})+\sqrt{2n}\right)^{2},$

where $x_{0}$ is a minimum point of $f$.

  2. (2)

([CLY11]) A quadratic lower bound of the scalar curvature (so long as the
shrinker is nonflat):

$R(x)\geq\frac{c}{f(x)-\mu_{g}}\qquad\text{ for all }\qquad x\in M.$

Here $c$ is a constant depending on the shrinker.

  3. (3)

([CMZ22]) A local gap theorem: if there is a large region in the shrinker with
almost-Euclidean $\mu$-functional, namely

$\mu\left(B(x_{0},\delta^{-1}),g,1\right)\geq-\delta,$

where $\delta=\delta(n)$ and $x_{0}$ is a minimum point of $f$, then the
shrinker is the Euclidean space.

  4. (4)

([LW20]) A global Sobolev inequality, or equivalently, Perelman’s
$\nu$-functional is finite on a shrinker:

$\nu(g)=\mu_{g}.$

  5. (5)

([MW12, LW20]) An optimal volume growth lower bound:

$|B(x_{0},r)|\geq c(n)e^{\mu_{g}}r\qquad\text{ for all }\qquad r\geq C(n).$

In the rest part of the current section, we shall introduce our main theorems.
It is therefore imperative to briefly introduce some basic notions which
appear in our statements. Our work is built upon Bamler’s theory [Bam20a,
Bam20b, Bam20c], which is a massive system containing quite a few definitions
and results. It is impossible to introduce every theorem and notion applied in
this paper. So we assume of the reader some basic familiarity with Bamler’s
work, and will refer to [Bam20a, Bam20b, Bam20c] whenever necessary.

Throughout this article, we consider an _$n$ -dimensional noncollapsed
$\mathbb{F}$-limit metric soliton_
$\left(\mathcal{X},(\nu_{t})_{t\in(-T,0)}\right)$ over the time-interval
$(-T,0)$. The terminology here implies that such a metric soliton arises as an
$\mathbb{F}$-limit of a sequence of $n$-dimensional Ricci flows with bounded
Nash entropy. Since a metric soliton is self-similar, it is modelled on some
measured metric space $(X,d,\nu)$, which is called its _model_. According to
[Bam20c], the model space is a metric completion of a smooth Remannian
manifold, called its _regular part_. We shall denote by
$(\mathcal{R}_{X},g,f_{0})$ the regular part of the model $(X,d,\nu)$. Note
that

$d\nu=(4\pi)^{-\frac{n}{2}}e^{-f_{0}}dg\quad\text{ on }\quad\mathcal{R}_{X}.$

The smooth function $f_{0}:\mathcal{R}_{X}\to\mathbb{R}$ is also called _the
potential function_ as in the smooth case. Another notion needed in the
statements below is called _center of the metric soliton_. It is a point
$x_{0}\in\mathcal{R}_{X}$ in the regular part which is almost equivalent to
the minimum point of the potential function in the smooth case. Henceforth,
the notation $x_{0}$ will be reserved for a center of a metric soliton.
Finally, we shall call

$W:=\int_{\mathcal{R}_{X}}f_{0}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg-\frac{n}{2}$

the _soliton entropy_. It is in fact the Nash entropy of the metric soliton
defined using the conjugate heat flow $\nu_{t}$ (which is a constant), and is
equivalent to the $\mu_{g}$ constant of a smooth shrinker. A more thorough
treatment of these notions is included in Section 2 and Section 3.

###  1.1 Basic properties of the potential function and the volume

To begin with, the potential function $f_{0}$ of a metric soliton also has
quadratic growth rate.

######  Theorem 1.1 (Potential function estimate).

Let $(X,d,\nu)$ be the model of an $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton and $(\mathcal{R}_{X},g,f_{0})$ the regular
part of the model. Let $x_{0}\in\mathcal{R}_{X}$ be a center of the soliton.
Then, for any $\varepsilon>0$, we have

$\displaystyle\frac{1}{8+\varepsilon}d^{2}_{g}(x_{0},x)-C(\varepsilon)\leq
f_{0}(x)-W\leq\frac{1}{4}\big{(}d_{g}(x_{0},x)+C(n)\big{)}^{2},$

where $C(\varepsilon)$ is a constant depending only on $\varepsilon$.

It is conceivable that the upper bound follows from integrating some classical
soliton identity (cf. the third equation in (2.8)) along almost-minimizing
curves contained in $\mathcal{R}_{X}$. So the key of proving the upper bound
is to show that $f_{0}(x_{0})-W$ is bounded by a dimensional constant. The
lower bound is in fact a consequence of Bamler’s conjugate heat kernel
estimate [Bam20a, Theorem 7.2] and the local smooth convergence property of
the $\mathbb{F}$-convergence [Bam20c, Theorem 2.5]. It goes without saying
that this idea can be applied to the volume. Therefore the following two
corollaries are immediately at hand.

######  Corollary 1.2 (Volume ratio upper bound at a center).

In the same settings as Theorem 1.1, we have

$\displaystyle\left|\mathcal{R}_{X}\cap B(x_{0},r)\right|\leq
C(n)e^{W}r^{n}\quad\text{ for all }\quad r\geq 1.$

######  Corollary 1.3 (Volume ratio upper bound).

In the same settings as Theorem 1.1, we have

$\displaystyle\left|\mathcal{R}_{X}\cap B(x,r)\right|\leq C(n)r^{n}\quad\text{
for all }\quad r\geq 1,$

where $x$ is any point in $X$.

The proofs of the results in this subsection can be found in Section 4.

###  1.2 A lower bound for the scalar curvature

A result of Chow-Lu-Yang [CLY11] shows that the scalar curvature of any
nonflat Ricci shrinker has at most quadratic decay. This estimate is sharp due
to the existence of asymptotically conical Ricci shrinkers [FIK03]. The Chow-
Lu-Yang estimate is generalized to ancient solutions by Chow and the authors
in [CCMZ23], with an additional assumption that the scalar curvature around
the $H_{n}$-centres of a fixed point is uniformly bounded from below. The
theorem below shows that the Chow-Lu-Yang estimate also holds on the regular
part of the model of a metric soliton, unless it is Ricci flat.

######  Theorem 1.4 (Scalar curvature lower bound).

Let $(X,d,\nu)$ be the model of an $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton and $(\mathcal{R}_{X},g,f_{0})$ its regular
part. Assume furthermore that $g$ is non-Ricci-flat. Let $W$ be the soliton
entropy and define

$\displaystyle c_{R}:=$ $\displaystyle\
\int_{\mathcal{R}_{X}}R\,(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg>0$ (1.1)

Then there is a constant $c=c(n,W,c_{R})$ depending only on $W$ and $c_{R}$,
such that

$\displaystyle R(x)\geq\frac{c}{f_{0}(x)-W+C_{0}}\quad\text{ for all }\quad
x\in M,$

where $C_{0}$ is a dimensional constant.

In combination with Theorem 1.1, the lower bound for the scalar curvature can
obviously be rewritten as

$R(x)\geq\frac{c(n,W,c_{R})}{d_{g}^{2}(x_{0},x)+C_{0}(n)}\qquad\text{ for all
}\qquad x\in\mathcal{R}_{X},$

where $x_{0}\in\mathcal{R}_{X}$ is a center of the metric soliton.

Recall that the method of [CLY11] is to apply the maximum principle to an
auxiliary function constructed with the scalar curvature and the potential
function, while in [CCMZ23], the auxiliary function implemented is a parabolic
version constructed with the scalar curvature and the conjugate heat kernel.
The proof of Theorem 1.4 is more similar to (and more complicated than) the
method of [CCMZ23]. In fact, we apply the non-Ricci-flat assumption to obtain
a local lower bound for the scalar curvatures along the $H_{n}$-centers of the
limiting sequence. And we argue that the sequence satisfies an almost
quadratic scalar curvature lower bound by applying the method in [CCMZ23].
Finally, the local smooth convergence property [Bam20c, Theorem 2.5] shows
that the lower bound of the scalar curvature can be carried to the regular
part of the metric soliton; the proof of Theorem 1.4 is presented in Section
5.

###  1.3 A local gap theorem

In [CMZ22], the authors proved a local gap theorem for Ricci shrinkers: if the
local $\mu$-functional of a large domain in the shrinker is almost zero, then
the shrinker must be an Euclidean space. We shall prove the same result for
metric solitons. Note that the region of the local $\mu$-functional must be
contained in the regular part, for otherwise the local $\mu$-functional is not
well-defined.

######  Theorem 1.5 (Local gap theorem).

There is a positive number $\delta=\delta(n)>0$ depending only on the
dimension $n$, with the following property. Let $(X,d,\nu)$ be the model of an
$n$-dimensional noncollapsed $\mathbb{F}$-limit metric soliton and
$(\mathcal{R}_{X},g,f_{0})$ its regular part. Let $x_{0}\in\mathcal{R}_{X}$ be
a center of the soliton. If

$\displaystyle\mu\Big{(}B(x_{0},\delta^{-1})\cap\mathcal{R}_{X},g,1\Big{)}\geq-\delta,$

where $\mu$ is Perelman’s local $\mu$-functional defined in Subsection 6.1,
then the metric soliton is the Gaussian soliton modelled on the Euclidean
space.

In [CMZ22], the authors presented three different proofs for the local gap
theorem of smooth shrinkers. The most easy but least intuitive one is to
modify the function $(4\pi)^{-\frac{n}{2}}e^{-f}$ by a cut-off method, and
implement it as the test function of the local $\mu$-functional. Then, the
smallness of the local $\mu$-functional implies the smallness of the soliton
entropy, and the flatness follows from the Carrillo-Ni-Yokota gap theorem
[CN09, Yo09, Yo12]. The other two approaches are more intuitive. One may apply
the pseudolocality theorem of [LW20] and show that under the smallness of the
local $\mu$-function, the scalar curvature remains bounded until the singular
time, which cannot happen to nonflat shrinkers. One may also apply the theory
of [CMZ23a] to see that the shrinker in question must have small Nash entropy
everywhere at arbitrary large scales, and the conclusion follows from Bamler’s
$\varepsilon$-regularity theorem [Bam20a, Theorem 10.2].

Our approach to Theorem 1.5 is more close to the first among the three
mentioned above. Although we do not have a Carrillo-Ni-Yokota gap theorem for
the soliton entropy, yet the soliton entropy, being the same as the Nash
entropy, is equal to the limit of the Nash entropy of the convergence
sequence. Thus, we can show a preliminary soliton entropy gap as presented in
Theorem 6.3. Next, to apply $(4\pi)^{-\frac{n}{4}}e^{-f_{0}}$ as the test
function of the local $\mu$-functional, one needs to modify it, such that it
is compactly supported not only on a compact ball, but also on the regular
part. Thus, given a proper cut-off function, the rest of the proof follows in
the same way as [CMZ22]. We shall modify Bamler’s cut-off function [Bam20c,
Lemma 15.27] for metric solitons. Due to its importance for proving other main
theorems, we shall include it here.

######  Proposition 1.6.

Let $(X,d,\nu)$ be the model of an $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton and $(\mathcal{R}_{X},g,f_{0})$ its regular
part. Let $x_{0}\in\mathcal{R}_{X}$ be a center of the soliton and $W$ the
soliton entropy. If

$\sigma>0,\quad A>\underline{A}(n),$

then for any $r\leq\overline{r}(A,W,\sigma)$ there is a continuous function
$\eta_{r}:X\to[0,1]$ satisfying the following properties.

  1. (1)

$\eta_{r}$ is supported on $\mathcal{R}_{X}$. Namely, $\eta_{r}\equiv 0$ on an
open neighborhood of $X\setminus\mathcal{R}_{X}$.

  2. (2)

$\eta_{r}$ is smooth on $\mathcal{R}_{X}$, satisfying
$|\nabla_{g}\eta_{r}|\leq C_{0}r^{-1}$, where $C_{0}$ is a dimensional
constant.

  3. (3)

For any $L>0$, $B(x_{0},L)\cap\\{\eta_{r}>0\\}$ is relatively compact in
$\mathcal{R}_{X}$. Here $x_{0}\in\mathcal{R}_{X}$ is a center of the soliton.

  4. (4)

$\displaystyle\int_{\mathcal{R}_{X}\cap
B(x_{0},A)\cap\\{|\nabla_{g}\eta_{r}|\neq 0\\}}dg\leq\int_{\mathcal{R}_{X}\cap
B(x_{0},A)\cap\\{0\leq\eta_{r}<1\\}}dg\leq Cr^{4-\sigma},$

where $C=C(A,W,\sigma)$.

  5. (5)

$\displaystyle\mathcal{R}_{X}=\bigcup_{r>0}\\{\eta_{r}=1\\}$.

The proofs of the results of this subsection are found in Section 6.

###  1.4 Global $\nu$-functional and Sobolev inequality

In [CN09], Carrillo-Ni made a nice observation: on a smooth shrinker,
Perelman’s global $\mu$-functional at scale $1$ $\mu(g,1)$ is equal to the
shrinker entropy $\mu_{g}$. This is tantamount to saying that the potential
function $f$ is exactly the minimizer of Perelman’s $\mathcal{W}$-functional
at scale $1$. The consequence is a sharp logarithmic Sobolev inequality. More
recently, Li-Wang [LW20] pushed this result one step more, and proved that the
shrinker entropy $\mu_{g}$ is equal to Perelman’s $\nu$-functional on all
scales. Hence, there is a global Sobolev inequality on each shrinker. We shall
prove that the same holds for metric solitons.

######  Theorem 1.7.

Let $(X,d,\nu)$ be the model of an $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton and $(\mathcal{R}_{X},g,f_{0})$ its regular
part. Let $W$ be the soliton entropy. Then we have

$\displaystyle\nu\left(\mathcal{R}_{X},g\right)=W,$

where $\nu$ is Perelman’s local $\nu$-functional defined in Subsection 6.1.

Note that according to the definition of the local $\nu$-functional, the test
functions of $\nu\left(\mathcal{R}_{X},g\right)$ must be compactly supported
in $\mathcal{R}_{X}$. Thus the logarithmic Sobolev inequalities and the
Sobolev inequality which follow from the theorem above can only be applied to
functions compactly supported on $\mathcal{R}_{X}$.

######  Corollary 1.8 (Logarithmic Sobolev inequalities).

In the same settings as Theorem 1.7, we have

$\displaystyle\ \int_{\mathcal{R}_{X}}u^{2}\log
u^{2}\,dg-\int_{\mathcal{R}_{X}}u^{2}\,dg\cdot\log\left(\int_{\mathcal{R}_{X}}u^{2}\,dg\right)$
(1.2) $\displaystyle\leq$ $\displaystyle\
\tau\int_{\mathcal{R}_{X}}\left(4|\nabla
u|^{2}+Ru^{2}\right)\,dg-\left(W+n+\frac{n}{2}\log(4\pi\tau)\right)\int_{\mathcal{R}_{X}}u^{2}\,dg$

for any $\tau>0$ and any $u\in C^{0,1}_{c}(\mathcal{R}_{X})$.

######  Corollary 1.9 (Sobolev inequality).

In the same settings as Theorem 1.7, we have

$\displaystyle\left(\int_{\mathcal{R}_{X}}|u|^{\frac{2n}{n-2}}\,dg\right)^{\frac{n-2}{n}}\leq
C(n)e^{-\frac{2W}{n}}\int_{\mathcal{R}_{X}}\left(4|\nabla
u|^{2}+Ru^{2}\right)\,dg$ (1.3)

for any $u\in C^{0,1}_{c}(\mathcal{R}_{X})$.

The proofs of the results in this subsection are found in Section 7.

###  1.5 Volume growth lower estimate

Lastly, we shall prove an optimal volume growth lower bound for metric
solitons, similar to the smooth case considered by Munteanu-Wang [MW12] and
Li-Wang [LW20]. The following theorem shows that a noncompact metric soliton,
similar to a Ricci shrinker, has at least linear volume growth. This volume
growth lower bound is obviously optimal since it is fulfilled by the standard
cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$.

######  Theorem 1.10.

Let $(X,d,\nu)$ be the model of an $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton and $(\mathcal{R}_{X},g,f_{0})$ its regular
part. Let $x_{0}\in\mathcal{R}_{X}$ be a center of the soliton and $W$ the
soliton entropy. If $X$ is noncompact (in the sense of infinite diameter),
then there exists positive dimensional constant $c(n)$ such that

$|B(x_{0},r)\cap\mathcal{R}_{X}|\geq c(n)e^{W}r\quad\text{ for all }\quad
r\geq c(n)^{-1}.$

There is not much to be commented on the proof of Theorem 1.10, since it
follows from the argument of [MW12, LW20]. However, we mention that the
prerequisite tools that make this argument possible are the cut-off functions
in Proposition 1.6 and the global Sobolev inequality in Theorem 1.7. The proof
of Theorem 1.10 is presented in Section 8. Acknowledgements. The authors would
like to thank Felix Schulze for asking the question of the scalar curvature
lower bound on metric solitons and for some helpful discussions.

##  2 Metric soliton as an $\mathbb{F}$-limit

According to Bamler [Bam20b, Definition 3.57], a metric soliton is no more
than a self-similar metric flow; this notion is too general, and could either
be related to or have nothing to do with the Ricci flow. For this reason, we
shall restrict our attention to those metric solitons that could possibly
arise in the singularity analysis of the Ricci flow, namely, the ones which
are $\mathbb{F}$-limits of noncollapsed sequences of smooth Ricci flows. In
this section, we shall survey the background and basic properties of such
metric solitons.

Let
$\displaystyle\left\\{(M^{i},g^{i}_{t},x_{i})_{t\in(-T_{i},0]}\right\\}_{i=1}^{\infty}$
be a sequence of $n$-dimensional Ricci flows, where $x_{i}\in M^{i}$ is a
fixed base point. To ensure that Bamler’s theory [Bam20a, Bam20b, Bam20c] is
applicable, we assume that _each Ricci flow in this sequence has bounded
curvature within each compact time-interval_ (but the bounds need not to be
uniform), see the appendix of [Bam21] for justifications. Assume that the
sequence satisfies the $(\tau_{0},Y_{0})$-noncollapsed condition: there are
positive numbers $\tau_{0}$ and $Y_{0}$, such that

$\displaystyle\mathcal{N}_{x_{i},0}(\tau_{0})\geq-Y_{0}\quad\text{ for all
}\quad i\in\mathbb{N}.$

It appears that the exact values of $\tau_{0}$ and $Y_{0}$ become
inconsequential if the limit is a metric soliton with finite entropy; the
reason will be explained below.

According to [Bam20b, Bam20c], after passing to a subsequence, we have the
following convergence

$\displaystyle\left((M^{i},g^{i}_{t})_{t\in(-T_{i},0]},(\nu_{x_{i},0\,|\,t})_{t\in(-T_{i},0]}\right)\xrightarrow[i\to\infty]{\quad\mathbb{F},\
\mathfrak{C}\quad}\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)},$ (2.1)

where $\mathfrak{C}$ is some correspondence, $\displaystyle
T=\lim_{i\to\infty}T_{i}\in(0,+\infty]$ (in case $T=+\infty$, the convergence
above is on compact intervals), $\mathcal{X}$ is a metric flow over $(-T,0]$
whose singular part $\mathcal{X}_{<0}\setminus\mathcal{R}_{<0}$ has space-time
codimension no less than $4$, and $\nu_{t}$ is a conjugate heat flow
satisfying

$\displaystyle\operatorname{Var}(\nu_{t})\leq H_{n}|t|.$ (2.2)

Here (and afterwards) $\operatorname{Var}$ represents the variance and $H_{n}$
stands for the constant $\frac{(n-1)\pi^{2}}{2}+4$. We point out that the
finial slice $\mathcal{X}_{0}$ of $\mathcal{X}$ consists of one point only,
and we will often regard $\mathcal{X}$ as a metric flow over the interval
$(-T,0)$.

The regular part $\mathcal{R}=\mathcal{R}_{<0}$ of $\mathcal{X}$ is a Ricci
flow space-time, and we shall always denote by $g_{t}$ the metric on
$\mathcal{R}_{t}$ for $t\in(-T,0)$. Then, on the regular part, the conjugate
heat flow $\nu_{t}$ can be written as ($\tau=-t$)

$\displaystyle d\nu_{t}=(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg_{t},$ (2.3)

where $f$ is a smooth function on $\mathcal{R}$. Since the singular part is a
null-set, the Nash entropy is well-defined for $\nu_{t}$.

$\displaystyle\mathcal{N}_{\nu}(\tau):=\int_{\mathcal{R}_{-\tau}}fd\nu_{-\tau}-\frac{n}{2}\quad\text{
for all }\quad\tau\in(0,T).$

Similar to a consequence of Perelman’s monotonicity formula, whenever
$\mathcal{N}_{\nu}(\tau)\equiv\operatorname{const}$ for all $\tau\in(0,T)$,
$\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$ is a metric soliton, which
is a self-similar metric flow in the sense of [Bam20b, Definition 3.57]; this
type of metric solitons are that which shall be studied in this article.
Henceforth, for a metric soliton obtained in this way we shall always define

$\displaystyle
W:\equiv\mathcal{N}_{\nu}(\tau)\in(-\infty,0],\quad\tau\in(0,T),$ (2.4)

and call this number $W$ the _soliton entropy_. The continuity of the Nash
entropy with respect to the $\mathbb{F}$-convergence [Bam20b, Theorem 2.10]
immediately implies that

$\displaystyle\mathcal{N}_{x_{i},0}(\tau)>W-1\quad\text{ for all
}\tau\in(0,T)\text{ whenever $i$ is large enough. }$ (2.5)

Thus we see that the $(\tau_{0},Y_{0})$-noncollapsing assumption can be
replaced by $(T/2,W-1)$-noncollapsing assumption, and we shall always refer to
such a soliton as an _$n$ -dimensional noncollapsed $\mathbb{F}$-limit metric
soliton_ (instead of a $(\tau_{0},Y_{0})$-noncollapsed $\mathbb{F}$-limit as
we did in [CMZ23b]).

At this point, it is helpful to recall the important properties of a metric
soliton as summarized in [Bam20c, Theorem 2.18,Theorem 15.69].

######  Theorem 2.1 ([Bam20c, Theorem 2.18, Theorem 15.69], basic properties
of the metric soliton).

Let $\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$ be the metric soliton
arising in the scenario of this section.

  1. 1.

On the metric part $\mathcal{R}$, we have

$\displaystyle\operatorname{Ric}+\nabla^{2}f-\tfrac{1}{2\tau}g=0,$ (2.6)
$\displaystyle\tau\left(2\Delta f-|\nabla f|^{2}+R\right)+f-n\equiv W,$
$\displaystyle-\tau\left(|\nabla f|^{2}+R\right)+f\equiv W,$
$\displaystyle\partial_{\mathfrak{t}}f=|\nabla f|^{2},$

where $f$ is defined in (2.3) and $W$ is the soliton entropy defined in (2.4).

  2. 2.

The heat kernel $K$ satisfies the following identity on $\mathcal{R}$

$\displaystyle(\tau\partial_{\mathfrak{t}}-\tau\nabla
f)_{x_{1}}K(x_{1}\,|\,x_{0})+(\tau\partial_{\mathfrak{t}}-\tau\nabla
f)_{x_{0}}K(x_{1}\,|\,x_{0})=\frac{n}{2}K(x_{1}\,|\,x_{0}).$ (2.7)

  3. 3.

There is a singular space $(X,d)$ of dimension $n$, a probability measure
$\nu$ on $X$, and an identification

$\mathcal{X}_{<0}=X\times(-T,0),$

such that the following holds for all $t\in(-T,0)$

    1. (a)

$(\mathcal{X}_{t},d_{t})=(X\times\\{t\\},|t|^{1/2}d)$.

    2. (b)

$\mathcal{R}=\mathcal{R}_{X}\times(-T,0)$ and $\partial_{\mathfrak{t}}-\nabla
f$ corresponds to the standard vector field on the second factor; here
$\mathcal{R}_{X}$ is the regular part of $X$.

    3. (c)

$(\mathcal{R}_{t},g_{t})=(\mathcal{R}_{X}\times\\{t\\},|t|g)$, where $g$ is
the Riemannian metric on $\mathcal{R}_{X}$.

    4. (d)

$\nu_{t}=\nu$.

Moreover, there is a unique family of probability measures
$(\nu^{\prime}_{x\,|\,t})_{x\in X,\,t\geq 0}$ such that the tuple
$(X,d,\nu,(\nu^{\prime}_{x\,|\,t})_{x\in X,\,t\geq 0})$ is a model for
$\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$ corresponding to the same
identification (in the sense of [Bam20b, Definition 3.57]).

  4. 4.

The singular part of $(X,d,\nu)$ is a null set with codimension no less than
$4$.

Henceforth, we shall call $(X,d,\nu)$ (where $\nu=\nu_{-1}$) the _model_ of
the metric soliton and $(\mathcal{R}_{X},g,f_{0})$ (where $f_{0}=f(\cdot,-1)$)
its _regular part_. It is clear that a metric soliton is completely determined
by the regular part of its model. The equations in (2.6) are constantly
applied in the proof of the main theorems, so it is helpful to restate them.

######  Corollary 2.2.

Let $(X,d,\nu)$ be the model of an $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton and $(\mathcal{R}_{X},g,f_{0})$ the regular
part of the model. Then the following equations hold on $\mathcal{R}_{X}$

$\displaystyle\operatorname{Ric}+\nabla^{2}f_{0}=\frac{1}{2}g,$ (2.8)
$\displaystyle 2\Delta f_{0}-|\nabla f_{0}|^{2}+R+f_{0}-n=W,$
$\displaystyle|\nabla f_{0}|^{2}+R=f_{0}-W.$

##  3 Metric soliton as an ancient $\mathbb{F}$-limit

It is well-known that a smooth skrinking gradient Ricci soliton always
generates an ancient solution, and this fact is essential to the study of
Ricci shrinkers. For instance, the positivity of the scalar curvature is a
consequence of the ancientness [Che09]. However, it is not immediately clear
from the limiting definition (2.1) that a metric soliton defined on a finite
interval $(-T,0)$ can be extended to an ancient metric flow. Even if it could,
it is not clear whether such ancient metric flow is a noncollapsed
$\mathbb{F}$-limit of a sequence of Ricci flows. In this section, we shall
apply a blow-up argument to verify this fact.

######  Theorem 3.1.

Let $\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$ be an $n$-dimensional
noncollapsed $\mathbb{F}$-limit metric soliton defined over a finite interval
$(-T,0)$. Then it can be extended to an ancient metric flow
$\big{(}\mathcal{X},(\nu_{t})_{t\in(-\infty,0)}\big{)}$, which is also a
metric soliton with the same model. Further more, there is a sequence of
$n$-dimensional smooth Ricci flows
$\\{(M^{i},g^{i}_{t},x_{i})_{t\in[-T_{i},0]}\\}_{i=1}^{\infty}$ with
$T_{i}\nearrow+\infty$ such that

$\displaystyle\lim_{i\to\infty}\mathcal{N}_{x_{i},0}(T_{i})=W,$ (3.1)

where $W$ is the soliton entropy, and that

$\displaystyle\left((M^{i},g^{i}_{t})_{t\in[-T_{i},0]},(\nu_{x_{i},0\,|\,t})_{t\in[-T_{i},0]}\right)\xrightarrow[i\to\infty]{\quad\mathbb{F}\quad}\big{(}\mathcal{X},(\nu_{t})_{t\in(-\infty,0)}\big{)},$
(3.2)

over compact intervals.

The strategy is to show that the blow-up limit of
$\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$ is a metric soliton with
the same model and defined over the interval $(-\infty,0)$. To begin with, the
following lemma is a trivial fact from [Bam20b, Theorem 7.4, Theorem 7.6].

######  Lemma 3.2.

There is a metric flow pair
$\left(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t<0}\right)$ defined over
$(-\infty,0)$, such that

$\displaystyle\left(\mathcal{X}^{k},(\nu^{k}_{t})_{t\in(-Tk^{2},0)}\right)\xrightarrow[k\to\infty]{\qquad\mathbb{F}\qquad}\left(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t<0}\right),$
(3.3)

on compact time intervals, where $\mathcal{X}^{k}$ and $\nu^{k}_{t}$ are the
scaling notations defined in [Bam20b, Definition 6.55].

The following proposition is the key ingredient of the proof of Theorem 3.1

######  Proposition 3.3.

The limit $\left(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t<0}\right)$ in
(3.3) is a metric soliton defined over $(-\infty,0)$ which is isometric to
$\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$ on the interval $(-T,0)$.

###### Proof.

Let $(X,d,\nu)$ be the model of the metric soliton
$\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$ and
$(\mathcal{R}_{X},g,f_{0})$ its regular part. Without of loss of generality,
we may assume that $T>1$. By Theorem 2.1, the vector field
$\partial_{\mathfrak{t}}-\nabla f$ is complete on $\mathcal{R}$, and the
$1$-parameter family of maps generated by it

$\displaystyle\phi_{t}$ $\displaystyle\
:\mathcal{R}_{X}(\cong\mathcal{R}_{-1})\to\mathcal{R},$
$\displaystyle\phi_{-1}$ $\displaystyle\ =\operatorname{id},$
$\displaystyle\tfrac{\partial}{\partial t}\phi_{t}$ $\displaystyle\
=\left(\partial_{\mathfrak{t}}-\nabla f\right)\circ\phi_{t}.$

defines isometries

$\phi_{t}:\big{(}\mathcal{R}_{X},|t|g,f_{0}\big{)}\longrightarrow\big{(}\mathcal{R}_{t},g_{t},f(\cdot,t)\big{)},\quad
t\in(-T,0),$

which can be extended to isometries

$\displaystyle\overline{\phi}_{t}:(X,|t|^{1/2}d,\nu)\longrightarrow(\mathcal{X}_{t},\nu_{t}),\quad
t\in(-T,0).$ (3.4)

Indeed, the fact that $\phi_{t}$ is an isometry between
$(\mathcal{R}_{X},|t|g,f_{0})$ and
$\big{(}\mathcal{R}_{t},g_{t},f(\cdot,t)\big{)}$ can also be verified by
direct computation:

$\displaystyle\tfrac{\partial}{\partial
t}|t|^{-1}\phi_{t}^{*}g_{t}=|t|^{-1}\phi_{t}^{*}\left(\mathcal{L}_{(\partial_{\mathfrak{t}}-\nabla
f)}g_{t}\right)+|t|^{-2}\phi_{t}^{*}g_{t}=-2|t|^{-1}\phi_{t}^{*}\left(\operatorname{Ric}+\nabla^{2}f-\frac{1}{2|t|}g_{t}\right)=0,$
$\displaystyle\tfrac{\partial}{\partial
t}f\circ\phi_{t}=\left(\partial_{\mathfrak{t}}f-|\nabla
f|^{2}\right)\circ\phi_{t}=0.$

Here we have applied (2.6).

Next, we define

$\displaystyle\overline{\phi}_{t}^{k}:(X,d,\nu)\to\left(\mathcal{X}^{k},(\nu^{k}_{t})_{t\in(-Tk^{2},0)}\right)$

by letting

$\displaystyle\overline{\phi}_{t}^{k}:=\overline{\phi}_{k^{-2}t}$

for all $t\in(-Tk^{2},0)$ and for all $k\in\mathbb{N}$; it is clear that each
$\overline{\phi}_{t}^{k}$ is an isometry onto $\mathcal{X}^{k}_{t}$. For any
interval of the form $I=[-S,-S^{-1}]$, where $S>1$, we may construct a
correspondence between
$\\{\left(\mathcal{X}^{k},(\nu^{k}_{t})_{t\in(-Tk^{2},0)}\right)\\}_{k>S}$ by

$\displaystyle\mathfrak{C}:=\left((X,|t|d)_{t\in
I},\left(\overline{\phi}^{k,-1}_{t}\right)_{t\in I}\right),$

where $\overline{\phi}^{k,-1}_{t}$ is the inverse of
$\overline{\phi}^{k}_{t}$. We observe that

$\displaystyle{\rm
dist}^{\mathfrak{C}}_{\mathbb{F}}\left(\left(\mathcal{X}^{k}_{I},(\nu^{k}_{t})_{t\in
I}\right),\left(\mathcal{X}^{j}_{I},(\nu^{j}_{t})_{t\in I}\right)\right)\equiv
0\qquad\text{ for all }\qquad k,\ j>S.$ (3.5)

To see this, we need only to verify the following claim.

Claim.  _For any $\lambda>1$,_

$\displaystyle\psi_{t}:=\overline{\phi}^{\lambda}_{t}\circ\overline{\phi}^{-1}_{t}:\mathcal{X}_{t}\to\mathcal{X}^{\lambda}_{t},\quad
t\in(-T,0)$

_defines an isometry between
$\left(\mathcal{X}^{\lambda},(\nu^{\lambda}_{t})_{t\in(-T\lambda^{2},0)}\right)$
and $\left(\mathcal{X},(\nu_{t})_{t\in(-T,0)}\right)$ over the interval
$(-T,0)$. _

###### Proof of the claim.

It is already clear that each $\psi_{t}$ is an isometry between metric spaces.
We need only to show that it preserves the conjugate heat kernels. Let
$(\nu^{\prime}_{x\,|\,t})_{x\in X,\,t\leq 0}$ be the family of probability
measures given in Theorem 2.1. Fix any
$x=\phi_{t}(x^{\prime})\in\mathcal{X}_{t}$ and any $-T<s<t<0$, we may compute
according to [Bam20b, Definition 3.57]

$\displaystyle(\psi_{s})_{*}\nu_{x\,|\,s}=$ $\displaystyle\
(\overline{\phi}^{\lambda}_{t})_{*}\nu^{\prime}_{x^{\prime}\,|\,\log(s/t)}=(\overline{\phi}_{\lambda^{-2}t})_{*}\nu^{\prime}_{x^{\prime}\,|\,\log(\lambda^{-2}s/\lambda^{-2}t)}$
$\displaystyle=$ $\displaystyle\
\nu_{\overline{\phi}_{\lambda^{-2}t}(x^{\prime})\,|\,\lambda^{-2}s}=\nu^{\lambda}_{\psi_{t}(x)\,|\,s}.$

Thus, $\psi_{t}$ satisfies [Bam20b, Definition 3.13]; this proves the claim.

∎

Finally, the proposition follows from (3.5).

∎

###### Proof of Theorem 3.1.

Since the metric soliton is obtained as an $\mathbb{F}$-limit (2.1), for each
$k$ we have that,

$\displaystyle\left((M^{i},k^{2}g^{i}_{k^{-2}t})_{t\in(-k^{2}T_{i},0]},(\nu_{x_{i},0\,|\,k^{-2}t})_{t\in(-k^{2}T_{i},0]}\right)\xrightarrow[i\to\infty]{\qquad\mathbb{F}\qquad}\left(\mathcal{X}^{k},(\nu^{k}_{t})_{t\in(-Tk^{2},0)}\right).$

We may, by a diagonal argument, find a new sequence of $n$-dimensional Ricci
flows with uniformly bounded Nash entropy, such that

$\left((\overline{M}^{i},\overline{g}^{i}_{t})_{t\in(-\overline{T}_{i},0]},(\overline{\nu}_{x_{i},0\,|\,t})_{t\in(-\overline{T}_{i},0]}\right)\xrightarrow[i\to\infty]{\qquad\mathbb{F}\qquad}\big{(}\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t<0}\big{)}.$

Since the soliton entropy depends only on (the regular part of) the model, we
have that

$\mathcal{N}_{\nu^{\infty}}(\tau)\equiv W\quad\text{ for all }\quad\tau>0.$

By the continuity of the Nash entropy with respect to the
$\mathbb{F}$-convergence [Bam20c, Theorem 2.10], we may, after discarding
finitely many terms and modifying $\overline{T}_{i}$, ensure that (3.1) holds;
this finishes the proof of the theorem. ∎

Since for the Ricci flow $(M^{i},g^{i}_{t})_{t\in[-T_{i},0]}$, the scalar
curvature always satisfies

$\displaystyle R_{g^{i}_{t}}\geq-\frac{n}{T_{i}}\qquad\text{ for all }\qquad
t\in[-T_{i}/2,0]$ (3.6)

by the maximum principle, and since the $\mathbb{F}$-convergence can be
upgraded to the smooth convergence on the regular part of the limit [Bam20c,
Theorem 2.5], we immediately have the following corollary (which is contained
in [Bam20c, Theorem 2.18]), generalizing [Che09].

######  Corollary 3.4.

Let $(X,d,\nu)$ be the model of any $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton, and let $(\mathcal{R}_{X},g,f_{0})$ be its
regular part. Then we have that

$\displaystyle R\geq 0\quad\text{ everywhere on }\quad\mathcal{R}_{X}.$

Furthermore, if $R=0$ is somewhere on $\mathcal{R}_{X}$, then $g$ is a Ricci-
flat metric.

###### Proof.

That $R$ is nonnegative follows from Theorem 3.1 and (3.6). That $R$ is
positive unless $g$ is Ricci flat follows from the strong maximim principle
(which is but a local argument). ∎

##  4 Center and the potential function

It is well known [CZ10] that the potential function of a smooth Ricci shrinker
has exactly quadratic growth, and its minimum point (which may not be unique)
is usually used as a center or an origin of the soliton. In like manner, it is
also convenient to find a center for a metric soliton. Although, as we shall
see below, on the regular part of the model of a metric soliton, the potential
function has almost quadratic growth, yet it is much more convenient to pick
an $H_{n}$-center of $\nu_{t}$ instead of a minimum point of the potential
function as an origin.

###  4.1 Center of metric soliton

Let $\big{(}\mathcal{X},(\nu_{t})_{t<0}\big{)}$ be an $n$-dimensional
noncollapsed $\mathbb{F}$-limit metric soliton that arises as a limit in
(2.1). Write $d\nu_{t}:=(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg_{t}$ on the regular
part $\mathcal{R}\subset\mathcal{X}$. By Theorem 3.1, we may assume that the
soliton is defined over the interval $(-\infty,0)$. Let $(X,d,\nu)$ be the
model of the metric soliton and $(\mathcal{R}_{X},g,f_{0})$ its regular part.

By (2.2), we have

$\displaystyle\int_{\mathcal{X}_{-1}}\operatorname{Var}(\delta_{x},\nu_{-1})d\nu_{-1}=\int_{\mathcal{X}_{-1}\times\mathcal{X}_{-1}}d_{-1}^{2}(x,y)\,d\nu_{-1}(x)d\nu_{-1}(y)=\operatorname{Var}(\nu_{-1})\leq
H_{n}.$

Thus, we can always find a point $x_{0}\in\mathcal{X}_{-1}$, such that

$\displaystyle\operatorname{Var}(\delta_{x_{0}},\nu_{-1})\leq H_{n}.$ (4.1)

Since $\mathcal{R}_{-1}$ is of full measure and $\mathcal{X}_{-1}$ is its
metric completion, we may, without loss of generality, assume that
$x_{0}\in\mathcal{R}_{-1}$. For otherwise we may replace it with a point
arbitrarily close by, such that (4.1) still holds with a slightly larger bound
(we shall not rewrite this bound, since the constant $H_{n}$ itself does not
have much significance except that it is a dimensional constant). Note that
$x_{0}$ can be identified with a point on $\mathcal{R}_{X}$ which we will also
call $x_{0}$. Henceforth, the point $x_{0}$ defined by (4.1) will be called a
_center (or origin) of the metric soliton_ ; a metric soliton may have
multiple centers, but their distance from each other cannot exceed
$\sqrt{2H_{n}}$.

Since $\partial_{\mathfrak{t}}-\nabla f$ is a complete vector field on
$\mathcal{R}=\mathcal{R}_{X}\times(-\infty,0)$ corresponding to the second
factor, we may find an integral curve $x_{0}(t)$ in $\mathcal{R}$ of
$\partial_{\mathfrak{t}}-\nabla f$ with $x_{0}(-1)=x_{0}$. Obviously,
$x_{0}(t)\in\mathcal{R}_{t}$ for all $t\in(-\infty,0)$. We can apply this
self-similarity to verify the following facts, which indicate why $x_{0}$ is
called a “center” by us.

######  Lemma 4.1.

$\displaystyle\operatorname{Var}\big{(}\nu_{t},\delta_{x_{0}(t)}\big{)}\leq
H_{n}|t|\quad\text{ for all }\quad t\in(-\infty,0).$ (4.2)

###### Proof.

Note that, by Theorem 2.1, $f$ is constant along the integral curvatures of
$\partial_{\mathfrak{t}}-\nabla f$ since

$\displaystyle\partial_{\mathfrak{t}}f-|\nabla f|^{2}=0.$

Therefore,

$\displaystyle\operatorname{Var}(\nu_{t},\delta_{x_{0}(t)})=$ $\displaystyle\
\int_{\mathcal{X}_{t}}d^{2}_{t}\big{(}x_{0}(t),y\big{)}\,d\nu_{t}(y)=\int_{\mathcal{R}_{t}}d_{t}^{2}\big{(}x_{0}(t),y\big{)}(4\pi|t|)^{-\frac{n}{2}}e^{-f(y)}\,dg_{t}(y)$
$\displaystyle=$ $\displaystyle\
\int_{\mathcal{R}_{X}}|t|d^{2}_{g}(x_{0},y)(4\pi)^{-\frac{n}{2}}e^{-f_{0}(y)}dg(y)$
$\displaystyle=$ $\displaystyle\
|t|\int_{\mathcal{R}_{-1}}d_{-1}^{2}(x_{0},y)\,d\nu_{-1}(y)$ $\displaystyle=$
$\displaystyle\ |t|\operatorname{Var}(\nu_{-1},\delta_{x_{0}}).$

Here we have applied the facts that $\mathcal{X}_{t}\setminus\mathcal{R}_{t}$
is a null set, that $\mathcal{X}_{t}$ is the metric completion of
$\mathcal{R}_{t}$, and that the integrand

$d_{t}^{2}\big{(}x_{0}(t),y\big{)}(4\pi|t|)^{-\frac{n}{2}}e^{-f(y)}\,dg_{t}(y)$

varies by scaling along integral curves of $\partial_{\mathfrak{t}}-\nabla f$.

∎

Similarly, we also have the following lemma. Its significance may not be clear
at this point, yet it is an important ingredient of the proof of Theorem 1.4.

######  Lemma 4.2.

We have

$\displaystyle|t|\int_{\mathcal{R}_{t}}R_{g_{t}}\,d\nu_{x_{0}(t/2)\,|\,t}\equiv\operatorname{const}\quad\text{
for all }\quad t\in(-\infty,0).$

###### Proof.

Let us write the integral as

$\displaystyle\int_{\mathcal{R}_{t}}|t|R_{g_{t}}(y)K(x_{0}(t/2)\,|\,y)\,dg_{t}(y).$

We shall verify that the integrand is invariant along the integral curves of
$\partial_{\mathfrak{t}}-\nabla f$. It is clear that both $|t|R_{g_{t}}$ and
$|t|^{-\frac{n}{2}}dg_{t}$ satisfy this property. We need only to consider
$|t|^{\frac{n}{2}}K(x_{0}(t/2)\,|\,y)$. Indeed, we have

$\displaystyle\ \left(\frac{\partial}{\partial t}-(\nabla
f)_{y}\right)\left(|t|^{\frac{n}{2}}K(x_{0}(t/2)\,|\,y)\right)$
$\displaystyle=$ $\displaystyle\
|t|^{\frac{n}{2}}\left(-\frac{n}{2|t|}K(x_{0}(t/2)\,|\,y)+(\partial_{\mathfrak{t}}-\nabla
f)_{y}K(x_{0}(t/2)\,|\,y)+\left.\frac{1}{2}\partial_{s}K(x_{0}(s)\,|\,y)\right|_{s=\frac{t}{2}}\right)$
$\displaystyle=$ $\displaystyle\
|t|^{\frac{n}{2}+1}\left(-\frac{n}{2}K(x_{0}(t/2)\,|\,y)+(\tau\partial_{\mathfrak{t}}-\tau\nabla
f)_{y}K(x_{0}(t/2)\,|\,y)+\left.(\tau\partial_{\mathfrak{t}}-\tau\nabla
f)_{x}K(x\,|\,y)\right|_{x=x_{0}(t/2)}\right)$ $\displaystyle=$
$\displaystyle\ 0,$

where we have applied (2.7). The lemma then follows in the same way as the
previous lemma. ∎

###  4.2 Estimates of the potential function

Next, we prove the quadratic growth estimate for the potential function of a
metric soliton. Note that the regular part is not a complete Riemannian
manifold, so the second variation along a geodesic is not available. This is
the reason why we have a weaker lower bound.

###### Proof of Theorem 1.1.

Let us consider the limiting definition of the metric soliton given by Theorem
3.1. Let $\\{(M^{i},g^{i}_{t},x_{i})_{t\in[-T_{i},0]}\\}_{i=1}^{\infty}$ be
the sequence in (3.2) that converges to the metric soliton
$(\mathcal{X},(\nu_{t})_{t<0})$ in the $\mathbb{F}$-sense.

The lower bound estimate of $f_{0}$ is simply a consequence of [Bam20c, Lemma
15.9(a)]. In fact, applying the second inequality of [Bam20c, Lemma 15.9(a)]
with $(x_{i}^{\prime},t_{i}^{\prime})=(x_{i},0)$ and $s=-1$, we immediately
have

$\displaystyle(4\pi)^{-\frac{n}{2}}e^{-f_{0}(x)}\leq
C(\varepsilon)\exp\left(-W-\frac{1}{8+\varepsilon}\big{(}d^{g}_{W_{1}}(\nu,\delta_{x})\big{)}^{2}\right)\qquad\text{
for all }\quad x\in\mathcal{R}_{X}(\cong\mathcal{R}_{-1}).$ (4.3)

On the other hand, the choice of $x_{0}$ (4.1) implies that (recall that
$\nu_{-1}=\nu$)

$\displaystyle d_{g}(x,x_{0})\leq
d^{g}_{W_{1}}(\nu,\delta_{x})+d^{g}_{W_{1}}(\nu,\delta_{x_{0}})\leq
d^{g}_{W_{1}}(\nu,\delta_{x})+\sqrt{H_{n}}\quad\text{ for all }\quad
x\in\mathcal{R}_{X}.$ (4.4)

The lower bound of $f_{0}$ follows from the combination of (4.3) and (4.4).

Next, we consider the upper bound. Note that the positivity of the scalar
curvature and the third equation in (2.8) imply that

$\displaystyle|\nabla_{g}f_{0}|^{2}\leq f_{0}-W.$ (4.5)

Thus, it remains to show that $f_{0}(x_{0})-W$ is bounded from above by a
constant depending only on $n$, and then the upper bound of $f_{0}$ follows
from integration along almost-distance-realizing curves on $\mathcal{R}_{X}$.

Since $x_{0}\in\mathcal{R}_{-1}(\cong\mathcal{R}_{X})$, where
$\mathcal{R}\subset\mathcal{X}$ is the region of smooth convergence, we may,
according to [Bam20b, Theorem 9.31], find a sequence $z^{\prime}_{i}\in
M^{i}$, such that

$(z^{\prime}_{i},-1)\xrightarrow{\quad i\to\infty\quad}x_{0}.$

This convergence is either via the local diffeomorphisms which define the
local smooth convergence, or is a convergence of points within a
correspondence. The points $(z_{i}^{\prime},-1)$ are “almost $H_{n}$-centers”
of $(x_{i},0)$. To see this, we argue in the same way as the proof of [Bam20a,
Proposition 3.13]. (4.1) implies that

$\displaystyle\int_{\mathcal{R}_{X}\cap
B(x_{0},\sqrt{2H_{n}})}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg\geq\frac{1}{2}.$
(4.6)

By Fatou’s lemma and the nature of local smooth convergence, we have

$\displaystyle\liminf_{i\to\infty}\nu_{x_{i},0\,|\,-1}\left(B_{-1}\left(z_{i}^{\prime},\sqrt{2H_{n}}\right)\right)\geq\int_{\mathcal{R}_{X}\cap
B(x_{0},\sqrt{2H_{n}})}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg\geq\frac{1}{2}.$

Let $(z_{i},-1)$ be an $H_{n}$-center of $(x_{i},0)$ with respect to the Ricci
flow $g^{i}_{t}$ for each $i$. Then, discarding finitely many terms if
necessary, we may apply [Bam20a, Proposition 3.13] again, to obtain that

$\displaystyle d_{g^{i}_{-1}}(z_{i},z_{i}^{\prime})\leq 2\sqrt{2H_{n}}\leq
3\sqrt{H_{n}}\quad\text{ for all }\quad i.$ (4.7)

(This is because, if (4.7) were false, then the total measure of
$\nu_{x_{i},0\,|\,-1}$ would exceed $1$). Consequently, we have

$\displaystyle
d^{g^{i}_{-1}}_{W_{1}}\left(\nu_{x_{i},0\,|\,-1},\delta_{z_{i}^{\prime}}\right)\leq$
$\displaystyle\
d^{g^{i}_{-1}}_{W_{1}}\left(\nu_{x_{i},0\,|\,-1},\delta_{z_{i}}\right)+d_{g^{i}_{-1}}(z_{i}^{\prime},z_{i})$
$\displaystyle\leq$ $\displaystyle\ 4\sqrt{H_{n}}.$

Taking into account [Bam20a, Corollary 5.11] and the scalar curvature lower
bound (3.6), we obtain

$\displaystyle\mathcal{N}_{z_{i}^{\prime},-1}(1)\leq\mathcal{N}_{x_{i},0}(2)+C(n)\quad\text{
whenever $i$ is large enough}.$

The fact that $\lim_{i\to\infty}\mathcal{N}_{x_{i},0}(2)=W$ then implies that

$\mathcal{N}_{z_{i}^{\prime},-1}(1)\leq W+C(n)\quad\text{ whenever $i$ is
large enough.}$

Thus, by [Bam20a, Theorem 8.1], we have

$\displaystyle\left|B_{g^{i}_{-1}}(z_{i}^{\prime},\sqrt{2H_{n}})\right|\leq
C(n)\exp(\mathcal{N}_{z_{i}^{\prime},-1}(2H_{n}))\leq
C(n)\exp(\mathcal{N}_{z_{i}^{\prime},-1}(1))\leq C(n)e^{W},$ (4.8)

for all large $i$. Here we have also applied the scalar curvature bound (3.6).
Taking this to the limit and applying Fatou’s lemma, we have

$\displaystyle\left|\mathcal{R}_{X}\cap B(x_{0},\sqrt{2H_{n}})\right|\leq
C(n)e^{W}.$

This estimate, combined with (4.6), implies that

$\displaystyle\frac{1}{2}\leq\sup_{\mathcal{R}_{X}\cap
B(x_{0},\sqrt{2H_{n}})}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\cdot\left|\mathcal{R}_{X}\cap
B(x_{0},\sqrt{2H_{n}})\right|\leq C(n)\sup_{\mathcal{R}_{X}\cap
B(x_{0},\sqrt{2H_{n}})}e^{-f_{0}+W}.$

Hence

$\inf_{\mathcal{R}_{X}\cap B(x_{0},\sqrt{2H_{n}})}(f_{0}-W)\leq C(n),$

and by (4.5),

$f_{0}(x_{0})-W\leq C(n).$

This finishes the proof of the upper bound. ∎

In the proof above, we have already shown the volume upper estimate Corollary
1.2.

###### Proof of Corollary 1.2.

In fact, when proving (4.8), we can apply [Bam20a, Theorem 8.1] to obtain a
stronger inequality

$\displaystyle\left|B_{-1}(z_{i}^{\prime},r)\right|\leq
C(n)e^{W}r^{n},\quad\text{ for all }\quad 1\leq r^{2}\leq T_{i}/4,$

whenever $i$ is sufficiently large; taking this to the limit, the corollary
follows.

∎

A similar argument can be applied to estimate the volume ratio at each point.
Nevertheless, the constant becomes worse if the base point is not a center. We
shall leave the proof of Corollary 1.3 to the reader.

###  4.3 Scalar curvature near a center

In this subsection, we shall estimate the constant in Lemma 4.2 in terms of

$\displaystyle c_{R}:=$ $\displaystyle\
\int_{\mathcal{R}_{X}}R_{g}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$

and the soliton entropy $W$ ($c_{R}$ is the constant defined in (1.1)). Note
that $c_{R}\geq 0$ because of Corollary 3.4, and $c_{R}\leq\frac{n}{2}$ due to
[Bam20a, Proposition 5.13], Fatou’s lemma, and the nature of local smooth
convergence on the regular part. The following lemma shall be used in the
proof of Theorem 1.4.

######  Lemma 4.3.

Whenever $c_{R}>0$, we have

$\displaystyle|2t|\int_{\mathcal{R}_{2t}}R_{g_{2t}}\,d\nu_{x_{0}(t)\,|\,2t}\geq
c(n,c_{R},W)>0\quad\text{ for all }\quad t\in(-\infty,0).$

###### Proof.

By Lemma (4.2), we need only to prove the lemma for $t=-1$. We argue by
contradiction. Let
$\\{(\mathcal{X}^{i},(\nu^{i}_{t})_{t<0})\\}_{i=1}^{\infty}$ be a sequence of
counterexamples (with models $(X^{i},d^{i},\nu^{i})$ and regular parts
$(\mathcal{R}_{X}^{i},g^{i},f_{0}^{i})$). Namely, we have

$\displaystyle c_{R}^{i}\geq c_{0}>0,\quad W^{i}\geq W_{0}>-\infty,\quad\text{
for each }i,$

but

$\displaystyle\int_{\mathcal{R}^{i}_{-2}}R_{g^{i}_{-2}}\,d\nu_{x_{0}^{i}\,|\,-2}\searrow
0.$ (4.9)

We may now extract a subsequence which $\mathbb{F}$-converges to a metric flow
$(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t<0})$. We also assume that the
convergence is time-wise at both $t=-1$ and $t=-2$. It is easy to see that the
limit is also a noncollapsed $\mathbb{F}$-limit metric soliton. Let
$(X^{\infty},d^{\infty},\nu^{\infty})$ be its model and
$(\mathcal{R}_{X}^{\infty},g^{\infty},f_{0}^{\infty})$ the regular part.

By [CMZ23b], we actually have a local smooth convergence from
$(\mathcal{R}_{X}^{i},g^{i},f_{0}^{i})$ to
$(\mathcal{R}_{X}^{\infty},g^{\infty},f_{0}^{\infty})$. Combining the
quadratic estimate of the potential function (Theorem 1.1), the quadratic
upper bound of $R$ given by the third equation in (2.8), and the volume upper
bound Corollary 1.2, we know that the integrals

$c_{R}^{i}:=\int_{\mathcal{R}_{X}^{i}}R_{g^{i}}(4\pi)^{-\frac{n}{2}}e^{-f_{0}^{i}}\,dg^{i}$

are uniformly negligible outside uniformly large compact sets. This then
implies that

$\displaystyle\int_{\mathcal{R}_{X}^{\infty}}R_{g^{\infty}}(4\pi)^{-\frac{n}{2}}e^{-f_{0}^{\infty}}\,dg^{\infty}=\lim_{i\to\infty}c_{R}^{i}\geq
c_{0}>0.$

Thus, the scalar curvature is positive everywhere on
$\mathcal{R}^{\infty}\subset\mathcal{X}^{\infty}$ by Corollary 3.4.

On the other hand, by Lemma 4.1 and [Bam20b, Theorem 6.49], the conjugate heat
kernels $(\nu_{x^{i}_{0}\,|\,t})_{t<-1}$ converge to a conjugate heat flow
$(\mu^{\infty}_{t})_{t<-1}$ on $\mathcal{X}^{\infty}$. The convergence can
also be regarded as a local smooth one according to [Bam20c, Theorem 9.31]. By
Fatou’s lemma and (4.9), we have

$\int_{\mathcal{R}^{\infty}_{-2}}R_{g^{\infty}_{-2}}\,d\mu^{\infty}_{-2}=0.$

Being a conjugate heat flow, $\mu^{\infty}_{-2}$ is fully-supported, which
contradicts the fact that $R_{g^{\infty}}$ is positive everywhere on
$\mathcal{R}^{\infty}$.

∎

##  5 A lower bound for the scalar curvature

We shall prove Theorem 1.4 in this section. Consider a non-Ricci-flat metric
soliton in the statement of the theorem. By Lemma 4.2 and Lemma 4.3, we fix
the constant $c_{0}$ by

$\displaystyle|2t|\int_{\mathcal{R}_{2t}}R_{g_{2t}}\,d\nu_{x_{0}(t)\,|\,2t}\equiv
2c_{0}\geq 2c(n,W,c_{R})>0\quad\text{ for all }\quad t\in(-\infty,0).$ (5.1)

By Theorem 3.1, we pick a sequence of $n$-dimensional Ricci flows
$\\{(M^{i},g_{i,t})_{t\in[-T_{i},0]}\\}_{i=1}^{\infty}$ satisfying (3.1) and
(3.2). In particular, by discarding finitely many terms, we assume

$\displaystyle\mathcal{N}_{x_{i},0}(\tau)\geq W-1\quad\text{ for all
}\quad\tau\in(0,T_{i}].$ (5.2)

By the standard scalar curvature lower estimate (3.6), we may, after replacing
each $T_{i}$ with $T_{i}/2n$, assume that for each $i$,

$\displaystyle R_{g^{i}_{t}}\geq-\frac{1}{T_{i}}\quad\text{ on }\quad
M^{i}\times[-T_{i},0].$ (5.3)

Let $z_{i,t}$ ($t\in[-T_{i},0)$) be the points defined by the property that

$\displaystyle(z_{i,t},t)\text{ \emph{is an $H_{n}$-center of $(x_{i},0)$ with
respect to the Ricci flow $g_{i,t}$.}}$

###  5.1 Local lower bound of the scalar curvature near $H_{n}$-centers

In order to apply the argument in [CCMZ23], we will need to obtain a uniform
lower bound near the $H_{n}$-centers of the limiting sequence. Precisely, we
shall prove the following theorem in this subsection.

######  Theorem 5.1.

By reducing $T_{i}$ if necessary, we have $T_{i}\nearrow+\infty$ satisfying
the following property. For any $D>0$, there is a positive number $a=a(D)$
(depending also on the constant $c_{0}$ in (5.1)), such that whenever $i$ is
large enough, we have

$\displaystyle(1+|t|)R_{g_{i,t}}(x)\geq a\quad\text{ for all }\quad x\in
B_{g_{i,t}}\left(z_{i,t},D\sqrt{1+|t|}\right)\quad\text{ and }\quad
t\in[-T_{i},0).$

The key technique of the subsection is presented in the proposition below.

######  Proposition 5.2.

For any $D>0$ there is a positive number $a=a(D)$ (depending also on the
constant $c_{0}$ in (5.1)) with the following property. For any $\tau>0$ and
any $\alpha\in(0,1/2]$, if $i\geq\underline{I}(D,c_{0},\tau)$, then

$\displaystyle|t|R_{g_{i,t}}(x)\geq a\quad\text{ for all }\quad x\in
B_{g_{i,t}}(z_{i,t},D\sqrt{|t|})\text{ and }\quad
t\in[-(1+\alpha)\tau,-(1-\alpha)\tau].$

###### Proof.

Let us fix $\tau>0$ and $\alpha\in(0,1/2]$. Assume that $i$ is large enough
such that $T_{i}\gg\tau$. By [Bam20c, Theorem 2.5] and [Bam20a, Theorem 9.31],
we can find a sequence of points $z_{i}^{\prime}\in M^{i}$, such that

$\displaystyle(z_{i}^{\prime},-2\tau)\xrightarrow{\quad
i\to\infty\quad}x_{0}(-2\tau).$

The convergence of points is either via the diffeomorphisms in the definition
of the local smooth convergence, or understood to be the convergence of points
within a correspondence. Note that, arguing as the proof of (4.7) (at scale
$\sqrt{\tau}$ instead of $1$), we also have

$\displaystyle{\rm dist}_{g_{i,-2\tau}}(z_{i,-2\tau},z^{\prime}_{i})\leq
C(n)\sqrt{\tau}\quad\text{ for all $i$ large enough}.$ (5.4)

Since, by [Bam20a, Theorem 9.31], we have

$R_{g_{i,-4\tau}}\longrightarrow R_{g_{-4\tau}},\qquad\qquad
K^{i}(z_{i}^{\prime},-2\tau\,|\,\cdot,-4\tau)\longrightarrow
K(x_{0}(-2\tau)\,|\ \cdot\ )\bigg{|}_{\mathcal{R}_{-4\tau}},$

locally smoothly on $\mathcal{R}_{-4\tau}$, Fatou’s lemma and (5.3) imply that

$\displaystyle\liminf_{i\to\infty}4\tau\int_{M^{i}}\left(R_{g^{i}_{-4\tau}}+\frac{1}{T_{i}}\right)\,d\nu_{z^{\prime}_{i},-2\tau\,|\,-4\tau}\geq
4\tau\int_{\mathcal{R}_{-2\tau}}R_{g_{-4\tau}}d\nu_{x_{0}(-2\tau)\,|\,-4\tau}=2c_{0}.$

It then follows that

$\displaystyle\int_{M^{i}}R_{g^{i}_{-4\tau}}d\nu_{z^{\prime}_{i},-2\tau\,|\,-4\tau}\geq\frac{c_{0}}{4\tau}\quad\text{
for all $i$ large enough.}$ (5.5)

Next, we define

$\displaystyle u_{i}:M^{i}\times[-4\tau,-2\tau]$ $\displaystyle\
\to\mathbb{R}_{+},$ $\displaystyle(x,t)$ $\displaystyle\
\mapsto\int_{M^{i}}R_{g^{i}_{-4\tau}}d\nu_{x,t\,|\,-4\tau}+\frac{1}{T_{i}}.$

Note that this is a positive solution to the heat equation. While the scalar
curvature is a supersolution to the heat equation, we always have
$R_{g_{i,t}}(x)\geq u_{i}(x,t)-\frac{1}{T_{i}}$ for $t\geq-4\tau$. [Bam20a,
Proposition 5.13] provides an upper bound for $u_{i}$, namely,

$\displaystyle
0<u_{i}\leq\frac{1}{\tau}\left(\frac{n}{2}+\frac{\tau}{T_{i}}\right)\quad\text{
on }\quad M^{i}\times[-3\tau,-2\tau].$

Since, by (5.5),
$u_{i}(z_{i}^{\prime},-2\tau)\geq\frac{1}{\tau}\left(\frac{c_{0}}{4}+\frac{\tau}{T_{i}}\right)$,
we can apply [Bam20a, Theorem 4.1] to $u_{i}$ on $M^{i}\times[-3\tau,-2\tau]$
to obtain a local lower bound. For any $D^{\prime}>0$, the following holds
whenever $i$ is large enough,

$\displaystyle u_{i}(x,-2\tau)\geq$ $\displaystyle\
\frac{1}{\tau}\left(\frac{n}{2}+\frac{\tau}{T_{i}}\right)\Phi\left(\Phi^{-1}\left(\frac{\frac{c_{0}}{4}+\frac{\tau}{T_{i}}}{\frac{n}{2}+\frac{\tau}{T_{i}}}\right)-D^{\prime}\right)$
$\displaystyle\geq$ $\displaystyle\
\frac{1}{\tau}\left(\frac{n}{2}+\frac{\tau}{T_{i}}\right)\Phi\left(\Phi^{-1}\left(\frac{c_{0}}{2n}\right)-D^{\prime}\right)\quad\text{
for all}\quad x\in B_{g_{i,-2\tau}}(z_{i}^{\prime},D^{\prime}\sqrt{\tau}).$

Here $\Phi$ is the function defined in [Bam20a, §4.1] In view of (5.4), this
can also be written as

$\displaystyle
R_{g_{i,-2\tau}}(x)\geq\frac{1}{\tau}\left(\frac{n}{2}+\frac{\tau}{T_{i}}\right)\Phi\left(\Phi^{-1}\left(\frac{c_{0}}{2n}\right)-D^{\prime}-C(n)\right)-\frac{1}{T_{i}}$
(5.6) $\displaystyle\text{ for all}\quad x\in
B_{g_{i,-2\tau}}(z_{i,-2\tau},D^{\prime}\sqrt{\tau}).$

Finally, it remains to extend the local estimate in space (5.6) to a space-
time neighborhood. To this end, let us fix an arbitrary point $(x,t)\in
M^{i}\times[-(1+\alpha)\tau,-(1-\alpha)\tau]$ such that $x\in
B_{g_{i,t}}(z_{i,t},D\sqrt{|t|})$. Recall that $\alpha\in(0,1/2]$. Let
$(z^{\prime},-2\tau)$ be an $H_{n}$-center of $(x,t)$ and let
$(z^{\prime\prime},-2\tau)$ be an $H_{n}$-center of $(z_{i,t},t)$. Since, by
[CMZ23a, Lemma 6.3], we have ${\rm
dist}_{g_{i,-2\tau}}(z_{i,-2\tau},z^{\prime\prime})\leq C_{0}(n)\sqrt{\tau}$,
it holds that

$\displaystyle{\rm dist}_{g_{i,-2\tau}}(z_{i,-2\tau},z^{\prime})\leq$
$\displaystyle\ {\rm dist}_{g_{i,-2\tau}}(z_{i,-2\tau},z^{\prime\prime})+{\rm
dist}_{g_{i,-2\tau}}(z^{\prime},z^{\prime\prime})$ $\displaystyle\leq$
$\displaystyle\ C_{0}(n)\sqrt{\tau}+{\rm
dist}_{W_{1}}^{g_{i,-2\tau}}(\delta_{z^{\prime}},\nu_{x,t\,|\,-2\tau})+{\rm
dist}_{W_{1}}^{g_{i,-2\tau}}(\delta_{z^{\prime\prime}},\nu_{z_{i,t},t\,|\,-2\tau})$
$\displaystyle\ +{\rm
dist}_{W_{1}}^{g_{i,-2\tau}}(\nu_{z_{i,t},t\,|\,-2\tau},\nu_{x,t\,|\,-2\tau})$
$\displaystyle\leq$ $\displaystyle\ (C_{1}(n)+D)\sqrt{\tau}.$

In combination with [Bam20a, Proposition 3.13], we have if
$D^{\prime}:=D+C_{1}(n)+\sqrt{2H_{n}}$, then

$\displaystyle\nu_{x,t\,|\,-2\tau}\left(B_{g_{i,-2\tau}}\left(z_{i,-2\tau},D^{\prime}\sqrt{\tau}\right)\right)\geq\tfrac{1}{2}.$
(5.7)

Taking into account (5.6), we have the following estimate whenever $i$ is
large enough.

$\displaystyle\int_{M^{i}}R_{g_{i,-2\tau}}\,d\nu_{x,t\,|\,-2\tau}=$
$\displaystyle\
\int_{B_{g_{i,-2\tau}}\left(z_{i,-2\tau},D^{\prime}\sqrt{\tau}\right)}R_{g_{i,-2\tau}}\,d\nu_{x,t\,|\,-2\tau}$
$\displaystyle\ +\int_{M^{i}\setminus
B_{g_{i,-2\tau}}\left(z_{i,-2\tau},D^{\prime}\sqrt{\tau}\right)}R_{g_{i,-2\tau}}\,d\nu_{x,t\,|\,-2\tau}$
$\displaystyle\geq$ $\displaystyle\
\frac{1}{2}\left(\frac{1}{\tau}\left(\frac{n}{2}+\frac{\tau}{T_{i}}\right)\Phi\left(\Phi^{-1}\left(\frac{c_{0}}{2n}\right)-D^{\prime}-C(n)\right)-\frac{1}{T_{i}}\right)-\frac{1}{2}\cdot\frac{1}{T_{i}}$
$\displaystyle=$ $\displaystyle\
\frac{1}{2\tau}\left(\frac{n}{2}+\frac{\tau}{T_{i}}\right)\Phi\left(\Phi^{-1}\left(\frac{c_{0}}{2n}\right)-D-C_{2}(n)\right)-\frac{1}{T_{i}}.$

Being a supersolution to the heat equation, $R$ always satisfies

$\displaystyle
R_{g_{i,t}}(x)\geq\int_{M^{i}}R_{g_{i,-2\tau}}\,d\nu_{x,t\,|\,-2\tau}.$

This finishes the proof of the proposition. ∎

Finally, to prove Theorem 5.1, it remains only to deal with the scalar
curvature near the $0$-time-slices. This is covered by the following
Proposition.

######  Proposition 5.3.

For any $D>0$ there is a positive number $a=a(D)$ (depending also on the
constant $c_{0}$ in (5.1)) with the following property. If
$i\geq\underline{I}(D,c_{0})$, then

$\displaystyle R_{g_{i,t}}(x)\geq a\quad\text{ for all }\quad x\in
B_{g_{i,t}}(z_{i,t},D)\text{ and }\quad t\in[-1,0).$

###### Proof.

The proof is a simpler version of the previous proposition. Fix $\tau=2$ in
Proposition 5.2. Then for any $D^{\prime}>0$ and
$i\geq\underline{I}(D^{\prime},c_{0})$, we have that

$\displaystyle R_{g_{i,-2}}(x)\geq a(D^{\prime})\quad\text{ for all }\quad
x\in B_{g_{i,-2}}(z_{i,-2},D^{\prime}).$

Arguing in the same way as the proof of (5.7), we have that, for any $(x,t)\in
B_{g_{i,t}}(z_{i,t},D)$, where $D>0$ and $t\in[-1,0)$, if $D^{\prime}=D+C(n)$,
then

$\displaystyle\nu_{x,t\,|\,-2}\left(B_{g_{i,-2}}(z_{i,-2},D^{\prime})\right)\geq\frac{1}{2}.$

Thus,

$\displaystyle\int_{M_{i}}R_{g_{i,-2}}\,d\nu_{x,t\,|\,-2}=$ $\displaystyle\
\int_{B_{g_{i,-2}}(z_{i,-2},D^{\prime})}R_{g_{i,-2}}\,d\nu_{x,t\,|\,-2}+\int_{M^{i}\setminus
B_{g_{i,-2}}(z_{i,-2},D^{\prime})}R_{g_{i,-2}}\,d\nu_{x,t\,|\,-2}$
$\displaystyle\geq$ $\displaystyle\ \frac{1}{2}a(D+C(n))-\frac{1}{2T_{i}}.$

This finishes the proof in the same way as Proposition 5.2. ∎

###  5.2 Proof of the scalar curvature lower bound

In this subsection, we conclude the proof of Theorem 1.4 following the
argument of [CCMZ23]. Let us recall the following conjugate heat kernel
estimate [Bam20a, Theorem 7.2].

$\displaystyle(4\pi|t|)^{-\frac{n}{2}}e^{-f_{i}(x,t)}:=K_{i}(x_{i},0\,|\,x,t)\leq\frac{C(n)}{(4\pi|t|)^{\frac{n}{2}}}\exp\left(-W-\frac{{\rm
dist}^{2}_{g_{i,t}}(z_{i,t},x)}{9|t|}\right)$

for all $(x,t)\in M^{i}\times[-T_{i},0)$. Note that we have applied (5.2) and
(5.3). This estimate is equivalent to

$\displaystyle f_{i}(x,t)\geq-C_{0}+1+W+\frac{{\rm
dist}^{2}_{g_{i,t}}(z_{i,t},x)}{9|t|}\quad\text{ for all }\quad(x,t)\in
M^{i}\times[-T_{i},0),$ (5.8)

where $C_{0}$ is a dimensional constant. Let

$\tau=-t=|t|$

throughout the rest of the subsection. Define the auxiliary function

$\displaystyle\rho_{i}(x,t):=\sqrt{1+\tau}\left(f_{i}(x,t)+C_{0}-W\right).$

Obviously, (5.8) implies that

$\displaystyle\rho_{i}(x,t)\geq\sqrt{1+\tau}\left(1+\frac{{\rm
dist}^{2}_{g_{i,t}}(z_{i,t},x)}{9\tau}\right)\quad\text{ for all
}\quad(x,t)\in M^{i}\times[-T_{i},0).$ (5.9)

Next, we define the function with which we shall apply the maximum principle.
Let

$A=A(n,W):=C_{0}-W+\tfrac{n}{2}$

and

$\displaystyle D=D(A,n)<+\infty,\quad\quad a_{0}=a(D)>0,\quad\quad
c=c(a_{0},D)>0$

to be fixed in the course of the proof; here $a(D)$ is the constant given by
Theorem 5.1. For each $i$ large enough, we define

$\displaystyle
F_{i}(x,t):=\sqrt{1+\tau}\,R_{g_{i,t}}(x)-c\rho_{i}^{-1}(x,t)-4cA\sqrt{1+\tau}\,\rho_{i}^{-2}(x,t),\quad(x,t)\in
M^{i}\times[-T_{i},0).$

Arguing in the same way as in [CCMZ23], we have:

######  Lemma 5.4.

The following differential inequality holds on $M^{i}\times[-T_{i},0]$.

$\displaystyle\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{i,t}}\right)F_{i}\geq-\tfrac{1}{2}F_{i}+cA\sqrt{1+\tau}\,\rho_{i}^{-3}\left(\rho_{i}-8A\sqrt{1+\tau}\,\right)-\frac{1}{2\sqrt{1+\tau}}\cdot\frac{1}{T_{i}}.$

###### Proof.

We shall suppress the indices in the proof of the Lemma. The conjugate heat
equation $\partial_{t}f=-\Delta f+|\nabla f|^{2}-R+\frac{n}{2\tau}$ implies
that

$\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)f=f-\tfrac{n}{2}-w,$

where

$w:=\tau(2\Delta f-|\nabla f|^{2}+R)+f-n\leq 0$

by Perelman’s Harnack inequality [Per02]. Thus

$\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\rho\geq-\frac{\tau}{2(1+\tau)}\rho+\sqrt{1+\tau}\left(f-\tfrac{n}{2}\right)\geq\tfrac{1}{2}\rho-A\sqrt{1+\tau},$

due to our choice of $A=C_{0}-W+\frac{n}{2}$ and (5.8). Then

$\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\rho^{-1}=-\tau\rho^{-2}\left(\tfrac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\rho-2\tau\rho^{-3}|\nabla\rho|^{2}\leq-\tfrac{1}{2}\rho^{-1}+A\sqrt{1+\tau}\rho^{-2}.$

Similarly, we have

$\displaystyle\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\rho^{-2}$ $\displaystyle\
\leq-\rho^{-2}+2A\sqrt{1+\tau}\rho^{-3},$
$\displaystyle\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\left(\sqrt{1+\tau}\rho^{-2}\right)$ $\displaystyle\
\leq-\sqrt{1+\tau}\rho^{-2}+2A(1+\tau)\rho^{-3}.$

In combination, we have

$\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\left(\rho^{-1}+4A\sqrt{1+\tau}\rho^{-2}\right)\leq-\tfrac{1}{2}\left(\rho^{-1}+4A\sqrt{1+\tau}\rho^{-2}\right)-A\sqrt{1+\tau}\rho^{-3}\left(\rho-8A\sqrt{1+\tau}\right).$

The lemma follows from combining the inequality above with the following fact.

$\displaystyle\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\left(\sqrt{1+\tau}R\right)$ $\displaystyle\
=-\frac{\tau}{2\sqrt{1+\tau}}R+2\tau\sqrt{1+\tau}|\operatorname{Ric}|^{2}$
$\displaystyle\
\geq-\frac{\tau}{2\sqrt{1+\tau}}\left(R+\frac{1}{T_{i}}\right)+\frac{\tau}{2\sqrt{1+\tau}}\cdot\frac{1}{T_{i}}$
$\displaystyle\
\geq-\tfrac{1}{2}\sqrt{1+\tau}R-\tfrac{1}{2}\sqrt{1+\tau}\cdot\frac{1}{T_{i}}+\frac{\tau}{2\sqrt{1+\tau}}\cdot\frac{1}{T_{i}}$
$\displaystyle\
=-\tfrac{1}{2}\sqrt{1+\tau}R-\frac{1}{2\sqrt{1+\tau}}\cdot\frac{1}{T_{i}},$

where we have applied (5.3). ∎

The following boundary conditions can be easily checked.

######  Lemma 5.5.

If

$D>0,\quad c<\frac{a_{0}}{1+4A},\quad\text{ and }\quad
i\geq\underline{I}(c,A,D),$

then

$\displaystyle\liminf_{t\to 0-}\left(\inf_{x\in
M}F_{i}(x,t)\right)\geq-\frac{1}{T_{i}},$ (5.10)
$\displaystyle\liminf_{x\to\infty}F_{i}(x,t)\geq-\frac{\sqrt{1+T_{i}}}{T_{i}}$
(5.11) $\displaystyle\inf_{x\in
M^{i}}F_{i}(x,-T_{i})\geq-\frac{\sqrt{1+T_{i}}}{T_{i}}-\frac{c(1+4A)}{\sqrt{1+T_{i}}}.$
(5.12)

###### Proof.

We first prove (5.10). We fix an $i$ that is large enough. Let
$t\in(-\tau_{i},0)$ be a fixed time. For any point $x\in M$, if $x\in
B_{g_{i,t}}(z_{i,t},D\sqrt{1+\tau}\,)$, then Theorem 5.1 and the choices of
$a_{0}$ and $c$ imply that

$\displaystyle
F_{i}(x,t)\geq\frac{a_{0}}{\sqrt{1+\tau}}-\frac{c(1+4A)}{\sqrt{1+\tau}}\geq
0.$

If $x\not\in B_{g_{i,t}}(z_{i,t},D\sqrt{1+\tau})$, then we have

$\rho_{i}(x,t)\geq\sqrt{1+\tau}\left(1+\frac{D^{2}(1+\tau)}{\tau}\right)\geq\frac{D^{2}}{\tau},$

and thus

$\displaystyle
F_{i}(x,t)\geq-\frac{1}{T_{i}}\sqrt{1+\tau}-\frac{c\tau}{D^{2}}-\frac{4cA\tau^{2}\sqrt{1+\tau}}{D^{4}},$

where we have applied (5.3). Therefore, we have

$\inf_{x\in
M}F_{i}(x,t)\geq-C(A,D,c)\tau-\frac{1}{T_{i}}\sqrt{1+\tau}\to-\frac{1}{T_{i}}\quad\text{
as }\quad t\to 0-.$

This shows (5.10).

(5.11) and (5.12) both follow directly from the definition of $F^{i}$, the
standard scalar curvature lower bound (5.3), and the lower bound of $\rho$
(5.9). ∎

Finally, Theorem 1.4 follows from taking the proposition below to the limit.

######  Proposition 5.6.

If

$\displaystyle D\geq\underline{D}(A),\quad c<\frac{a_{0}}{1+4A},\quad\text{
and }\quad i\geq\underline{I}(c,A,D),$

then

$\displaystyle\inf_{M^{i}\times[-T_{i},0)}F_{i}\geq-2\cdot\left(\frac{\sqrt{1+T_{i}}}{T_{i}}+\frac{c(1+4A)}{\sqrt{1+T_{i}}}\right).$

###### Proof.

Assume by contradition that for some $i$ large enough (such that Lemma 5.5 is
applicable), we have

$\displaystyle\inf_{M^{i}\times[-\tau_{i},0)}F_{i}<-2\cdot\left(\frac{\sqrt{1+T_{i}}}{T_{i}}+\frac{c(1+4A)}{\sqrt{1+T_{i}}}\right)<0.$

By the boundary conditions (5.10)—(5.12), the infimum must be attained at an
interior point $(x_{0},t_{0})\in M^{i}\times(-T_{i},0)$. Then, by Lemma 5.4,

$\displaystyle 0\geq\tau\left(\frac{\partial}{\partial
t}-\Delta_{g_{i,t}}\right)F_{i}\geq-\tfrac{1}{2}F_{i}+cA\sqrt{1+\tau}\,\rho_{i}^{-3}\left(\rho_{i}-8A\sqrt{1+\tau}\,\right)-\frac{1}{2\sqrt{1+\tau}}\cdot\frac{1}{T_{i}}\quad\text{at}\quad(x_{0},t_{0}).$

Thus, the following holds true at the point $(x_{0},t_{0})$.

$\displaystyle
cA\sqrt{1+\tau}\,\rho_{i}^{-3}\left(\rho_{i}-8A\sqrt{1+\tau}\,\right)$
$\displaystyle\
\leq\frac{1}{2\sqrt{1+\tau}}\cdot\frac{1}{T_{i}}+\frac{1}{2}F_{i}$
$\displaystyle\leq\frac{1}{2\sqrt{1+\tau}}\cdot\frac{1}{T_{i}}-\frac{\sqrt{1+T_{i}}}{T_{i}}-\frac{c(1+4A)}{\sqrt{1+T_{i}}}$
$\displaystyle<0.$

Therefore,

$\displaystyle
8A\sqrt{1+|t_{0}|}\geq\rho_{i}(x_{0},t_{0})\geq\sqrt{1+|t_{0}|}\left(1+\frac{{\rm
dist}^{2}_{g_{i,t_{0}}}(z_{i,t_{0}},x_{0})}{9|t_{0}|}\right)$

It follows that

$\displaystyle{\rm
dist}_{g_{i,t_{0}}}(z_{i,t_{0}},x_{0})\leq\sqrt{72A(1+|t_{0}|)}.$

Now, if we take $D\geq\sqrt{100A}$, then Theorem 5.1 implies that

$\displaystyle
F_{i}(x_{0},t_{0})\geq\frac{a_{0}}{\sqrt{1+|t_{0}|}}-\frac{c(1+4A)}{\sqrt{1+|t_{0}|}}\geq
0.$

This is a contradiction; the proof of the proposition is complete.

∎

##  6 A local gap theorem

In this section, we prove Theorem 1.5 by constructing a cut-off function and
applying the method of [CMZ22]. Note that the cut-off function in Proposition
1.6 is also very helpful in the proof of Theorem 1.10.

###  6.1 Perelman’s functionals

In this subsection, we recall some of Perelman’s functionals, particularly
their local versions. A systematic treatment of Perelman’s local functionals
can be found in [W18]. Let $M^{n}$ be any smooth manifold and $\Omega\Subset
M$ any precompact open subset. For any Riemannian metric $g$ and real number
$\tau>0$, define

$\displaystyle\mu(\Omega,g,\tau):=\inf\left\\{\overline{\mathcal{W}}(g,u,\tau)\,\left|\,u\in
C^{\infty}_{c}(\Omega),\ \int_{M}u^{2}\,dg=1\right\\}\right.,$

where

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

The local $\nu$-functional is defined to be the infimum of the local
$\mu$-functional (on all scales or up to a certain scale):

$\displaystyle\nu(\Omega,g)$ $\displaystyle\
:=\inf_{\tau>0}\nu(\Omega,g,\tau),$ $\displaystyle\nu(\Omega,g,\tau)$
$\displaystyle\ :=\inf_{\tau\geq s>0}\nu(\Omega,g,s).$

It is clear that the local $\mu$-functional is a local logarithmic Sobolev
constant.

######  Proposition 6.1.

If $\mu(\Omega,g,\tau)>-\infty$, then

$\displaystyle\tau\int_{M}(4|\nabla u|^{2}+Ru^{2})\,dg\leq\int_{M}u^{2}\log
u^{2}\,dg+\frac{n}{2}\log(4\pi\tau)+n-\mu(\Omega,g,\tau),$

for any $u\in C_{c}^{0,1}(\Omega)$ satisfying

$\int_{M}u^{2}\,dg=1.$

A standard argument in analysis (cf. [Zh10]) also implies the following
Sobolev inequality.

######  Proposition 6.2.

If $\nu_{0}:=\nu(\Omega,g,\tau)>-\infty$ and $R\geq R_{\min}$ on $\Omega$,
then there is a positive number $c=c(n)$, such that

$\displaystyle\left(\int_{M}|u|^{\frac{2n}{n-2}}\,dg\right)^{\frac{n-2}{n}}\leq
ce^{-\frac{2\nu_{0}}{n}-c\tau
R_{\operatorname{min}}}\int_{\Omega}\bigg{(}4|\nabla
u|^{2}+\left(R_{g_{0}}-R_{\operatorname{min}}+\frac{c}{\tau}\right)u^{2}\bigg{)}\,dg,$

for any $u\in C_{c}^{0,1}(\Omega)$.

###  6.2 A preliminary soliton entropy gap

The following gap theorem of the soliton entropy is a straightforward
consequence of Bamler’s $\varepsilon$-regularity theorem. The critical idea of
the proof of Theorem 1.5 is to convert the local gap problem to the entropy
gap problem.

######  Theorem 6.3.

There is a constant $\varepsilon=\varepsilon(n)$ with the following property.
Let $\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$ be a noncollapsed
$\mathbb{F}$-limit metric soliton defined over interval $(-T,0)$, where
$T\in(0,+\infty]$. Let $W$ be its soliton entropy defined in (2.4). If
$W\geq-\varepsilon$, then $\big{(}\mathcal{X},(\nu_{t})_{t\in(-T,0)}\big{)}$
is the flat Gaussian shrinker.

###### Proof.

Let $\varepsilon(n)=\frac{1}{2}\varepsilon_{n}$, where $\varepsilon_{n}$ is
the constant in [Bam20a, Theorem 10.2]. By Theorem 3.1, there is a sequence
$\\{(M^{i},g^{i}_{t},x_{i})_{t\in[-T_{i},0]}\\}_{i=1}^{\infty}$ with
$T_{i}\nearrow+\infty$ such that (3.1) and (3.2) hold. By [Bam20a, Theorem
10.2] and (3.1), we have that

$\displaystyle\left|\operatorname{Rm}_{g^{i}}\right|\leq\frac{1}{\varepsilon^{2}T_{i}}\quad\text{
on }\quad
B_{g^{i}_{0}}\left(x_{i},\varepsilon\sqrt{T_{i}}\right)\times[-\varepsilon^{2}T_{i},0].$

It follows that the metric soliton is a smooth and flat shrinking gradient
Ricci soliton; this finishes the proof of the Theorem. ∎

###  6.3 Construction of the cut-off functions

In this subsection is the proof of Proposition 1.6. We shall construct cut-off
functions for the regular part of a metric soliton. Precisely, we shall
convert the cut-off function [Bam20c, Lemma 15.27] to one that can be applied
to metric solitons.

Let $(\mathcal{X},\nu_{t})$ be an $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton. Denote by $(X,d,\nu)$ its model and
$(\mathcal{R}_{X},g,f_{0})$ the regular part of the model. Let
$x_{0}\in\mathcal{R}_{X}$ be a center of the soliton and $W$ the soliton
entropy. Let $x_{0}(t)$ be the integral curve of
$\partial_{\mathfrak{t}}-\nabla f$ with $x_{0}(-1)=x_{0}\in\mathcal{R}_{-1}$.
By Theorem 3.1, we may assume that the soliton is defined over $(-\infty,0)$,
and that there is a sequence
$\\{(M^{i},g^{i}_{t},x_{i})_{t\in[-T_{i},0]}\\}_{i=1}^{\infty}$ satisfying
$T_{i}\nearrow+\infty$, (3.1), and (3.2). Since in [Bam20c, Lemma 15.27], the
$P^{*}$-parabolic neighborhood is defined using the limit of
$\nu_{x_{i},0\,|\,t}$, which is $\nu_{t}$ in our case, let us define

$\displaystyle
P^{*}_{\nu}(A,-T):=\left\\{x\in\mathcal{X}_{[-T,0)}\,\left|\,d^{-T}_{W_{1}}(\nu_{x\,|\,-T},\nu_{-T})<A\right\\}\right.$

######  Lemma 6.4.

For any $x\in B_{t}\big{(}x_{0}(t),A\big{)}\subset\mathcal{X}_{t}$, where
$t\in[-T,0)$, we have

$x\in P^{*}_{\nu}\left(A+\sqrt{H_{n}T},-T\right).$

###### Proof.

By Lemma 4.1, the fact $d_{W_{1}}\leq\sqrt{\operatorname{Var}}$, and the
monotonicity of the $W_{1}$-Wasserstein distance (cf. [Bam20c, Proposition
4.17]), we have

$\displaystyle d^{-T}_{W_{1}}(\nu_{x\,|\,-T},\nu_{-T})$ $\displaystyle\ \leq
d^{t}_{W_{1}}(\delta_{x},\nu_{t})$ $\displaystyle\ \leq
d_{t}(x_{0}(t),x)+d^{t}_{W_{1}}(\delta_{x_{0}(t)},\nu_{t})$ $\displaystyle\
\leq A+\sqrt{H_{n}T}.$

This proves the lemma. ∎

###### Proof of Proposition 1.6.

For fixed $\sigma>0$ and $A>\underline{A}(n)$, Lemma 6.4 implies that,

$\displaystyle B_{t}(x_{0}(t),A\sqrt{|t|})\subset
P^{*}_{\nu}(2A,-2)\quad\text{ for any }\quad t\in[-2,0).$ (6.1)

Now, if $r\leq\overline{r}(A,W,\sigma)$, we may consider the cut-off function
$\eta_{r}$ defined by Bamler in [Bam20c, Lemma 15.27]. Note that in the
statement thereof, $\Delta=4$, and the parameters $Y_{0}$ and $\tau_{0}$ are
absorbed into the parameter $W$ in our case due to (3.1).

By (6.1) and [Bam20c, Lemma 15.27], we have

$\displaystyle\ \int_{-2}^{-\frac{1}{2}}\left(\int_{\mathcal{R}_{t}\cap
B_{t}(x_{0}(t),A\sqrt{|t|})\cap\\{0\leq\eta_{r}<1\\}}dg_{t}\right)dt$
$\displaystyle\leq$ $\displaystyle\
\int_{-2}^{0}\int_{\\{0<\tilde{r}_{\operatorname{Rm}}\leq 2r\\}\cap
P^{*}(2A,-2)\cap\mathcal{R}_{t}}dg_{t}dt$ $\displaystyle\leq$ $\displaystyle\
Cr^{4-\sigma},$

where $C=C(A,W,\sigma)$. Thus, we may find a time $t\in[-2,-1/2]$, such that

$\int_{\mathcal{R}_{t}\cap
B_{t}(x_{0}(t),A\sqrt{|t|})\cap\\{0\leq\eta_{r}<1\\}}dg_{t}\leq
2Cr^{4-\sigma}.$

Since the geometry of $\mathcal{R}_{t}\subset\mathcal{X}_{t}$ and
$\mathcal{R}_{X}\subset X$ differ only by a scaling with bounded factor
$|t|\in[1/2,2]$, and since $B_{t}(x_{0}(t),A\sqrt{|t|})\subset\mathcal{X}_{t}$
corresponds to $B(x_{0},A)\subset X$, we see that the time-$t$-slice of
$\eta_{r}$, when identified as a function on $X$ via the isometry
$\overline{\phi}_{t}$ defined in (3.4), satisfies Properties (1)—(4) in the
statement of the proposition.

Finally, to see Property (5), let us fix an arbitrary $x\in\mathcal{R}_{X}$
and let $x(t)$ ($t\in[-2,-1/2]$) be its trajectory. We may find a positive
number $c$, such that for all $t\in[-2,-1/2]$, it holds that the parabolic
neighborhood $B_{t}(x(t),2c)\times[t-4c^{2},t+4c^{2}]\subset\mathcal{R}$ is
unscathed and

$\displaystyle|\operatorname{Rm}_{g_{t}}|\leq(2c)^{-2}\quad\text{ on }\quad
B_{t}(x(t),2c)\times[t-4c^{2},t+4c^{2}].$

On the other hand, for each $t\in[-2,-1/2]$, we can also find
$x_{i}^{\prime}\in M^{i}$, such that

$\displaystyle(x^{\prime}_{i},t)\xrightarrow{\quad i\to\infty\quad}x(t).$

This convergence is either via the local diffeomorphisms which define the
smooth convergence, or is a convergence of points within a correspondence. By
the local smooth convergence property [Bam20c, Theorem 2.5] at $x(t)$, we have
that

$r_{\operatorname{Rm}}(x_{i}^{\prime},t)\geq c\qquad\text{ for all $i$ large
enough},$

where $r_{\operatorname{Rm}}$ is the curvature scale defined in [Bam20a,
Definition 10.1]. Thus, we have $\tilde{r}_{\operatorname{Rm}}(x(t))\geq c$
for all $t\in[-2,-1/2]$ due to [Bam20c, Lemma 15.16]. By Bamler’s definition
of $\eta_{r}$, we have $\eta_{r}(x(t))=1$ for all $t\in[-2,-1/2]$ whenever
$r<c/2$. Thus, in our choice of $\eta_{r}$ above, no matter which time slice
it is, we always have $x\in\\{\eta_{r}=1\\}$ if $r<c/2$; this shows property
(5).

∎

###  6.4 Proof of the local gap theorem

In this subsection, we prove Theorem 1.5. The settings are the same as that of
the previous subsection. Let

$\sigma>0,\quad 0<\delta\leq\overline{\delta}(n),\quad
0<r\leq\overline{r}(\delta,W,\sigma)$

to be determined. Let $\eta$ be a standard cutoff function on $\mathbb{R}$
such that

$\eta|_{(-\infty,1/2]}=1,\quad\eta|_{[1,\infty)}=0,\quad-3\sqrt{\eta}\leq\eta^{\prime}\leq
0.$

Define

$\displaystyle\phi_{\delta}(x)=\eta\left(\frac{d(x,x_{0})}{\delta^{-1}}\right)=\eta\left(\frac{d_{g}(x,x_{0})}{\delta^{-1}}\right).$
(6.2)

Then we obviously have that

$\displaystyle\phi_{\delta}\in
C^{0,1}\big{(}B(x_{0},1/\delta)\cap\mathcal{R}_{X}\big{)},$ (6.3)
$\displaystyle|\nabla\phi_{\delta}|\leq 3\delta\quad\text{ on
}\quad\mathcal{R}_{X},$ $\displaystyle\phi_{\delta}(x)=1\quad\text{ on }\quad
B(x_{0},1/2\delta)\cap\mathcal{R}_{X}.$

We shall apply the logarithmic Sobolev inequality given by the local
$\mu$-functional (Proposition 6.1) to the test function
$\eta_{r}^{2}\phi_{\delta}^{2}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\in
C_{c}^{0,1}\big{(}B(x_{0},1/\delta)\cap\mathcal{R}_{X}\big{)}$, where
$\eta_{r}$ is the cut-off function given by Proposition 1.6. Let us first of
all normalize its integral. If $\delta^{-1}>2\sqrt{2H_{n}}$, then we have

$\displaystyle V$
$\displaystyle:=\int_{\mathcal{R}_{X}}\eta_{r}^{2}\phi_{\delta}^{2}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
(6.4)
$\displaystyle=\int_{\mathcal{R}_{X}}\phi_{\delta}^{2}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg-\int_{\mathcal{R}_{X}}\phi_{\delta}^{2}(1-\eta_{r}^{2})(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle\geq\int_{B\big{(}x_{0},\frac{1}{2}\delta^{-1}\big{)}}\,d\nu-\int_{\\{0\leq\eta_{r}<1\\}\cap\mathcal{R}_{X}\cap
B(x_{0},\delta^{-1})}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle\geq\nu\left(B\left(x_{0},\tfrac{1}{2}\delta^{-1}\right)\right)-C(n)e^{-W}\int_{\\{0\leq\eta_{r}<1\\}\cap\mathcal{R}_{X}\cap
B(x_{0},\delta^{-1})}\,dg$ $\displaystyle\geq
1-4H_{n}\delta^{2}-C(\delta,W,\sigma)r^{4-\sigma},$

where we have applied Lemma 4.1 (in combination with [Bam20a, Proposition
3.13]) and Theorem 1.1. Define

$u:=\phi_{\delta}\eta_{r}\sqrt{(4\pi)^{-\frac{n}{2}}e^{-f_{0}}/V},$

and we shall use it as the test function for the local $\mu$-functional. Note
that

$\int_{\mathcal{R}_{X}}u^{2}\,dg=1\qquad\text{ and }\qquad u\in
C^{0,1}_{c}(\mathcal{R}_{X}).$

By Proposition 6.1, we have

$\displaystyle\mu\big{(}B(x_{0},\delta^{-1})\cap\mathcal{R}_{X},g,1\big{)}$
(6.5) $\displaystyle\leq\int_{\mathcal{R}_{X}}\left(4|\nabla
u|^{2}+Ru^{2}\right)\,dg-\int_{\mathcal{R}_{X}}u^{2}\log
u^{2}\,dg-\tfrac{n}{2}\log(4\pi)-n$
$\displaystyle=\int_{\mathcal{R}_{X}}\left(4\left|\nabla\log(\phi_{\delta}\eta_{r})-\tfrac{1}{2}\nabla
f_{0}\right|^{2}u^{2}+Ru^{2}\right)\,dg$
$\displaystyle\qquad-\int_{\mathcal{R}_{X}}\left(\log(\phi_{\delta}\eta_{r})^{2}-\tfrac{n}{2}\log
4\pi-f_{0}-\log V\right)u^{2}\,dg-\tfrac{n}{2}\log(4\pi)-n$
$\displaystyle=\frac{1}{V}\int_{\mathcal{R}_{X}}\left(4\left|\nabla(\phi_{\delta}\eta_{r})\right|^{2}-4(\phi_{\delta}\eta_{r})\left\langle\nabla(\phi_{\delta}\eta_{r}),\nabla
f_{0}\right\rangle\right)(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle\qquad+\int_{\mathcal{R}_{X}}\left(\left|\nabla
f_{0}\right|^{2}+R+f_{0}-n\right)u^{2}\,dg+\log
V-\frac{1}{V}\int_{\mathcal{R}_{X}}\left((\phi_{\delta}\eta_{r})^{2}\log(\phi_{\delta}\eta_{r})^{2}\right)(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle=\frac{1}{V}\int_{\mathcal{R}_{X}}\left(4\left|\nabla(\phi_{\delta}\eta_{r})\right|^{2}+2(\phi_{\delta}\eta_{r})^{2}\Delta
f_{0}-2(\phi_{\delta}\eta_{r})^{2}\left|\nabla
f_{0}\right|^{2}\right)(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle\qquad+\int_{\mathcal{R}_{X}}\left(\left|\nabla
f_{0}\right|^{2}+R+f_{0}-n\right)u^{2}\,dg+\log
V-\frac{1}{V}\int_{\mathcal{R}_{X}}\left((\phi_{\delta}\eta_{r})^{2}\log(\phi_{\delta}\eta_{r})^{2}\right)(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle=\int_{\mathcal{R}_{X}}\left(2\Delta f_{0}-\left|\nabla
f_{0}\right|^{2}+R+f_{0}-n\right)u^{2}\,dg+\log V$
$\displaystyle\qquad+\frac{4}{V}\int_{\mathcal{R}_{X}}|\nabla(\phi_{\delta}\eta_{r})|^{2}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg-\frac{1}{V}\int_{\mathcal{R}_{X}}\left((\phi\eta_{r})^{2}\log(\phi\eta_{r})^{2}\right)(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle=W+\log
V+\frac{4}{V}\int_{\mathcal{R}_{X}}|\nabla(\phi_{\delta}\eta_{r})|^{2}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg-\frac{1}{V}\int_{\mathcal{R}_{X}}\left((\phi_{\delta}\eta_{r})^{2}\log(\phi_{\delta}\eta_{r})^{2}\right)(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg.$

We estimate the last two terms in the above formula. First of all, by (6.3)
and Proposition 1.6, we have

$\displaystyle\
\frac{4}{V}\int_{\mathcal{R}_{X}}|\nabla(\phi_{\delta}\eta_{r})|^{2}(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
(6.6) $\displaystyle\leq$ $\displaystyle\
\frac{8}{V}\int_{\mathcal{R}_{X}}\left(\eta_{r}^{2}|\nabla\phi_{\delta}|^{2}+\phi_{\delta}^{2}|\nabla\eta_{r}|^{2}\right)(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle\leq$ $\displaystyle\
\frac{8}{V}\sup_{\mathcal{R}_{X}}|\nabla\phi_{\delta}|^{2}\cdot\nu(X)+\frac{8}{V}\int_{B(x_{0},\delta^{-1})\cap\mathcal{R}_{X}}|\nabla\eta_{r}|^{2}\,dg$
$\displaystyle\leq$ $\displaystyle\
\frac{72\delta^{2}}{V}+\frac{C(n)}{V}\int_{\\{|\nabla\eta_{r}|\neq 0\\}\cap
B(x_{0},\delta^{-1})\cap\mathcal{R}_{X}}r^{-2}\,dg$ $\displaystyle\leq$
$\displaystyle\frac{1}{V}\left(72\delta^{2}+C(\delta,W,\sigma)r^{2-\sigma}\right).$

On the other hand, by Jensen’s inequality, we have

$\displaystyle\frac{1}{V}\int_{M}\left((\phi_{\delta}\eta_{r})^{2}\log(\phi_{\delta}\eta_{r})^{2}\right)(4\pi)^{-\frac{n}{2}}e^{-f_{0}}\,dg$
$\displaystyle=\frac{1}{V}\int_{M}(\phi_{\delta}\eta_{r})^{2}\log(\phi_{\delta}\eta_{r})^{2}d\nu$
(6.7)
$\displaystyle\geq\frac{1}{V}\left(\int_{M}(\phi_{\delta}\eta_{r})^{2}\,d\nu\right)\log\left(\int_{M}(\phi_{\delta}\eta_{r})^{2}\,d\nu\right)$
$\displaystyle=\frac{1}{V}\cdot V\log V$ $\displaystyle=\log V.$

By fixing $\sigma=1/2$, $\delta\leq\overline{\delta}(n)$, and
$r\leq\overline{r}(\delta,W,\sigma)$, we have by (6.4) that

$V\geq 1-C_{n}\delta^{2}\geq\tfrac{1}{2}.$

Thus, combining (6.5), (6.6), and (6.7), we obtain

$\displaystyle
W\geq\mu\big{(}B(x_{0},\delta^{-1})\cap\mathcal{R}_{X},g,1\big{)}-\delta-
C_{n}\delta^{2}.$ (6.8)

Finally, letting $\delta=\delta(n)$ be small enough, we then have that if

$\displaystyle\mu\big{(}B(x_{0},\delta^{-1})\cap\mathcal{R}_{X},g,1\big{)}\geq-\delta,$

then

$\displaystyle W\geq-\varepsilon(n),$

where $\varepsilon(n)$ is the constant in Theorem 6.3; this finishes the proof
of Theorem 1.5.

Taking $\delta\to 0$ in (6.8), we also have the following estimate.

######  Corollary 6.5.

On a metric soliton, we have

$\displaystyle W\geq\mu(\mathcal{R}_{X},g,1).$

##  7 Global $\nu$-functional and Sobolev inequality

In this section, we shall prove Theorem 1.7. In view of Corollary 6.5, we need
only to show the lower estimate of the $\nu$-functional; the whole section is
devoted to the proof of the following proposition.

######  Proposition 7.1.

Let $(X,d,\nu)$ be the model of an $n$-dimensional noncollapsed
$\mathbb{F}$-limit metric soliton and $(\mathcal{R}_{X},g,f_{0})$ its regular
part. Let $W\in(-\infty,0]$ be the soliton entropy. Then we have

$\mu(\mathcal{R}_{X},g,\tau)\geq W\quad\text{ for all }\quad\tau>0.$

In particular, we have

$\nu(\mathcal{R}_{X},g)\geq W$

where $\mu$ and $\nu$ are Perelman’s functionals defined in Subsection 6.1.

Let us consider a metric soliton $(\mathcal{X},(\nu_{t})_{t<0})$ in the
statement of the proposition. By Theorem 3.1 and the reasoning at the
beginning of Section 5, we may pick a sequence
$\\{(M^{i},g^{i}_{t},x_{i})\\}_{i=1}^{\infty}$ satisfying
$T_{i}\nearrow+\infty$, (3.1), (3.2), and (5.3). By [Bam20b, Theorem 9.31] and
[Bam20c, Theorem 2.5], there is an open-set-exhaustion $U_{1}\subset
U_{2}\subset\ldots\subset
U_{i}\subset\ldots\subset\mathcal{R}\subset\mathcal{X}$ and a sequence of
smooth time-preserving diffeomorphisms $\psi_{i}$

$\displaystyle\psi_{i}:U_{i}\longrightarrow V_{i}\subset
M^{i}\times[-T_{i},0],$

and $\varepsilon_{i}\to 0$ such that [Bam20b, Theorem 9.31(a)—(f)] hold. In
particular, we have

$\displaystyle\left\|\psi_{i}^{*}g^{i}_{t}-g_{t}\right\|_{C^{[\varepsilon_{i}^{-1}]}(U_{i})}\leq\varepsilon_{i}.$
(7.1)

Let $x_{0}=x_{0}(-1)\in\mathcal{R}_{-1}$ be a center of the metric soliton,
and let

$(z_{i},-1)=\psi_{i}(x_{0})\in M^{i}\times\\{-1\\}$

be the images of $x_{0}$. Arguing in the same way as the proof of (4.7), we
have that whenever $i$ is large enough, each $(z_{i},-1)$ is at most
$3\sqrt{H_{n}}$-away from an $H_{n}$-center of $(x_{i},0)$. Thus, in view of
[Bam20a, Definition 3.10], we have

$\displaystyle
d^{g^{i}_{-1}}_{W_{1}}\left(\delta_{z_{i}},\nu_{x_{i},0\,|\,-1}\right)\leq
4\sqrt{H_{n}}.$ (7.2)

By the nature of local smooth convergence, the geometry of $\mathcal{R}_{X}$
near a center is close to that of $(M^{i},g^{i}_{-1})$ near an $H_{n}$-center
of $(x_{i},0)$. In spite of the formation of singularity along the converging
sequence, the Sobolev inequality is actually unaffected. The idea is to send
functions compactly supported on $\mathcal{R}_{X}(\cong\mathcal{R}_{-1})$ to
functions on $(M^{i},g^{i}_{-1})$, and apply the Sobolev inequality therein.
However, since $(M^{i},g^{i}_{t})$ is not necessarily ancient, we cannot
directly apply the uniform Sobolev inequality [CMZ23], but the local Sobolev
inequality [CMZ23a] is sufficient for our purpose.

######  Proposition 7.2 (Local Sobolev inequality [CMZ23a, Theorem 1.1]).

For any $\varepsilon>0$ and for each $i$, we have

$\displaystyle\nu\left(B_{g^{i}_{-1}}\left(z_{i},\varepsilon\sqrt{T_{i}/2}\right),g^{i}_{-1},\varepsilon
T_{i}/2\right)\geq\mathcal{N}_{z_{i},-1}(T_{i}/2)-\sqrt{n}\varepsilon-\tfrac{n}{2}\varepsilon-\tfrac{n}{2}\log(1+\varepsilon).$

For any fixed positive scale $\tau>0$, let us consider an arbitrary test
function $u\in C^{\infty}_{c}(\mathcal{R}_{X})$ satisfying

$\displaystyle\int_{\mathcal{R}_{X}}u^{2}\,dg=1.$ (7.3)

Write $K=\operatorname{\operatorname{spt}}u$ and fix an $A>0$ such that

$K\Subset\mathcal{R}_{X}\cap B(x_{0},A)\subset\mathcal{R}_{-1}.$

Obviously, whenever $i$ is large enough, we have

$\displaystyle K\Subset U_{i}\quad\text{ and }\quad\psi_{i,-1}(K)\subset
B_{g^{i}_{-1}}(z_{i},2A).$

We may then consider the pull-backs of $u$ via $\psi_{i,-1}$. Define

$\displaystyle u_{i}:=u\circ\psi_{i,-1}^{-1}:M^{i}\to\mathbb{R}.$

The following lemma is a straightforward consequence of (7.1).

######  Lemma 7.3.

Whenever $i$ is large enough, we have

  1. (1)

$u_{i}$ is compactly supported in $B_{g^{i}_{-1}}(z_{i},2A)$.

  2. (2)

Let $\displaystyle V_{i}^{2}:=\int_{M^{i}}u_{i}^{2}\,dg^{i}_{-1}$, then
$V_{i}=1+\Psi(\varepsilon_{i})$.

  3. (3)

$\displaystyle\int_{M^{i}}u_{i}^{2}\log u_{i}^{2}\,dg^{i}_{-1}$
$\displaystyle\ =(1+\Psi(\varepsilon_{i}))\int_{\mathcal{R}_{X}}u^{2}\log
u^{2}\,dg,$ (7.4)
$\displaystyle\int_{M^{i}}\tau(4|\nabla_{g^{i}_{-1}}u_{i}|^{2}+R_{g^{i}_{-1}}u_{i}^{2})\,dg^{i}_{-1}$
$\displaystyle\ =(1+\Psi(\varepsilon_{i}))\int_{\mathcal{R}_{X}}\tau(4|\nabla
u|^{2}+Ru^{2})\,dg.$ (7.5)

Here (and in the following) $\Psi(\varepsilon)$ stands for a general constant
that converges to zero as $\varepsilon\to 0$, which depends on $\tau$ and $u$
and varies from line to line; $\varepsilon_{i}$ is the constant in (7.1).

Taking $u_{i}/V_{i}$ as the test function of
$\mu\big{(}B_{g^{i}_{-1}}(z_{i},2A),g^{i}_{-1},\tau\big{)}$, we have

######  Lemma 7.4.

$\displaystyle\ \int_{\mathcal{R}_{X}}\left(\tau(4|\nabla
u|^{2}+Ru^{2})-u^{2}\log
u^{2}\right)\,dg-\left(\frac{n}{2}\log(4\pi\tau)+n\right)$ $\displaystyle\geq$
$\displaystyle\
(1+\Psi(\varepsilon_{i}))\mu\Big{(}B_{g^{i}_{-1}}(z_{i},2A),g^{i}_{-1},(1+\Psi(\varepsilon_{i}))\tau\Big{)}+\Psi(\varepsilon_{i}).$

###### Proof.

Let $v_{i}=u_{i}/V_{i}$, by (7.4) we have

$\displaystyle\int_{M^{i}}v_{i}^{2}\log v_{i}^{2}\,dg^{i}_{-1}$
$\displaystyle\ =\frac{1}{V_{i}^{2}}\int_{M^{i}}u_{i}^{2}\log
u_{i}^{2}\,dg^{i}_{-1}-\frac{\log
V_{i}^{2}}{V_{i}^{2}}\int_{M^{i}}u_{i}^{2}\,dg^{i}_{-1}$ $\displaystyle\
=(1+\Psi(\varepsilon_{i}))\int_{\mathcal{R}_{X}}u^{2}\log
u^{2}\,dg+\Psi(\varepsilon_{i}),$

or, equivalently

$\displaystyle\int_{\mathcal{R}_{X}}u^{2}\log
u^{2}\,dg=(1+\Psi(\varepsilon_{i}))\int_{M^{i}}v_{i}^{2}\log
v_{i}^{2}\,dg^{i}_{-1}+\Psi(\varepsilon_{i}).$ (7.6)

On the other hand, (7.5) implies that

$\displaystyle\int_{\mathcal{R}_{X}}\tau(4|\nabla
u|^{2}+Ru^{2})\,dg=(1+\Psi(\varepsilon_{i}))\int_{M^{i}}\tau(4|\nabla_{g^{i}_{-1}}v_{i}|^{2}+R_{g^{i}_{-1}}v_{i}^{2})\,dg^{i}_{-1}.$
(7.7)

The lemma follows from combining (7.6) and (7.7).

∎

It turns out that one needs only to bound the local $\mu$-functional at
$(z_{i},-1)$, which in turn depends on the Nash entropy at $(z_{i},-1)$ by
Proposition 7.2. The latter can be easily estimated via [Bam20a, Corollary
5.11].

######  Lemma 7.5.

For any $\varepsilon>0$, we have $\mathcal{N}_{z_{i},-1}(T_{i}/2)\geq
W-\varepsilon$ whenever $i$ is large enough.

###### Proof.

Let us apply [Bam20a, Corollary 5.11] with

$\displaystyle(x_{1},t_{1})=(x_{i},0),\quad(x_{2},t_{2})=(z_{i},-1),\quad
t^{*}=-1,\quad s=-1-T_{i}/2,\quad R_{\min}=-\tfrac{1}{T_{i}}.$

Note that the last term above is due to (5.3), and that we have the closeness
between $(x_{1},t_{1})$ and $(x_{2},t_{2})$ (7.2). Thus,

$\displaystyle\mathcal{N}_{z_{i},-1}(T_{i}/2)\geq\mathcal{N}_{x_{i},0}(T_{i}/2+1)-\frac{C(n)}{T_{i}}.$

The lemma follows from the above inequality and the convergence of the Nash
entropy (3.1). ∎

Finally, let us fix any positive number $\varepsilon>0$ and take $i$ large
enough such that $\varepsilon T_{i}/2\gg\tau$. Proposition 7.2, Lemma 7.4, and
Lemma 7.5 imply that

$\displaystyle\int_{\mathcal{R}_{X}}\left(\tau(4|\nabla
u|^{2}+Ru^{2})-u^{2}\log
u^{2}\right)\,dg-\left(\frac{n}{2}\log(4\pi\tau)+n\right)\geq(1+\Psi(\varepsilon_{i}))W-\Psi(\varepsilon_{i})-\Psi(\varepsilon).$

First taking $i\to\infty$ and then $\varepsilon\to 0$, we have

$\int_{\mathcal{R}_{X}}\left(\tau(4|\nabla u|^{2}+Ru^{2})-u^{2}\log
u^{2}\right)\,dg-\left(\frac{n}{2}\log(4\pi\tau)+n\right)\geq W.$

Proposition 7.1 follows since the test function $u$ is arbitrarily chosen. As
a consequence, we have proved Theorem 1.7.

##  8 Volume growth lower estimate

This section is devoted to the proof of Theorem 1.10. The idea of the proof,
according to [MW12, LW20], consists of two steps. The first step is to show
that a noncompact (in the sense of infinite diameter) metric soliton has
infinite volume. In this step, we apply the Sobolev inequality (Corollary 1.9)
to show that the volume of unit balls does not decay at a rate higher than
$r^{-n}$, where $r$ is the distance from the origin. While an argument similar
to [MW12] shows that a metric soliton with finite volume must have exponential
decay rate for the volume of unit balls. The second step is to apply the
argument of [LW20], and show that sublinear volume growth contradicts the
infinity of the volume.

As always, we consider an $n$-dimensional noncollapsed $\mathbb{F}$-limit
metric soliton. Let $(X,d,\nu)$ be its model and $(\mathcal{R}_{X},g,f_{0})$
the regular part. Let $x_{0}\in\mathcal{R}_{X}$ be the center of the soliton
and $W$ the soliton entropy.

###  8.1 Preliminary volume decay lower bound

######  Lemma 8.1.

For any $x\in X$, if $R\leq r^{-2}$ on $B(x,r)\cap\mathcal{R}_{X}$, then

$\displaystyle|B(x,r)\cap\mathcal{R}_{X}|\geq c(n)e^{W}r^{n}$

###### Proof.

This is a straightforward consequence of Corollary 1.9 and an iteration
argument. First of all, we consider the case $x\in\mathcal{R}_{X}$. Fix
$\sigma=1/2$, $A\gg r+d_{g}(x_{0},x)$, and any $\delta>0$ (corresponding to
$r$ in Proposition 1.6), let $\eta_{\delta}$ be the cut-off function given by
Proposition 1.6. Letting

$\displaystyle u:=\left(r-d_{g}(x,\cdot)\right)_{+}:\
\mathcal{R}_{X}\to\mathbb{R}$

(note that $u$ is $1$-Lipschitz on $\mathcal{R}_{X}$) and applying the Sobolev
inequality (1.3) to $u\eta_{\delta}$, we have

$\displaystyle\left(\int_{\mathcal{R}_{X}}(u\eta_{\delta})^{\frac{2n}{n-2}}\,dg\right)^{\frac{n-2}{2}}\leq$
$\displaystyle\
C(n)e^{-\frac{2W}{n}}\int_{\mathcal{R}_{X}}\left(4|\nabla(u\eta_{\delta})|^{2}+R(u\eta_{\delta})^{2}\right)\,dg$
$\displaystyle\leq$ $\displaystyle\
C(n)e^{-\frac{2W}{n}}\int_{\mathcal{R}_{X}}\left(8|\nabla
u|^{2}\eta_{\delta}^{2}+8u^{2}|\nabla\eta_{\delta}|^{2}+\eta_{\delta}^{2}\right)\,dg$
$\displaystyle\leq$ $\displaystyle\
C(n)e^{-\frac{2W}{n}}\left(9|B(x,r)\cap\mathcal{R}_{X}|+C(A,W,\sigma)\delta^{2-\sigma}\right).$

Taking $\delta\to 0$ and applying Proposition 1.6 as well as the dominated
convergence theorem, we have

$\displaystyle\left(\tfrac{r}{2}|B(x,r/2)\cap\mathcal{R}_{X}|\right)^{\frac{n-2}{n}}\leq
C(n)e^{-\frac{2W}{n}}|B(x,r)\cap\mathcal{R}_{X}|.$

Since $x\in\mathcal{R}_{X}$, the geometry in a very small disk around $x$ is
almost-Euclidean. Thus, iterating the inequality above, we obtain

$|B(x,r)\cap\mathcal{R}_{X}|\geq c(n)e^{W}r^{n}$

as in the classical case.

Finally, if $x\in X\setminus\mathcal{R}_{X}$, we can then apply the former
result to a point $x^{\prime}\in\mathcal{R}_{X}\cap B(x,r/100)$, and the
conclusion follows in like manner. ∎

######  Lemma 8.2.

If $d(x,x_{0})\geq 1$, then

$|B(x,1)\cap\mathcal{R}_{X}|\geq
c(n)|B(x_{0},1)\cap\mathcal{R}_{X}|\cdot\frac{1}{d^{n}(x_{0},x)}.$

###### Proof.

The third equation in (2.8) and Theorem 1.1 implies that

$\displaystyle R(x)\leq\frac{1}{4}d^{2}(x_{0},x)+C(n)\quad\text{ for all
}\quad x\in\mathcal{R}_{X}.$ (8.1)

Thus, if we can show

$\displaystyle e^{W}\geq c(n)|B(x_{0},1)\cap\mathcal{R}_{X}|,$

the lemma follows from Lemma 8.1; but this is simply Corollary 1.2. ∎

###  8.2 Differential equation of the volume and the scalar curvature

Similar to [CZ10], let

$\rho:=2\sqrt{f_{0}-W}\geq 0,$

and we shall consider the volumes of the sub-level-sets of $\rho$ and the
integrals of the scalar curvature therein. It follows from Theorem 1.1 that

$2d_{g}(x,x_{0})/3-c_{n}\leq\rho(x)\leq d_{g}(x,x_{0})+c_{n},$ (8.2)

where $c_{n}>0$ is a dimensional constant. We also define

$\displaystyle D(s)$ $\displaystyle:=$ $\displaystyle\\{x\in X:\,\rho(x)\leq
s\\}$ $\displaystyle V(s)$ $\displaystyle:=$
$\displaystyle\int_{D(s)\cap\mathcal{R}_{X}}\,dg$ $\displaystyle\chi(s)$
$\displaystyle:=$ $\displaystyle\int_{D(s)\cap\mathcal{R}_{X}}R\,dg.$

In spite of the existence of singular points, $V(s)$ and $\chi(s)$ are still
nice in the measure-theoretic sense.

######  Lemma 8.3.

$V(s)$ and $\chi(s)$ are absolutely continuous, and

$\displaystyle V^{\prime}(s)$ $\displaystyle\
=\int_{\\{\rho=s\\}\cap\mathcal{R}_{X}}\frac{1}{|\nabla\rho|}\,dA=\frac{s}{2}\int_{\\{\rho=s\\}\cap\mathcal{R}_{X}}\frac{1}{|\nabla
f_{0}|}\,dA,$ (8.3) $\displaystyle\chi^{\prime}(s)$ $\displaystyle\
=\int_{\\{\rho=s\\}\cap\mathcal{R}_{X}}\frac{R}{|\nabla\rho|}\,dA=\frac{s}{2}\int_{\\{\rho=s\\}\cap\mathcal{R}_{X}}\frac{R}{|\nabla
f_{0}|}\,dA,$

for almost every $s\geq 0$.

###### Proof.

Thanks to Lemma 4.2, Theorem 1.1, and Corollary 1.2, both $V$ and $\chi$ are
nondecreasing finite real-valued functions in $s$. In view of the regularity
result [Kot13, Corolary 1.3], both $g$ and $\nabla f_{0}$ are real analytic in
the geodesic normal coordinates. Consequently, both the soliton potential
$f_{0}=W+R+|\nabla f_{0}|^{2}$ and the scalar curvature $R$ are real analytic
function on $\mathcal{R}_{X}$. Applying [M20] to the analytic function
$|\nabla f_{0}|^{2}=f_{0}-R-W$, we have that the singular set
$\mathcal{C}:=\\{x\in\mathcal{R}_{X}:|\nabla\rho|(x)=0\\}$ has zero measure.
Hence, $\frac{1}{|\nabla\rho|}$ is a nonnegative measurable function on
$\mathcal{R}_{X}\setminus\mathcal{C}$. Furthermore, by the Morse-Sard Theorem
[SS72], the singular value set $\rho(\mathcal{C})$ is at most countable.

By applying the coarea formula to $\rho$ on
$\mathcal{R}_{X}\setminus\mathcal{C}$, we also have

$\displaystyle V(s)$ $\displaystyle=$
$\displaystyle|D(s)\cap\mathcal{R}_{X}\cap\mathcal{C}|+|D(s)\cap\mathcal{R}_{X}\setminus\mathcal{C}|=|D(s)\cap\mathcal{R}_{X}\setminus\mathcal{C}|$
$\displaystyle=$
$\displaystyle\int_{D(s)\cap\mathcal{R}_{X}\setminus\mathcal{C}}\,dg=\int_{0}^{s}\int_{\\{x\in\mathcal{R}_{X}\setminus\mathcal{C}:\,\rho(x)=t\\}}\frac{1}{|\nabla\rho|}\,dA\,dt.$

Similarly,

$\chi(s)=\int_{D(s)\cap\mathcal{R}_{X}}R\,dg=\int_{D(s)\cap\mathcal{R}_{X}\setminus\mathcal{C}}R\,dg=\int_{0}^{s}\int_{\\{x\in\mathcal{R}_{X}\setminus\mathcal{C}:\,\rho(x)=t\\}}\frac{R}{|\nabla\rho|}\,dA\,dt.$

Hence the nonnegative functions

$\displaystyle t$ $\displaystyle\mapsto$
$\displaystyle\int_{\\{x\in\mathcal{R}_{X}\setminus\mathcal{C}:\,\rho(x)=t\\}}\frac{1}{|\nabla\rho|}\,dA$
$\displaystyle t$ $\displaystyle\mapsto$
$\displaystyle\int_{\\{x\in\mathcal{R}_{X}\setminus\mathcal{C}:\,\rho(x)=t\\}}\frac{R}{|\nabla\rho|}\,dA$

are integrable on $[0,s]$, and $V$ and $\chi$ are absolutely continous
functions with

$\displaystyle V^{\prime}(s)$ $\displaystyle=$
$\displaystyle\int_{\\{x\in\mathcal{R}_{X}\setminus\mathcal{C}:\,\rho(x)=s\\}}\frac{1}{|\nabla\rho|}\,dA,$
$\displaystyle\chi^{\prime}(s)$ $\displaystyle=$
$\displaystyle\int_{\\{x\in\mathcal{R}_{X}\setminus\mathcal{C}:\,\rho(x)=s\\}}\frac{R}{|\nabla\rho|}\,dA$

for almost every $s$. Since for each regular value $s$ of $\rho$, it holds
that

$\\{x\in\mathcal{R}_{X}\setminus\mathcal{C}:\,\rho(x)=s\\}=\\{x\in\mathcal{R}_{X}:\,\rho(x)=s\\},$

so the integral on both sets are equal. By The Morse-Sard Theorem again, the
singular value consists of a null set; (8.3) follows immediately.

∎

We then show a statement analogous to [CZ10, Lemma 3.1].

######  Lemma 8.4.

[CZ10] For almost every positive number $s$,

$\frac{n}{2}V(s)-\frac{s}{2}V^{\prime}(s)=\chi(s)-\frac{2}{s}\chi^{\prime}(s),$
(8.4)

and

$\chi(s)\leq\frac{n}{2}V(s).$ (8.5)

###### Proof.

Let $s$ be a positive regular value of $f_{0}$ at which both $V$ and $\chi$
are differentiable. By (8.2)

$D(s)\subseteq B_{g}(x_{0},3(s+c_{n})/2).$

We apply Proposition 1.6 with $A:=3(s+c_{n})+\underline{A}(n)$ and $\sigma<2$
to get a cutoff function $\eta_{r}$ for each $r\leq\overline{r}(A,W,\sigma)$;
here $0<\sigma<2$ is arbitrarily fixed. By taking the trace of the soliton
equation (2.8), we have

$\Delta f_{0}+R=\frac{n}{2}.$

We multiply the equation by $\eta_{r}^{2}$ and integrate it over
$D(s)\cap\mathcal{R}_{X}$:

$\int_{D(s)\cap\mathcal{R}_{X}}\eta_{r}^{2}\Delta
f_{0}\,dg+\int_{D(s)\cap\mathcal{R}_{X}}\eta^{2}_{r}R\,dg=\frac{n}{2}\int_{D(s)\cap\mathcal{R}_{X}}\eta^{2}_{r}\,dg.$
(8.6)

We shall estimate the equation term by term.

First of all, we have

$0\leq\frac{n}{2}\int_{D(s)\cap\mathcal{R}_{X}}(1-\eta^{2}_{r})\,dg\leq\frac{n}{2}\int_{\mathcal{R}_{X}\cap
B(x_{0},A)\cap\\{0\leq\eta_{r}<1\\}}\,dg\leq C(A,W,\sigma)r^{4-\sigma}\to 0\
\text{ as }\ r\to 0.$

Hence

$\frac{n}{2}\int_{D(s)\cap\mathcal{R}_{X}}\eta^{2}_{r}\,dg\to\frac{n}{2}V(s)\
\text{ as }\ r\to 0.$

Using also the third equation of (2.8), it holds that
$R\leq\frac{1}{4}\rho^{2}$. Thus

$0\leq\int_{D(s)\cap\mathcal{R}_{X}}(1-\eta^{2}_{r})R\,dg\leq\frac{s^{2}}{4}\int_{D(s)\cap\mathcal{R}_{X}}(1-\eta^{2}_{r})\,dg\leq
C(A,W,\sigma)s^{2}r^{4-\sigma}\to 0$

and

$\int_{D(s)\cap\mathcal{R}_{X}}\eta^{2}_{r}R\,dg\to\int_{D(s)\cap\mathcal{R}_{X}}R\,dg\
\text{ as }\ r\to 0.$

By Proposition 1.6, when applying the divergence theorem [Lee13, Theorem
16.32, Theorem 16.48] to $\eta_{r}^{2}\nabla f_{0}$ on
$D(s)\cap\mathcal{R}_{X}$, the boundary integral occurs only on
$\\{x\in\mathcal{R}_{X}:\rho(x)=s\\}$, which is a smooth hypersurface (recall
that $s$ is a regular value of $f_{0}$). Thus

$\displaystyle\int_{D(s)\cap\mathcal{R}_{X}}\eta_{r}^{2}\Delta f_{0}\,dg$
$\displaystyle=$
$\displaystyle-2\int_{D(s)\cap\mathcal{R}_{X}}\eta_{r}\langle\nabla\eta_{r},\nabla
f_{0}\rangle\,dg+\int_{\\{x\in\mathcal{R}_{X}:\rho(x)=s\\}}\eta_{r}^{2}\frac{\langle\nabla_{0}f,\nabla\rho\rangle}{|\nabla\rho|}\,dA$
$\displaystyle=$
$\displaystyle-2\int_{D(s)\cap\mathcal{R}_{X}}\eta_{r}\langle\nabla\eta_{r},\nabla
f_{0}\rangle\,dg+\int_{\\{x\in\mathcal{R}_{X}:\rho(x)=s\\}}\eta_{r}^{2}|\nabla
f_{0}|\,dA$ $\displaystyle
2\int_{D(s)\cap\mathcal{R}_{X}}\eta_{r}\left|\langle\nabla\eta_{r},\nabla
f_{0}\rangle\right|\,dg$ $\displaystyle\leq$ $\displaystyle
C_{0}sr^{-1}\int_{\mathcal{R}_{X}\cap
B(x_{0},A)\cap\\{|\nabla_{g}\eta_{r}|\neq 0\\}}dg$ $\displaystyle\leq$
$\displaystyle C(A,W,\sigma)sr^{3-\sigma}\to 0\ \text{ as }\ r\to 0.$

Moreover, Lemma 8.3 also implies that, for almost every $s\geq 0$,

$\displaystyle\int_{\mathcal{R}_{X}\cap\\{\rho=s\\}}|\nabla
f_{0}|\,dA\leq\frac{s^{2}}{4}\int_{\mathcal{R}_{X}\cap\\{\rho=s\\}}\frac{1}{|\nabla
f_{0}|}dA<+\infty.$

So, for such an $s$, Proposition 1.6(5) implies that

$\displaystyle\lim_{r\to
0}\int_{\\{x\in\mathcal{R}_{X}:\rho(x)=s\\}}\eta_{r}^{2}|\nabla
f_{0}|\,dA=\int_{\mathcal{R}_{X}\cap\\{\rho=s\\}}|\nabla f_{0}|\,dA$

due to the dominated convergence theorem.

Finally, (8.5) follows from letting $r\to 0$ in (8.6) and the fact

$\displaystyle\frac{n}{2}V(s)-\int_{D(s)\cap\mathcal{R}_{X}}R\,dg=\int_{\\{x\in\mathcal{R}_{X}:\rho(x)=s\\}}|\nabla
f_{0}|\,dA\geq 0.$

In like manner, (8.4) also follows from taking $r\to 0$; note that the right-
hand-side of (8.4) comes from the following computation as in [CZ10] using
(2.8)

$\int_{\\{x\in\mathcal{R}_{X}:\rho(x)=s\\}}|\nabla
f_{0}|\,dA=\int_{\\{x\in\mathcal{R}_{X}:\rho(x)=s\\}}\frac{f_{0}-W-R}{|\nabla
f_{0}|}\,dA=\frac{s}{2}V^{\prime}(s)-\frac{2}{s}\chi^{\prime}(s).$

∎

###  8.3 Noncompact metric soliton has infinite volume

Next we show that a noncompact metric soliton must have infinite volume.
Taking $\tau=1$ in (1.2), we get the following logarithmic Sobolev inequality
which shall be applied in this subsection.

$\displaystyle\overline{W}\int_{\mathcal{R}_{X}}u^{2}\,dg+\int_{\mathcal{R}_{X}}u^{2}\log
u^{2}\,dg\leq\int_{\mathcal{R}_{X}}\left(4|\nabla
u|^{2}+Ru^{2}\right)\,dg+\int_{\mathcal{R}_{X}}u^{2}\,dg\cdot\log\left(\int_{\mathcal{R}_{X}}u^{2}\,dg\right),$
(8.7)

for $u\in C^{0,1}_{c}(\mathcal{R}_{X})$. Here we have defined

$\overline{W}:=\min\left\\{W+n+\frac{n}{2}\log(4\pi),0\right\\}$

for notational simplicity. The following theorem is prove by an argument
similar to that of [MW12].

######  Proposition 8.5.

Suppose that $\mathcal{R_{X}}$ has infinite diameter. Then
$\left|\mathcal{R_{X}}\right|=\infty$.

###### Proof.

Suppose on the contrary that $\left|\mathcal{R_{X}}\right|<\infty$. We will
show that the volume of the unit ball decays at least exponentially. This
contradicts the polynomial decay in Lemma 8.2. Similar to [MW12], we consider
the Lipschitz function

$u_{t}(s)=\int^{s}_{-\infty}\chi_{[t-1,t]}(\tau)\,d\tau=\left\\{\begin{array}[]{rl}1&\text{
if }t\leq s;\\\ s-t+1&\text{ if }t-1\leq s\leq t;\\\ 0&\text{ if }s\leq
t-1.\end{array}\right.$

By the definition of $u_{t}$, for all $s\geq 0$

$|u_{t_{1}}(s)-u_{t_{2}}(s)|\leq 4|t_{1}-t_{2}|.$

We write $u_{t}(x)=u_{t}(\rho(x))\in W^{1,2}(\mathcal{R}_{X})$ and let

$y(t)=\int_{\mathcal{R}_{X}}u_{t}^{2}\,dg.$

Since $u_{t_{2}}\leq u_{t_{1}}$ if $t_{1}\leq t_{2}$ and $\mathcal{R}_{X}$ has
finite volume, $y(t)$ is a nonincreasing Lipschitz function in $t$. We claim
that there exists positive constant $C=C(n,W)$ such that

$ty^{\prime}(t)-2y(t)\log y(t)\leq Cy(t-1).$ (8.8)

Assuming the claim at the moment, the same argument in [MW12] shows that there
are small positive constants $\varepsilon$ and $\delta$ such that

$y(t)\leq\delta e^{-\varepsilon t}\quad\text{ for all $t\geq 1/\varepsilon$}.$

For any $t\geq 1/\varepsilon$, choose $x\in\mathcal{R}_{X}$ such that
$\rho(x)=t+1$. Using (8.2) and the fact that $|\nabla\rho|=\tfrac{|\nabla
f_{0}|}{\sqrt{f_{0}-W}}\leq 1$ on $B(x,1)\cap\mathcal{R}_{X}$, for each $z\in
B(x,1)\cap\mathcal{R}_{X}$, we may integrate $\nabla\rho$ along almost
minimizing curve joining $x$ and $z$ to obtain

$t\leq\rho(z)\leq t+2$

and

$d_{g}(z,x_{0})\leq c_{n}(t+1)\quad\text{ for all }\quad z\in
B(x,1)\cap\mathcal{R}_{X}.$

Hence by Lemma 8.2

$c(n)|B(x_{0},1)\cap\mathcal{R}_{X}|\cdot\frac{1}{(t+1)^{n}}\leq|B(x,1)\cap\mathcal{R}_{X}|\leq
y(t)\leq\delta e^{-\varepsilon t},$

which is impossible when $t$ is large; it remains to prove (8.8) under the
assumption $|\mathcal{R}_{X}|<+\infty$.

Since (8.7) only holds for functions compactly supported on $\mathcal{R}_{X}$,
we first verify that indeed (8.7) also holds when $u=u_{t}(x)$. Fix
$\sigma\in(0,1/2)$ and let $\delta$ be a small positive number, we define
cutoff functions $\phi_{\delta}$ and $\eta_{r}$ in the same way as in Section
6.4, that is, $\phi_{\delta}$ is the cut-off function defined by (6.2), and
$\eta_{r}$ is the cut-off function given by Proposition 1.6 with
$A=\delta^{-1}$ and $r\leq\overline{r}(\delta,W,\sigma)$. We substitute
$u=\phi_{\delta}\eta_{r}u_{t}(x)$ into (8.7). Since

$\int_{\mathcal{R}_{X}}\phi_{\delta}^{2}(1-\eta_{r}^{2})u_{t}^{2}(x)\,dg\leq\int_{\mathcal{R}_{X}\cap
B(x_{0},\delta^{-1})\cap\\{0\leq\eta_{r}<1\\}}\,dg\leq
C(\delta,W,\sigma)r^{4-\sigma}\to 0\ \text{ as }\ r\to 0,$

by first letting $r\to 0$ and then $\delta\to 0$, we have

$\int_{\mathcal{R}_{X}}\phi^{2}_{\delta}\eta_{r}^{2}u_{t}^{2}(x)\,dg\to\int_{\mathcal{R}_{X}}u_{t}^{2}(x)\,dg=y(t),$
$\int_{\mathcal{R}_{X}}\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2}(x)\,dg\log\left(\int_{\mathcal{R}_{X}}\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2}(x)\,dg\right)\to
y(t)\log y(t).$

By the similar computations as in (6.5), (6.6) and (6.7), we have

$\displaystyle\int_{\mathcal{R}_{X}}\left(4\left|\nabla(\phi_{\delta}\eta_{r}u_{t})\right|^{2}+R\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2}\right)\,dg-\int_{\mathcal{R}_{X}}\phi^{2}_{\delta}\eta_{r}^{2}u_{t}^{2}\log\left(\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2}\right)\,dg$
$\displaystyle=$
$\displaystyle\int_{\mathcal{R}_{X}}\left(4\phi_{\delta}^{2}\eta_{r}^{2}\left|\nabla
u_{t}\right|^{2}+R\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2}\right)\,dg-\int_{\mathcal{R}_{X}}\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2}\log(\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2})\,dg$
$\displaystyle+\int_{\mathcal{R}_{X}}\left(4u_{t}^{2}\left|\nabla(\phi_{\delta}\eta_{r})\right|^{2}+8\phi_{\delta}\eta_{r}u_{t}\langle\nabla(\phi_{\delta}\eta_{r}),\nabla
u_{t}\rangle\right)\,dg.$

Since $0\leq u_{t}(x)\leq 1$ and $u_{t}(x)$ is $1$-Lipschitz in $x$,
$\left|\mathcal{R}_{X}\right|<\infty$, and $-e^{-1}\leq u_{t}^{2}\log
u_{t}^{2}\leq 0$, we have

$\displaystyle\bigg{|}\int_{\mathcal{R}_{X}}\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2}\log\left(\phi_{\delta}^{2}\eta_{r}^{2}u_{t}^{2}\right)\,dg-\int_{\mathcal{R}_{X}}u_{t}^{2}\log
u_{t}^{2}\,dg\bigg{|}$ $\displaystyle=$
$\displaystyle\bigg{|}\int_{\mathcal{R}_{X}}\Big{(}u_{t}^{2}\phi_{\delta}^{2}\eta_{r}^{2}\log(\phi_{\delta}^{2}\eta_{r}^{2})+\phi_{\delta}^{2}(\eta_{r}^{2}-1)u_{t}^{2}\log
u_{t}^{2}+(\phi_{\delta}^{2}-1)u_{t}^{2}\log u_{t}^{2}\Big{)}\,dg\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left|\int_{\\{\phi_{\delta}>0\\}\cap(\\{\phi_{\delta}<1\\}\cup\\{0<\eta_{r}<1\\})}e^{-1}\,dg\right|+\left|\int_{\mathcal{R}_{X}}\phi_{\delta}^{2}(1-\eta_{r}^{2})e^{-1}\,dg\right|+\left|\int_{\mathcal{R}_{X}}(1-\phi_{\delta}^{2})e^{-1}\,dg\right|$
$\displaystyle\leq$ $\displaystyle
Cy(1/2\delta)+C(\delta,W,\sigma)r^{4-\sigma}\to 0,$
$\displaystyle\int_{\mathcal{R}_{X}}4u_{t}^{2}\left|\nabla(\phi_{\delta}\eta_{r})\right|^{2}$
$\displaystyle\leq$
$\displaystyle\int_{\mathcal{R}_{X}}8u_{t}^{2}(\left|\nabla\phi_{\delta}\right|^{2}+\left|\nabla\eta_{r}\right|^{2})\,dg$
$\displaystyle\leq$ $\displaystyle
C(n)\left|\mathcal{R}_{X}\right|\delta^{2}+C(\delta,W,\sigma)r^{2-\sigma}\to
0,$
$\displaystyle\int_{\mathcal{R}_{X}}8\phi_{\delta}\eta_{r}u_{t}\left|\langle\nabla(\phi_{\delta}\eta_{r}),\nabla
u_{t}\rangle\right|\,dg$ $\displaystyle\leq$ $\displaystyle
C\int_{\mathcal{R}_{X}}8\phi_{\delta}\eta_{r}u_{t}(\left|\nabla\phi_{\delta}\right|+\left|\nabla\eta_{r}\right|)\,dg$
$\displaystyle\leq$ $\displaystyle
C(n)\left|\mathcal{R}_{X}\right|\delta+C(\delta,W,\sigma)r^{3-\sigma}\to 0,$

the limits above are all obtained by first taking $r\to 0$ and then $\delta\to
0$. Hence from (8.7) and the definition of $u_{t}$, we have

$y(t)\leq C(\left|\mathcal{R}_{X}\right|-V(t-1))$

and

$\displaystyle(C\overline{W}-e^{-1})(\left|\mathcal{R}_{X}\right|-V(t-1))$
$\displaystyle\leq\overline{W}y(t)-e^{-1}\int_{\mathcal{R}_{X}\setminus
D(t-1)}\,dg$ (8.9)
$\displaystyle\leq\overline{W}y(t)+\int_{\mathcal{R}_{X}}u_{t}^{2}\log
u_{t}^{2}\,dg$ $\displaystyle\leq\int_{\mathcal{R}_{X}}(4|\nabla
u_{t}|^{2}+Ru_{t}^{2})\,dg+y(t)\log y(t)$ $\displaystyle\leq
C(\left|\mathcal{R}_{X}\right|-V(t-1))+\int_{\mathcal{R}_{X}}Ru_{t}^{2}\,dg+y(t)\log
y(t).$

Moreover, as in [MW12], we have

$\displaystyle\langle\nabla(u_{t}^{2}),\nabla f_{0}\rangle\leq\rho u_{t}\leq
tu_{t}\text{ on }\mathcal{R}_{X}\cap D(t)\setminus D(t-1)\text{ and vanishes
elsewhere.}$ (8.10)

The following differential equation can also be verified straightforwardly.

Claim.

$\frac{d}{dt}y(t)=-2\int_{\mathcal{R}_{X}\cap D(t)\setminus D(t-1)}u_{t}\,dg$

_for every $t\geq 0$_.

###### Proof of the claim.

For each $t\geq 0$ and $\varepsilon>0$, we may compute as follows.

$\displaystyle y(t+\varepsilon)-y(t)=$ $\displaystyle\
\int_{\mathcal{R}_{X}}\left(u_{t+\varepsilon}^{2}-u_{t}^{2}\right)\,dg$
$\displaystyle=$ $\displaystyle\ -\int_{\mathcal{R}_{X}\cap
D(t+\varepsilon-1)\setminus D(t-1)}\left(\rho-(t-1)\right)^{2}\,dg$
$\displaystyle\ -\int_{\mathcal{R}_{X}\cap D(t+\varepsilon)\setminus
D(t)}\big{(}1-(\rho-(t+\varepsilon-1))^{2}\big{)}\,dg$ $\displaystyle\
-\varepsilon\int_{\mathcal{R}_{X}\cap D(t)\setminus
D(t+\varepsilon-1)}\big{(}2(\rho-(t-1))-\varepsilon\big{)}\,dg.$

The first observation obtained from the above formula is that $y(t)$ is
locally Lipschitz (and hence absolutely continuous), since each term on the
right-hand-side can be bounded with $C(n)|\mathcal{R}_{X}|\varepsilon$.
Secondly, we have

$\displaystyle\frac{1}{\varepsilon}\left|\int_{\mathcal{R}_{X}\cap
D(t+\varepsilon-1)\setminus
D(t-1)}\left(\rho-(t-1)\right)^{2}\,dg\right|\leq\varepsilon|\mathcal{R}_{X}\cap
D(t+\varepsilon-1)\setminus D(t-1)|\to 0,$
$\displaystyle\frac{1}{\varepsilon}\left|\int_{\mathcal{R}_{X}\cap
D(t+\varepsilon)\setminus
D(t)}\big{(}1-(\rho-(t+\varepsilon-1))^{2}\big{)}\,dg\right|\leq
2|\mathcal{R}_{X}\cap D(t+\varepsilon)\setminus D(t)|\to 0$
$\displaystyle\frac{1}{\varepsilon}\left(-\varepsilon\int_{\mathcal{R}_{X}\cap
D(t)\setminus
D(t+\varepsilon-1)}\big{(}2(\rho-(t-1))-\varepsilon\big{)}\,dg\right)\to-2\int_{\mathcal{R}_{X}\cap
D(t)\setminus D(t-1)}u_{t}\,dg,$

as $\varepsilon\to 0$.

∎

We continue the proof of the proposition. For any regular value $s>0$ of
$\rho$, picking a cut-off function $\eta_{r}$ in Proposition 1.6 with $A\geq
3s+\underline{A}(n)$, $\sigma<3$, and $r\leq\overline{r}(A,W,\sigma)$, we may
integrate $\Delta f_{0}+R=\frac{n}{2}$ on $\mathcal{R}_{X}\cap D(s)$ and apply
the divergence theorem as in the proof of Lemma 8.4.

$\displaystyle 0\leq\int_{\mathcal{R}_{X}\cap D(s)}R\eta_{r}^{2}u_{t}^{2}\,dg$
$\displaystyle=$ $\displaystyle\frac{n}{2}\int_{\mathcal{R}_{X}\cap
D(s)}\eta_{r}^{2}u_{t}^{2}\,dg-\int_{\mathcal{R}_{X}\cap D(s)}\Delta
f_{0}\eta_{r}^{2}u_{t}^{2}\,dg$ $\displaystyle=$
$\displaystyle\frac{n}{2}\int_{\mathcal{R}_{X}\cap
D(s)}\eta_{r}^{2}u_{t}^{2}\,dg+2\int_{D(s)\cap\mathcal{R}_{X}}u_{t}^{2}\eta_{r}\langle\nabla\eta_{r},\nabla
f_{0}\rangle\,dg$
$\displaystyle+\int_{D(s)\cap\mathcal{R}_{X}}\eta_{r}^{2}\langle\nabla(u_{t}^{2}),\nabla
f_{0}\rangle\,dg-\int_{\\{x\in\mathcal{R}_{X}:\rho(x)=s\\}}\eta_{r}^{2}u_{t}^{2}\frac{\langle\nabla_{0}f,\nabla\rho\rangle}{|\nabla\rho|}\,dA$
$\displaystyle\leq$
$\displaystyle\frac{n}{2}y(t)+\int_{D(s)\cap\mathcal{R}_{X}}\eta_{r}^{2}\langle\nabla(u_{t})^{2},\nabla
f_{0}\rangle\,dg+C(A,W,\sigma)r^{3-\sigma}.$

By letting $r\to 0$ and then $s\to+\infty$, we have

$0\leq\int_{\mathcal{R}_{X}}Ru_{t}^{2}\,dg\leq\frac{n}{2}y(t)-\frac{t}{2}y^{\prime}(t),$

where we have applied (8.10), the claim above, and the dominated convergence
theorem; (8.8) then follows by $\left|\mathcal{R}_{X}\right|-V(t-1)\leq
y(t-1)$ and substituting the above inequality into (8.9). This completes the
proof of the proposition. ∎

###  8.4 Noncompact metric soliton has at least linear volume growth

###### Proof of Theorem 1.10.

In view of the arguments in [MW12, Theorem 6.1] and [LW20, Proposition 6], it
suffices to verify the following differential inequalities for $t\geq c(n)$

$V(t+1)\leq 2V(t),$ (8.11) $V(t+1)-V(t)\leq C_{1}\frac{V(t)}{t},$ (8.12)
$|D(t+1)\cap\mathcal{R}_{X}\setminus D(t)|^{\frac{n-2}{n}}\leq
C_{2}e^{-\frac{2W}{n}}\Big{[}|D(t)\cap\mathcal{R}_{X}\setminus
D(t-1)|+|D(t+2)\cap\mathcal{R}_{X}\setminus
D(t+1)|+\chi(t+2)-\chi(t-1)\Big{]},$ (8.13)

where $C_{1}$ and $C_{2}$ are dimensional constants, c.f. equations (266),
(267), and (271) in [LW20]. By the absolute continuity of $V$ and $\chi$, we
can mulitply both sides of (8.4) by $s^{-n}$ and integrate the equation over
$[t,t+1]$ as in [CZ10] to get

$(t+1)^{-n}V(t+1)-t^{-n}V(t)\leq 4(t+1)^{-(n+2)}\chi(t+1)\leq
2n(t+1)^{-(n+2)}V(t+1)$ (8.14)

for all $t\geq\sqrt{2(n+2)}$; here we also used (8.5). Hence

$V(t+1)\left(1-\frac{2n}{(t+1)^{2}}\right)\leq\frac{(t+1)^{n}}{t^{n}}V(t)$

which implies (8.11) for all $t\geq c(n)$. Substitute (8.11) into (8.14) and
use $((t+1)^{n}-t^{n})/t^{n}\leq 2^{n}/t$ as in [MW12], we have

$V(t+1)-V(t)\leq\frac{2^{n}}{t}V(t)+\frac{4n}{(t+1)^{2}}V(t).$

This gives (8.12). To prove (8.13), we make use of the global Sobolev
inequality in Corollary 1.9. For any fixed $t\geq c(n)$ Let

$w_{t}(s)=\left\\{\begin{array}[]{rl}0&\text{ if }s\leq t-1;\\\ s-t+1&\text{
if }t-1\leq s\leq t;\\\ 1&\text{ if }t\leq s\leq t+1;\\\ t+2-s&\text{ if
}t+1\leq s\leq t+2;\\\ 0&\text{ if }s\geq t+2.\end{array}\right.$

We also write $w_{t}(x):=w_{t}(\rho(x))$. Using (8.2), we may fix $A\geq
3t+\underline{A}(n)$ in Proposition 1.6 to get a cutoff function $\eta_{r}$
for all $r\leq\overline{r}(t,W,\sigma)$, where $0<\sigma<2$ is arbitrarily
fixed. We substitute $u:=\eta_{r}w_{t}\in C^{0,1}_{c}(\mathcal{R}_{X})$ into
the Sobolev inequality of Corollary 1.9 to get

$\left(\int_{\mathcal{R}_{X}}|\eta_{r}w_{t}|^{\frac{2n}{n-2}}\,dg\right)^{\frac{n-2}{n}}\leq
C(n)e^{-\frac{2W}{n}}\int_{\mathcal{R}_{X}}\left(4|\nabla(\eta_{r}w_{t})|^{2}+R\eta_{r}^{2}w_{t}^{2}\right)\,dg.$
(8.15)

By Proposition 1.6 and the definition of $w_{t}$, as $r\to 0$

$\int_{\mathcal{R}_{X}}|(1-\eta_{r})w_{t}|^{\frac{2n}{n-2}}\,dg\leq\int_{\mathcal{R}_{X}\cap
B(x_{0},A)\cap\\{0\leq\eta_{r}<1\\}}\,dg\leq C(A,W,\sigma)r^{4-\sigma}\to 0$

and thus by the triangle inequality

$\left(\int_{\mathcal{R}_{X}}|\eta_{r}w_{t}|^{\frac{2n}{n-2}}\,dg\right)^{\frac{n-2}{n}}\to\left(\int_{\mathcal{R}_{X}}|w_{t}|^{\frac{2n}{n-2}}\,dg\right)^{\frac{n-2}{n}}\geq|D(t+1)\cap\mathcal{R}_{X}\setminus
D(t)|^{\frac{n-2}{n}}.$

Using (8.1), we see that,

$\displaystyle 0$ $\displaystyle\
\leq\int_{\mathcal{R}_{X}}R(1-\eta_{r}^{2})w_{t}^{2}\,dg\leq
c_{n}(t+2)^{2}\int_{\mathcal{R}_{X}}(1-\eta_{r}^{2})w_{t}^{2}$ $\displaystyle\
\leq c_{n}(t+2)^{2}\int_{\mathcal{R}_{X}\cap
B(x_{0},A)\cap\\{0\leq\eta_{r}<1\\}}\,dg\leq C(A,W,\sigma)\cdot(t+2)^{2}\cdot
r^{4-\sigma}\to 0$

as $r\to 0$, and

$\int_{\mathcal{R}_{X}}R\eta_{r}^{2}w_{t}^{2}\,dg\to\int_{\mathcal{R}_{X}}Rw_{t}^{2}\,dg\leq\int_{D(t+2)\cap\mathcal{R}_{X}\setminus
D(t-1)}R\,dg=\chi(t+2)-\chi(t-1).$

On the other hand,

$\int_{\mathcal{R}_{X}}4|\nabla(\eta_{r}w_{t})|^{2}\,dg=\int_{\mathcal{R}_{X}}\left(4\eta_{r}^{2}|\nabla
w_{t}|^{2}+4w_{t}^{2}|\nabla\eta_{r}|^{2}+8\eta_{r}w_{t}\langle\nabla\eta_{r},\nabla
w_{t}\rangle\right)\,dg.$

Thanks to the fact $|\nabla\rho|\leq 1$ and Proposition 1.6, when $r\to 0$,

$\displaystyle\int_{\mathcal{R}_{X}}\left(4w_{t}^{2}|\nabla\eta_{r}|^{2}+8\eta_{r}w_{t}\left|\langle\nabla\eta_{r},\nabla
w_{t}\rangle\right|\right)\,dg$ $\displaystyle\leq$ $\displaystyle
C\int_{\mathcal{R}_{X}\cap\\{|\nabla\eta_{r}|\neq
0\\}}w_{t}^{2}r^{-2}\,dg+C\int_{D(t+2)\cap\mathcal{R}_{X}}|\nabla\eta_{r}|\,dg$
$\displaystyle\leq$ $\displaystyle C(A,W,\sigma)(r^{2-\sigma}+r^{3-\sigma})\to
0,$ $\int_{\mathcal{R}_{X}}4(1-\eta_{r}^{2})|\nabla w_{t}|^{2}\leq
C\int_{\mathcal{R}_{X}\cap B(x_{0},A)\cap\\{0\leq\eta_{r}<1\\}}\,dg\leq
C(A,W,\sigma)r^{4-\sigma}\to 0.$

Hence as $r\to 0$,

$\displaystyle\int_{\mathcal{R}_{X}}4|\nabla(\eta_{r}w_{t})|^{2}\,dg$
$\displaystyle=$ $\displaystyle\int_{\mathcal{R}_{X}}4\eta_{r}^{2}|\nabla
w_{t}|^{2}+4w_{t}^{2}|\nabla\eta_{r}|^{2}+8\eta_{r}w_{t}\langle\nabla\eta_{r},\nabla
w_{t}\rangle\,dg$ $\displaystyle\to$
$\displaystyle\int_{\mathcal{R}_{X}}4|\nabla w_{t}|^{2}\,dg$ $\displaystyle=$
$\displaystyle\int_{D(t)\cap\mathcal{R}_{X}\setminus D(t-1)}4|\nabla
w_{t}|^{2}\,dg+\int_{D(t+2)\cap\mathcal{R}_{X}\setminus D(t+1)}4|\nabla
w_{t}|^{2}\,dg$ $\displaystyle\leq$ $\displaystyle
4|D(t)\cap\mathcal{R}_{X}\setminus
D(t-1)|+4|D(t+2)\cap\mathcal{R}_{X}\setminus D(t+1)|.$

(8.13) then follows by letting $r\to 0$ in (8.15). The linear volume growth is
now a consequence of (8.11),(8.12), (8.13), Proposition 8.5, and the
contradiction arguments in [MW12, LW20]. ∎

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Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4
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Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
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School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai,
China, 200240
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