Definition 2.11]). Let $\mathbb{M}_{r}(V,b)\subset\mathbb{M}$ be the subset
consisting of $(X,d,\mu)$ satisfying

$\text{Var}(\mu)\leq Vr^{2}\quad\text{and}\quad\mu\left(\left\\{x\in
X\mid\mu(D(x,\epsilon
r))<b(\epsilon)\right\\}\right)\leq\epsilon,\quad\forall\epsilon\in(0,1].$
(6.4)

Here, $V,r$ are two positive constants and $b:(0,1]\to(0,1]$ is a function.
Moreover, $D(x,\epsilon r)$ denotes a closed ball with center $x$ and radius
$\epsilon r$. It is proved by [3, Theorem 2.27] that $\mathbb{M}_{r}(V,b)$ is
compact.

  2. 2.

For any metric flow pair $(\mathcal{X},(\mu_{t})_{t\in I})$ defined in (6.1).
It is clear by $H_{n}$-concentration that $\text{Var}(\mu_{t})\leq H_{n}|t|$.
It can be proved (cf. [3, Proposition 4.1] with
$\tau=\frac{\epsilon^{3}}{8H_{n}}$) that for any $t<0$,

$(\mathcal{X}_{t},d_{t},\mu_{t})\in\mathbb{M}_{r}(V,b),$ (6.5)

where $V=1/8$, $r=\sqrt{8H_{n}|t|}$,
$b(\epsilon)=\Phi(\epsilon^{-2}\sqrt{8H_{n}})/2$ and $\Phi$ is given by
(3.31). The proof of (6.5) uses Definition 6.1(6)(7) in an essential way.
Therefore, for any $t\leq 0$, $(\mathcal{X}^{i},d^{i}_{t},\mu^{i}_{t})$
subconverges in $GW_{1}$ to a limit metric measure space.

  3. 3.

To compare different time-slices of $(\mathcal{X},(\mu_{t})_{t\in I})$, one
considers the function

$D(t):=\int_{\mathcal{X}_{t}}\int_{\mathcal{X}_{t}}d_{t}\,d\mu_{t}d\mu_{t}$
(6.6)

for $t\in I$. It is not hard to prove (cf. [3, Lemma 4.7]) that for any $s\leq
t\in I$,

$-\sqrt{H_{n}(t-s)}\leq
D(t)-D(s)\leq\sqrt{\text{Var}(\mu_{t})-\text{Var}(\mu_{s})+H_{n}(t-s)}+2\sqrt{H_{n}(t-s)}.$
(6.7)

It follows immediately from (6.7) that $D(t)$ is continuous on a complement of
a countable subset of $I$. In addition, it is proved (cf. [3, Theorem 4.31])
that for any $t_{0}\leq 0$, $D(t)$ is continuous at $t_{0}$ is equivalent to
the statement that $(\mathcal{X}_{t},d_{t},\mu_{t})$ is continuous at $t_{0}$
in the $GW_{1}$ sense. In this case, one can construct an isometric embedding
of $(\mathcal{X}_{t},d_{t})$ and $(\mathcal{X}_{t_{0}},d_{t_{0}})$ into a
metric space $(Z_{t},d^{Z}_{t})$ with an explicit coupling $q_{t}$ between
$\mu_{t}$ and $\mu_{t_{0}}$ for $t$ close to $t_{0}$. Therefore, one concludes
that the metric flow $(\mathcal{X}_{t},\mu_{t})$ is continuous on $I$ except
at a countable set of times.

  4. 4.

For the sequence $(\mathcal{X}^{i},(\mu^{i}_{t})_{t\in I})$ in (6.3), we
consider the limit

$D^{\infty}(t):=\lim_{t\to\infty}D^{i}(t),$ (6.8)

which exists for any $t\in I$ by taking a subsequence. Indeed, by (6.7), we
may assume that $D^{\infty}(t)$ exists for $t\in I\cap\mathbb{Q}$ and
$D^{\infty}(t)-D^{\infty}(s)\geq-\sqrt{H_{n}(t-s)}$ for any $s,t\in
I\cap\mathbb{Q}$ with $s\leq t$, after taking a subsequence if necessary.
Therefore, there exists a countable set $S\subset I$ such that $D^{\infty}$ is
continuous on $I\setminus S$, by extending the definition of $D^{\infty}$.
Moreover, (6.8) holds for any $t\in I\setminus S$. Now, (6.8) also holds for
$t\in S$, by further taking a subsequence.

The $\mathbb{F}$-convergence of $(\mathcal{X}^{i},(\mu^{i}_{t})_{t\in I})$ can
be constructed as follows. We assume $D^{\infty}(t)$ is continuous at
$I\setminus S$ for a countable set $S$. For a large $k>0$, we take a compact
set $I_{1}\subset[-k,0]\setminus S$ so that $|[-k,0]\setminus I_{1}|$ is
small. Then $I_{1}$ is finitely covered by compact intervals $I_{t_{i}}$
centered at $t_{i}\in I_{1}$ such that $|I_{t_{i}}|$ and the oscillation of
all $D^{i}$ and $D^{\infty}$ on each $I_{t_{i}}$ are sufficiently small. By
steps 1 and 2 above, one can construct a correspondence $\mathfrak{C}_{0}$
that is fully defined on the finite set $I_{0}:=\\{t_{i}\\}$ between
$\mathcal{X}^{i}$, so that

$d_{\mathbb{F}}^{\,\mathfrak{C}_{0},I_{0}}\big{(}(\mathcal{X}^{i},(\mu^{i}_{t})_{t\in
I_{0}}),(\mathcal{X}^{j},(\mu^{j}_{t})_{t\in I_{0}})\big{)}<\epsilon$ (6.9)

for any $\epsilon>0$, if $i,j$ are sufficiently large. Then by using the small
oscillation of $D^{i}$ on $I_{t_{i}}$, one can extend the correspondence
$\mathfrak{C}_{0}$ to $\mathfrak{C}_{1}$ over $I_{1}$ so that
$(\mathcal{X}^{i},(\mu^{i}_{t})_{t\in I_{1}})$ forms a Cauchy sequence over
$I_{1}$ in the sense of (6.9) with respect to
$d_{\mathbb{F}}^{\,\mathfrak{C}_{1},I_{1}}$ (cf. [3, Lemma 7.24]). By letting
$k\to\infty$ and taking a diagonal sequence, we obtain from the completeness
of $(\mathbb{F}_{I},d_{\mathbb{F}})$ a limit metric flow pair
$(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I\setminus S})$, which has an
extended definition for all $t\in I$ by the future completion (cf. [3, Section
4.4]) so that $(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ is right
continuous for $t\in I$. Notice that Definition 6.1(1)-(7) for
$(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ are inherited from
$(\mathcal{X}^{i},(\mu^{i}_{t})_{t\in I})$. In addition, one can construct a
correspondence $\mathfrak{C}$ so that

$(\mathcal{X}^{i},(\mu^{i}_{t})_{t\in
I})\xrightarrow[i\to\infty]{\quad\mathbb{F},\mathfrak{C}\quad}(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in
I})$

on compact time intervals and the convergence is uniform over the set on which
$(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ is continuous. Moreover,
$(\mathcal{X}^{\infty}_{t^{\prime}},d^{\infty}_{t^{\prime}},\mu^{\infty}_{t^{\prime}})\subset\mathbb{M}_{r}(V,b)$
as (6.5) and $\text{Var}(\mu^{\infty}_{t})\leq H_{n}|t|$ for any $t\in I$.
Notice that $\mathcal{X}^{\infty}_{0}$ consists of a single point from which
$\mu^{\infty}_{t}$ is the conjugate heat measure.

######  Remark 6.11.

In _[3, Theorem 7.4]_, a general compactness for a subset
$\mathbb{F}_{I}^{J}(H,V,b,r)\subset\mathbb{F}_{I}^{J}$ is proved by the same
method as described above.

It follows from [3, Theorem 8.2, 8.4] that the limit metric flow pair
$(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ obtained in (6.3) is a
length space for any $t\in I$. In general, further geometric information
contained in $(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ is scarce.
However, if $(M_{i}^{n},g_{i}(t))$ are induced by Ricci shrinkers in
$\mathcal{M}(A)$, then, in particular, their Nash entropies are uniformly
bounded by Corollary 3.22. In this case, one obtains much more concrete
structure theorem regarding the limit metric flow obtained in (6.3); see [4,
Theorem 2.3, 2.4, 2.5, 2.6, 2.46] and [3, Theorem 9.31].

######  Theorem 6.12.

Let $(M_{i}^{n},g_{i}(t),x_{i})_{t\in I}$ be a sequence of pointed Ricci flows
induced by Ricci shrinkers in $\mathcal{M}(A)$ and
$(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ the limit metric flow
pair obtained in Theorem 6.10. Then the following properties hold.

  1. (1)

There exists a decomposition

$\mathcal{X}^{\infty}_{0}=\\{x_{\infty}\\},\quad\mathcal{X}^{\infty}_{t<0}=\mathcal{R}\sqcup\mathcal{S},$
(6.10)

such that $\mathcal{R}$ is given by an $n$-dimensional Ricci flow spacetime
$(\mathcal{R},\mathfrak{t},\partial^{\infty}_{\mathfrak{t}},g^{\infty})$, in
the sense of _[3, Definition 9.1]_ and
$\emph{\text{dim}}_{\mathcal{M}^{*}}(\mathcal{S})\leq n-2$, where
$\emph{\text{dim}}_{\mathcal{M}^{*}}$ denotes the $*$-Minkowski dimension in
_[3, Definition 3.42]_. Moreover, $\mu^{\infty}_{t}(\mathcal{S}_{t})=0$ for
any $t<0$.

  2. (2)

Every tangent flow $(\mathcal{X}^{\prime},(v_{x_{\infty}^{\prime};t})_{t\leq
0})$ at every point $x\in\mathcal{X}^{\infty}$ is a metric soliton in the
sense of _[3, Definition 3.57]_. Moreover, $\mathcal{X}^{\prime}$ is the
Gaussian soliton iff $x\in\mathcal{R}$. If $x\in\mathcal{S}$, the singular set
of $(\mathcal{X}^{\prime},(v_{x_{\infty}^{\prime};t})_{t\leq 0})$ on each
$t<0$ has Minkowski dimension at most $n-4$. In particular, if $n=3$, the
metric soliton is a smooth Ricci flow associated with a $3$-dimensional Ricci
shrinker. If $n=4$, each slice of the metric soliton is a smooth Ricci
shrinker orbifold with isolated singularities.

  3. (3)

$\mathcal{R}_{t}=\mathcal{R}\cap\mathcal{X}^{\infty}_{t}$ is open such that
the restriction of $d_{t}$ on $\mathcal{R}_{t}$ agrees with the length metric
of $g_{t}$.

  4. (4)

The convergence (6.3) is smooth on $\mathcal{R}$, in the following sense.
There exists an increasing sequence $U_{1}\subset
U_{2}\subset\ldots\subset\mathcal{R}$ of open subsets with
$\bigcup_{i=1}^{\infty}U_{i}=\mathcal{R}$, open subsets $V_{i}\subset
M_{i}\times I$, time-preserving diffeomorphisms $\phi_{i}:U_{i}\to V_{i}$ and
a sequence $\epsilon_{i}\to 0$ such that the following holds:

    1. (a)

We have

$\displaystyle\|\phi_{i}^{*}g^{i}-g^{\infty}\|_{C^{[\epsilon_{i}^{-1}]}(U_{i})}$
$\displaystyle\leq\epsilon_{i},$
$\displaystyle\|\phi_{i}^{*}\partial^{i}_{\mathfrak{t}}-\partial^{\infty}_{\mathfrak{t}}\|_{C^{[\epsilon_{i}^{-1}]}(U_{i})}$
$\displaystyle\leq\epsilon_{i},$
$\displaystyle\|w^{i}\circ\phi_{i}-w^{\infty}\|_{C^{[\epsilon_{i}^{-1}]}(U_{i})}$
$\displaystyle\leq\epsilon_{i},$

where $g^{i}$ is the spacetime metric induced by $g_{i}(t)$, and $w^{i}$ is
the conjugate heat kernel defined by $d\mu^{i}=w^{i}dg^{i}$,
$i\in\mathbb{N}\cup\\{\infty\\}$.

    2. (b)

Let $y_{\infty}\in\mathcal{R}$ and $y_{i}\in M_{i}\times(-\infty,0)$. Then
$y_{i}$ converges to $y_{\infty}$ within $\mathfrak{C}$ _(cf.[3, Definition
6.18])_ if and only if $y_{i}\in V_{i}$ for large $i$ and
$\phi_{i}^{-1}(y_{i})\to y_{\infty}$ in $\mathcal{R}$.

    3. (c)

If the convergence (6.3) is uniform at some time $t\in I$, then for any
compact subset $K\subset\mathcal{R}_{t}$ and for the same subsequence we have

$\sup_{x\in K\cap
U_{i}}d^{Z}_{t}(\varphi^{i}_{t}(\phi_{i}(x)),\varphi^{\infty}_{t}(x))\longrightarrow
0.$

Theorem 6.12 is a flow version of the Cheeger-Colding theory (cf. [11], [12]
and [13]). Its proof shares similar strategy as its elliptic counterparts.
Many concepts also have counterparts. For example, tangent flow corresponds to
tangent space, metric soliton corresponds to metric cone. We recall their
definitions. See [3, Definition 6.55, 3.57].

######  Definition 6.13 (Tangent flow).

Let $\mathcal{X}$ be a metric flow over $I$ and $x_{0}\in\mathcal{X}_{t_{0}}$
a point. We say that a metric flow pair
$(\mathcal{X}^{\prime},(v^{\prime}_{x_{\max};t})_{t\in I})$ is a tangent flow
of $\mathcal{X}$ at $x_{0}$ if there is a sequence of scales $\lambda_{k}>0$
with $\lambda_{k}\to\infty$ such that for any $T>0$ the parabolic rescalings

$\big{(}\mathcal{X}^{-t_{0},\lambda_{k}}_{[-T,0]},(v_{x_{0};t}^{-t_{0},\lambda_{k}})_{\lambda_{k}^{-2}t+t_{0}\in
I^{\prime},t\in[-T,0]}\big{)}$

$\mathbb{F}$-converge to
$(\mathcal{X}^{\prime}_{[-T,0]},(v^{\prime}_{x_{\max};t})_{t\in[-T,0]})$.

######  Definition 6.14 (Metric soliton).

A metric flow pair $(\mathcal{X},(\mu_{t})_{t\in I})$ is called a metric
soliton if there is a tuple

$\big{(}X,d,\mu,(v^{\prime}_{x;t})_{x\in X;t\leq 0}\big{)}$

and a map $\phi:\mathcal{X}\to X$ such that the following holds:

  1. 1.

For any $t\in I$, the map
$\phi_{t}:(\mathcal{X}_{t},d_{t},\mu_{t})\to(X,\sqrt{t}d,\mu)$ is an isometry
between metric measure spaces.

  2. 2.

For any $x\in\mathcal{X}_{t}$, $s\in I$ with $s\leq t$, we have
$(\phi_{s})_{*}v_{x;s}=v^{\prime}_{\phi_{t}(x);\log(s/t)}$.

Roughly speaking, a metric soliton is a metric flow pair induced by a metric
measure space in a shrinking way. In general, a tangent flow of a metric flow
may not be a metric soliton. In the setting of Theorem 6.12, every tangent
flow of $(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ is also an
$\mathbb{F}$-limit of a sequence of Ricci flows induced by Ricci shrinkers in
$\mathcal{M}(A)$ (cf. [3, Theorem 6.58]).

Notice that the limit metric flow
$(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ in (6.3) always admits a
regular-singular decomposition

$\mathcal{X}^{\infty}_{t<0}=\mathcal{R}\sqcup\mathcal{S},$

so that $\mathcal{R}$ is given by a Ricci flow spacetime (cf. [3, Definition
9.1]). The key point is to control the size of the singular part in the
appropriate sense. To avoid the distance distortion at different time-slices,
one can redefine the Hausdorff and Minkowski dimensions (denoted by
$\mathcal{H}^{*}$ and $\mathcal{M}^{*}$ respectively) by using the
$P^{*}$-parabolic balls instead of the conventional ones; see [3, Definition
3.41, 3.42].

One can control the size of $\mathcal{S}$ quantitatively. Let
$(M^{n},g(t))_{t\in I}$ be the Ricci flow induced by a Ricci shrinker in
$\mathcal{M}(A)$. We fix a point $(x_{0},t_{0})\in M\times I$ and define
$\tau=t_{0}-t$ and
$H(x_{0},t_{0},\cdot,\cdot)=(4\pi\tau)^{-\frac{n}{2}}e^{-b}$. We next recall
the following definitions from [4, Definition 5.1, 5.5, 5.6, 5.7], which
indicate the extent to which the local geometry around $(x_{0},t_{0})$ is a
Ricci shrinker, Ricci flat space or splitting off an $\mathbb{R}^{k}$.

######  Definition 6.15 (Almost self-similarity).

Let $(M^{n},g(t))_{t\in I}$ be the Ricci flow induced by a Ricci shrinker. The
point $(x_{0},t_{0})\in M\times I$ is called $(\epsilon,r)$-selfsimilar if the
following holds:

$\displaystyle\int_{t_{0}-\epsilon^{-1}r^{2}}^{t_{0}-\epsilon
r^{2}}\int_{M}\tau\Big{|}Rc+\nabla^{2}b-\frac{1}{2\tau}g\Big{|}^{2}dv_{x_{0},t_{0};t}dt\leq\epsilon,$
$\displaystyle\int_{M}\big{|}\tau(2\triangle b-|\nabla
b|^{2}+R)+b-n-\mathcal{N}_{x_{0},t_{0}}(r^{2})\big{|}dv_{x_{0},t_{0};t}\leq\epsilon,\quad\forall
t\in[t_{0}-\epsilon^{-1}r^{2},t_{0}-\epsilon r^{2}].$

######  Definition 6.16 (Almost static).

The point $(x_{0},t_{0})$ is called $(\epsilon,r)$-static if the following
holds:

$\displaystyle r^{2}\int_{t_{0}-\epsilon^{-1}r^{2}}^{t_{0}-\epsilon
r^{2}}\int_{M}|Rc|^{2}dv_{x_{0},t_{0};t}dt\leq\epsilon,$ $\displaystyle
r^{2}\int_{M}R\,dv_{x_{0},t_{0};t}\leq\epsilon,\quad\forall
t\in[t_{0}-\epsilon^{-1}r^{2},t_{0}-\epsilon r^{2}].$

######  Definition 6.17 (Weak splitting).

$(x_{0},t_{0})$ is called weakly $(k,\epsilon,r)$-split if there exists a
vector-valued function
$\vec{y}=(y_{1},\ldots,y_{k}):M\times[t_{0}-\epsilon^{-1}r^{2},t_{0}-\epsilon
r^{2}]\to\mathbb{R}^{k}$ with the following properties for all
$i,j=1,\ldots,k$:

  1. (1)

We have

$r^{-1}\int_{t_{0}-\epsilon^{-1}r^{2}}^{t_{0}-\epsilon r^{2}}\int_{M}|\square
y_{i}|dv_{x_{0},t_{0};t}dt\leq\epsilon.$

  2. (2)

We have

$r^{-2}\int_{t_{0}-\epsilon^{-1}r^{2}}^{t_{0}-\epsilon r^{2}}\int_{M}|\nabla
y_{i}\cdot\nabla y_{j}-\delta_{ij}|dv_{x_{0},t_{0};t}dt\leq\epsilon.$

######  Definition 6.18 (Strong splitting).

$(x_{0},t_{0})$ is called strongly $(k,\epsilon,r)$-split if there exists a
vector-valued function
$\vec{y}=(y_{1},\ldots,y_{k}):M\times[t_{0}-\epsilon^{-1}r^{2},t_{0}-\epsilon
r^{2}]\to\mathbb{R}^{k}$ with the following properties for all
$i,j=1,\ldots,k$:

  1. (1)

$y_{i}$ solves the heat equation $\square y_{i}=0$ on
$M\times[t_{0}-\epsilon^{-1}r^{2},t_{0}-\epsilon r^{2}]$.

  2. (2)

We have

$r^{-2}\int_{t_{0}-\epsilon^{-1}r^{2}}^{t_{0}-\epsilon r^{2}}\int_{M}|\nabla
y_{i}\cdot\nabla y_{j}-\delta_{ij}|dv_{x_{0},t_{0};t}dt\leq\epsilon.$

  3. (3)

For all $t\in[t_{0}-\epsilon^{-1}r^{2},t_{0}-\epsilon r^{2}]$ we have

$\int_{M}y_{i}\,dv_{x_{0}t_{0};t}=0.$

It can be proved (cf. [4, Proposition 12.1]) that if $(x_{0},t_{0})$ is weakly
$(k,\epsilon,r)$-split, then it is strongly $(k,\delta(\epsilon),r)$-split.
With these definitions, one can consider the following quantitative
stratification.

######  Definition 6.19.

For $\epsilon>0$ and $0<r_{1}<r_{2}\leq\infty$ the effective strata

$\widetilde{\mathcal{S}}^{\epsilon,0}_{r_{1},r_{2}}\subset\widetilde{\mathcal{S}}^{\epsilon,1}_{r_{1},r_{2}}\subset\widetilde{\mathcal{S}}^{\epsilon,2}_{r_{1},r_{2}}\subset\ldots\subset\widetilde{\mathcal{S}}^{\epsilon,n+2}_{r_{1},r_{2}}\subset
M\times I$

are defined as follows:
$(x^{\prime},t^{\prime})\in\widetilde{\mathcal{S}}^{\epsilon,k}_{r_{1},r_{2}}$
if and only for all $r^{\prime}\in(r_{1},r_{2})$ none of the following two
properties hold:

  1. 1.

$(x^{\prime},t^{\prime})$ is $(\epsilon,r^{\prime})$-selfsimilar and weakly
$(k+1,\epsilon,r^{\prime})$-split.

  2. 2.

$(x^{\prime},t^{\prime})$ is $(\epsilon,r^{\prime})$-selfsimilar,
$(\epsilon,r^{\prime})$-static and weakly $(k-1,\epsilon,r^{\prime})$-split.

By a delicate choice of the covering by $P^{*}$-parabolic balls, it can be
proved, see [4, Proposition 11.2], that for any $0<\sigma<\epsilon$, there are
points
$(x_{1},t_{1}),\ldots,(x_{N},t_{N})\in\widetilde{\mathcal{S}}^{\epsilon,k}_{\sigma
r,\epsilon r}\cap P^{*}(x_{0},t_{0};r)$ with $N\leq
C(A,\epsilon)\sigma^{-k-\epsilon}$ and

$\widetilde{\mathcal{S}}^{\epsilon,k}_{\sigma r,\epsilon r}\cap
P^{*}(x_{0},t_{0};r)\subset\bigcup_{i=1}^{N}P^{*}(x_{i},t_{i};\sigma r).$
(6.11)

Notice that (6.11) can be regarded as a parabolic version of the covering in
[12] by Cheeger and Naber.

On the complement of $\widetilde{\mathcal{S}}^{\epsilon,n-2}_{\sigma
r,\epsilon r}$, the following $\epsilon$-regularity theorem is proved (cf. [4,
Proposition 17.1]), which can be viewed as a parabolic analogue of Cheeger-
Naber’s codimension $4$ theorem in [13]. Roughly speaking, one needs to rule
out the tangent flows which are Ricci-flat, and split off an
$\mathbb{R}^{n-3}$.

######  Proposition 6.20.

There exists a constant $\epsilon=\epsilon(n,A)>0$ such that the following
holds. Let $(M^{n},g(t))_{t\in I}$ be the Ricci flow induced by a Ricci
shrinker in $\mathcal{M}(A)$. Suppose that $(x_{0},t_{0})$ is strongly
$(n-1,\epsilon,r)$-split or strongly $(n-3,\epsilon,r)$-split and
$(\epsilon,r)$-static. Then $r_{\emph{Rm}}(x_{0},t_{0})\geq\epsilon r$.

There are many implications of Proposition 6.20. Notice that one has the
following decomposition:

$\mathcal{X}^{\infty}_{t<0}=\mathcal{R}^{*}\sqcup\mathcal{S}^{*},$

where $\mathcal{R}^{*}\subset\mathcal{R}$ is the set of points where the
convergence (6.3) is smooth as defined in [3, Section 9.4]. Since
$\mathcal{S}\subset\mathcal{S}^{*}$, one can obtain the estimate of
$*$-Minkowski dimension of $\mathcal{S}$ by that of $\mathcal{S}^{*}$ from
(6.11) and Proposition 6.20 (cf. [4, Theorem 15.28 (a)]). Moreover, it can be
proved that $\mathcal{S}^{*}\cap\mathcal{X}^{\infty}_{t}$ has measure $0$ for
any $t<0$ (cf. [4, Theorem 15.28 (b)]). Therefore, Theorem 6.12 (1) is
obtained.

Since $\mathcal{S}$ has measure $0$ on each time-slice, one can extend the
definition of the Nash entropy on $\mathcal{X}^{\infty}$. Therefore, the Nash
entropy at the base point $x^{\prime}$ of any tangent flow
$(\mathcal{X}^{\prime},(v_{x^{\prime};t})_{t\in I})$ of
$(\mathcal{X}^{\infty},(\mu^{\infty}_{t})_{t\in I})$ is a constant. By the
relation between the Nash entropy and the almost self-similarity (cf. [4,
Proposition 7.1]), one concludes that
$(\mathcal{X}^{\prime},(v_{x^{\prime};t})_{t\in I})$ is a metric soliton since
its regular part admits an incomplete Ricci shrinker and the tangent flow
itself is determined by its regular part due to the high codimension of the
singular part (cf. [4, Theorem 15.60, 15.69]). Moreover, the singular set on
each time-slice of $(\mathcal{X}^{\prime},(v_{x^{\prime};t})_{t\in I})$ has
Minkowski dimension $\leq n-4$ (cf. [4, Theorem 2.16]). Furthermore, the fact
that $x\in\mathcal{R}$ iff $\mathcal{X}^{\prime}$ is the Gaussian soliton
follows from the $\epsilon$-regularity theorem 5.15 and the convergence of the
Nash entropies under (6.3) (cf. [4, Theorem 2.11, 2.14]). Notice that if
$n=4$, each time-slice of $(\mathcal{X}^{\prime},(v_{x^{\prime};t})_{t\in I})$
is a smooth orbifold with isolated singularities since each tangent flow at
any singular point of $(\mathcal{X}^{\prime},(v_{x^{\prime};t})_{t\in I})$ is
a flat cone (cf. [4, Theorem 2.46]). Therefore, we obtain Theorem 6.12 (2).

For Theorem 6.12 (3), the inequality $d_{t}\leq d_{g_{t}}$ is clear. The
opposite inequality is proved by showing that any $u\in
C^{0}(\mathcal{R}_{t})$ that is $1$-Lipschitz with respect to $d_{g_{t}}$ is
also $1$-Lipschitz with respect to $d_{t}$ (cf. [4, Theorem 15.28 (c)]). The
argument uses the high codimension of $\mathcal{S}$, the fact that
$\mathcal{X}^{\infty}$ is future continuous at $t$, and the fact that
$\mathcal{R}=\mathcal{R}^{*}$, which can be proved by using the
$\epsilon$-regularity theorem and the convergence of the Nash entropies (cf.
[4, Corollary 15.47]).

Once we know $\mathcal{R}=\mathcal{R}^{*}$, the diffeomorphisms in Theorem
6.21 (4) can be obtained by patching all local conventional Ricci flows into a
Ricci flow spacetime by a center of mass construction (cf. [3, Theorem 9.31]).
Notice that similar constructions are well-known for the Cheeger-Gromov
convergence (cf. Remark 7.7 of [24] and references therein). All assertions
Theorem 6.21 (4) are proved by smooth convergence. Therefore, Theorem 6.12 (4)
is obtained.

As an application of the theory of $\mathbb{F}$-convergence, we have the
following backward pseudolocality theorem; see [4, Theorem 2.47]. Earlier
backward pseudolocality can be found in [34, Corollary 11.6(b)] [15, Lemma
4.2] [16, Theorem 4.7] [6, Theorem 1.5].

######  Theorem 6.21 (Backward pseudolocality theorem).

For any $n\in\mathbb{N}$ and $\alpha>0$ there is an $\epsilon(n,\alpha)>0$
such that the following holds.

Let $(M^{n},g(t))_{t\in I}$ be a Ricci flow induced by a Ricci shrinker. Given
$(x_{0},t_{0})\in M\times I$ and $r>0$, if

$|B_{t_{0}}(x_{0},r)|\geq\alpha r^{n},\qquad|Rm|\leq(\alpha
r)^{-2}\quad\text{on}\quad B_{t_{0}}(x_{0},r),$

then

$|Rm|\leq(\epsilon r)^{-2}\quad\text{on}\quad
P(x_{0},t_{0};(1-\alpha)r,-(\epsilon r)^{2},0).$

Note that the combination of the above theorem with the forward pseudolocality
(cf. Theorem 24 of [28]), we arrive at the two-sided pseudolocality. Thus
Theorem 1.6 is proved.

Combining Theorem 6.21 and (5.23), we have

######  Corollary 6.22 (Comparison of the curvature radii).

There exists a constant $C(n,A)>1$ such that the following holds.

Let $(M^{n},g(t))_{t\in I^{\prime}}$ be a Ricci flow induced by a Ricci
shrinker in $\mathcal{M}(A)$. Then for any $(x,t)\in M\times I^{\prime}$,

$r_{\emph{Rm}}(x,t)\leq r_{\emph{Rm}}^{-}(x,t)\leq r_{\emph{Rm}}^{s}(x,t)\leq
Cr_{\emph{Rm}}(x,t).$

Another application is the following integral estimate using the quantitative
stratification; see [4, Theorem 2.28].

######  Theorem 6.23.

Let $(M^{n},g(t))_{t<1}$ be a Ricci flow associated with a Ricci shrinker in
$\mathcal{M}(A)$. Then for any $(x_{0},t_{0})\in M\times(-\infty,1)$, $r>0$
and $\epsilon>0$,

$\displaystyle\int_{[t_{0}-r^{2},t_{0}+r^{2}]\cap(-\infty,1)}\int_{P^{*}(x_{0},t_{0};r)\cap
M\times\\{t\\}}|Rm|^{2-\epsilon}\,dV_{t}dt$ $\displaystyle\leq$
$\displaystyle\int_{[t_{0}-r^{2},t_{0}+r^{2}]\cap(-\infty,1)}\int_{P^{*}(x_{0},t_{0};r)\cap
M\times\\{t\\}}r_{\emph{Rm}}^{-4+2\epsilon}\,dV_{t}dt\leq
C(n,A,\epsilon)r^{n-2+2\epsilon}.$ (6.12)

As a corollary, we prove

######  Corollary 6.24.

Let $(M^{n},g,f,p)$ be a Ricci shrinker in $\mathcal{M}(A)$. Then

$\displaystyle\int_{d(p,\cdot)\leq r}|Rm|^{2-\epsilon}\,dV\leq$
$\displaystyle\int_{d(p,\cdot)\leq r}r_{\emph{Rm}}^{-4+2\epsilon}\,dV\leq
Cr^{n+2\epsilon-2},$ (6.13) $\displaystyle\int_{d(p,\cdot)\geq
1}\frac{|Rm|^{2-\epsilon}}{d^{n+2\epsilon-2}(p,\cdot)}\,dV\leq$
$\displaystyle\int_{d(p,\cdot)\geq
1}\frac{r_{\emph{Rm}}^{-4+2\epsilon}}{d^{n+2\epsilon-2}(p,\cdot)}\,dV\leq C$
(6.14)

for any $\epsilon>0$ and $r\geq 1$, where
$r_{\emph{Rm}}(\cdot)=r_{\emph{Rm}}(\cdot,0)$ and $C=C(n,A,\epsilon)$.

###### Proof.

We consider the Ricci flow $(M,g(t))_{t<1}$ associated with the given Ricci
shrinker. It follows from Proposition 5.7 that

$\displaystyle Q(p,0;1,0,1)\subset P^{*}(p,0;C_{1},0,1)\subset
P^{*}(p,0;C_{1})$

for some constant $C_{1}=C_{1}(n,A)>1$. Therefore, it follows from Theorem
6.23 that with $(x_{0},t_{0})=(p,0)$ and $r=C_{1}$ that

$\displaystyle\int_{0}^{1}\int_{d_{t}(p,\cdot)<1}|Rm|^{2-\epsilon}\,dV_{t}dt\leq
C(n,A,\epsilon).$ (6.15)

Since $g(t)=(1-t)(\psi^{t})^{*}g$ and $\psi^{t}$ is defined by (2.3), we have
$d_{t}(x,p)=\sqrt{1-t}d(x^{t},p)$ and $|Rm|(x,t)=|Rm|(x^{t})/(1-t)$, where
$x^{t}=\psi^{t}(x)$. Therefore, we have

$\displaystyle\int_{d_{t}(p,x)<1}|Rm|^{2-\epsilon}(x,t)\,dV_{t}(x)=(1-t)^{\frac{n}{2}-2+\epsilon}\int_{d(x,p)<\frac{1}{\sqrt{1-t}}}|Rm|^{2-\epsilon}(x)\,dV(x).$
(6.16)

By a change of variable with $t=1-r^{-2}$, it follows from (6.15) and (6.16)
that

$\displaystyle\int_{1}^{\infty}r^{1-2\epsilon-n}m(r)\,dr\leq C(n,A,\epsilon),$
(6.17)

where

$\displaystyle m(r):=\int_{d(\cdot,p)<r}|Rm|^{2-\epsilon}\,dV.$

We claim that there exists a sequence $r_{i}\to\infty$ such that

$\displaystyle\lim_{i\to\infty}\frac{m(r_{i})}{r_{i}^{n+2\epsilon-2}}=0.$
(6.18)

Otherwise, there exists a constant $\delta>0$ such that $m(r)\geq\delta
r^{n+2\epsilon-2}$ for sufficiently large $r$. However, it contradicts (6.17).

We apply the integration by parts to (6.17) from $1$ to $r_{i}$ to (6.17) and
obtain

$\displaystyle\int_{1\leq d(p,\cdot)\leq
r_{i}}\frac{|Rm|^{2-\epsilon}}{d^{n+2\epsilon-2}(p,\cdot)}\,dV\leq
C(n,A,\epsilon)+m(r_{i})r_{i}^{2-2\epsilon+n}.$

By letting $i\to\infty$, we have from (6.18) that

$\displaystyle\int_{d(p,\cdot)\geq
1}\frac{|Rm|^{2-\epsilon}}{d^{n+2\epsilon-2}(p,\cdot)}\,dV\leq
C(n,A,\epsilon).$ (6.19)

In addition, we have for any $r\geq 1$,

$\displaystyle r^{2-2\epsilon-n}\int_{1\leq d(p,\cdot)\leq
r}|Rm|^{2-\epsilon}\,dV\leq\int_{1\leq d(p,\cdot)\leq
r}\frac{|Rm|^{2-\epsilon}}{d^{n+2\epsilon-2}(p,\cdot)}\,dV\leq
C(n,A,\epsilon).$

Therefore, for any $r\geq 1$,

$\displaystyle\int_{d(p,\cdot)\leq r}|Rm|^{2-\epsilon}\,dV\leq
C(n,A,\epsilon)r^{n+2\epsilon-2},$ (6.20)

since $m(1)$ is bounded by (6.17). In sum, the inequalities involving $|Rm|$
in (6.13) and (6.14) are proved.

Notice that for any $(x_{0},t_{0})\in M\times(-\infty,1)$, it follows from the
definition of $\psi^{t}$ (2.3) that

$\displaystyle\psi^{\theta(t)}\circ\psi^{t_{0}}=\psi^{t},$

where $\theta(t):=\frac{t-t_{0}}{1-t_{0}}$. Therefore, for any $t<1$,

$\displaystyle
g(t)=(1-t)(\psi^{t})^{*}g=(1-t_{0})(1-\theta(t))(\psi^{t_{0}})^{*}(\psi^{\theta(t)})^{*}g=(1-t_{0})(\psi^{t_{0}})^{*}g(\theta(t)).$

Therefore,

$\displaystyle
r_{\text{Rm}}(x_{0},t_{0})=\sqrt{1-t_{0}}r_{\text{Rm}}(x_{0}^{t_{0}},0).$

Now, the conclusion regarding $r_{\text{Rm}}$ can be proved similarly. ∎

We end this section by proving a gap property for the volume ratio at
infinity.

######  Corollary 6.25.

Let $(M^{n},g,f,p)$ be a Ricci shrinker in $\mathcal{M}(A)$. Suppose

$\displaystyle\liminf_{r\to\infty}\frac{|B(p,r)|}{r^{n}}=0.$ (6.21)

Then

$\displaystyle|B(p,r)|\leq Cr^{n-2+\epsilon}$ (6.22)

for any $r\geq 1$ and some $C=C(n,A,\epsilon)$.

###### Proof.

We claim that $r_{\text{Rm}}(x)<2$ for any $x$. Indeed, if
$r_{\text{Rm}}(x)\geq 2$, we have

$\displaystyle|Rm|(y,t)<1$

for any $y\in B(x,1)$ and $t<1$. By the same argument as in [28, Corollary 9],
we obtain that

$\displaystyle\psi^{t}\left(B(x,1)\right)\subset
B\left(p,\frac{c_{1}}{\sqrt{1-t}}\right)\setminus
B\left(p,\frac{c_{2}}{\sqrt{1-t}}\right)$ (6.23)

for $c_{1}>c_{2}>0$, if $t$ is sufficiently close to $1$. From the standard
distance distortion and Theorem 4.1, we obtain that

$\displaystyle|\psi^{t}\left(B(x,1)\right)|\geq c_{3}(1-t)^{-\frac{n}{2}}.$
(6.24)

However, (6.23) and (6.24) contradict (6.21). Thus the desired inequality
(6.22) follows immediately from (6.13). ∎

##  Appendix A Integral estimates for the conjugate heat kernel

In this appendix, we generalize some integral estimates regarding the
conjugate heat kernel from [4, Section 6] to Ricci flows associated with Ricci
shrinkers. These estimates also hold for Ricci flows induced by Ricci
shrinkers since they are scaling-invariant.

Throughout this appendix, let $(M^{n},g(t))_{t<1}$ be the Ricci flow
associated with a Ricci shrinker in $\mathcal{M}(A)$. We fix a spacetime point
$(x_{0},t_{0})\in M\times(-\infty,1)$ and set $dv_{t}=dv_{x_{0},t_{0};t}$ and
$\tau=t_{0}-t$. Moreover, we define $w=w(x,t)$ and $b=b(x,t)$ by
$w=H(x_{0},t_{0},x,t)=(4\pi(t_{0}-t))^{-\frac{n}{2}}e^{-b}$.

######  Lemma A.1.

There exists a constant $C=C(n,A)>1$ such that for any $0<\tau_{0}<\tau_{1}$,

$\displaystyle\int_{t_{0}-\tau_{1}}^{t_{0}-\tau_{0}}\int_{M}\left\\{|Rc|^{2}+|\nabla^{2}b|^{2}\right\\}dv_{t}\,dt\leq
C\tau_{0}^{-1}\left(1+\log\frac{\tau_{1}}{\tau_{0}}\right).$ (A.1)

###### Proof.

Without loss of generality, we assume $t_{0}=0$.

From Corollary 3.20, we have for any $\tau>0$ that

$\displaystyle\int_{M}|\nabla b|^{2}+R\,dv_{-\tau}\leq\frac{n}{2\tau}.$ (A.2)

Direct calculation shows that

$\displaystyle\partial_{t}\int_{M}Rw\phi^{r}\,dV_{t}$
$\displaystyle=\int_{M}\left\\{(\square
R)w\phi^{r}-R\square^{*}(w\phi^{r})\right\\}dV_{t}$
$\displaystyle=\int_{M}2|Rc|^{2}w\phi^{r}+R\left\\{w(\Delta\phi^{r}+\phi^{r}_{t})+2\langle\nabla
w,\nabla\phi^{r}\rangle\right\\}\,dV_{t}$
$\displaystyle=\int_{M}\left\\{2|Rc|^{2}\phi^{r}+R(\Delta\phi^{r}+\phi^{r}_{t})-2R\langle\nabla
b,\nabla\phi^{r}\rangle\right\\}\,dv_{t}.$

Integrating the above equation from $-\tau_{1}$ to $-\tau_{0}$, we obtain

$\displaystyle\int_{-\tau_{1}}^{-\tau_{0}}\int_{M}2|Rc|^{2}\phi^{r}\,dv_{t}dt$
$\displaystyle\leq$
$\displaystyle\int_{M}R\phi^{r}\,dv_{-\tau_{0}}+\int_{-\tau_{1}}^{-\tau_{0}}\int_{M}\left\\{R(|\Delta\phi^{r}|+|\phi_{t}^{r}|)+R^{2}|\nabla\phi^{r}|+|\nabla
b|^{2}|\nabla\phi^{r}|\right\\}dv_{t}dt.$ (A.3)

From (2.8) and Lemma 2.3, $R$ increases at most quadratically. Combining
(2.12), (2.13), (2.14) and (A.2), it follows that the last integral in (A)
tends to $0$ as $r\to\infty$. Therefore, we obtain

$\displaystyle\int_{-\tau_{1}}^{-\tau_{0}}\int_{M}|Rc|^{2}\,dv_{t}dt\leq\frac{1}{2}\int_{M}R\,dv_{-\tau_{0}}\leq\frac{n}{4\tau_{0}}.$
(A.4)

On the other hand, it follows from (3.40) and Corollary 3.22 that

$\displaystyle\int_{-\tau_{1}}^{-\tau_{0}}2\tau\int_{M}\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}\,dv_{t}dt\leq-\mathcal{W}_{(x_{0},t_{0})}(\tau_{1})\leq
A.$ (A.5)

By virtue of the elementary identity $(x-y)^{2}\geq x^{2}/2-y^{2}$, it follows
from (A.5) that

$\displaystyle\tau_{0}\int_{-\tau_{1}}^{-\tau_{0}}\int_{M}|Rc+\nabla^{2}b|^{2}\,dv_{t}dt\leq\int_{-\tau_{1}}^{-\tau_{0}}\tau\int_{M}|Rc+\nabla^{2}b|^{2}\,dv_{t}dt\leq
A+\frac{n}{2}\log\frac{\tau_{1}}{\tau_{0}}.$ (A.6)

Combining (A.4) and (A.6), the conclusion follows immediately. ∎

######  Lemma A.2.

There exists a constant $C=C(n,A)>1$ such that the following estimates hold
for any $t<t_{0}$ and $0\leq s\leq 1/4$.

$\displaystyle\int_{M}\left\\{1+|b|+\tau(|\Delta b|+|\nabla
b|^{2}+R)\right\\}e^{sb}\,dv_{t}\leq C.$ (A.7)

###### Proof.

We compute

$\displaystyle\frac{d}{ds}\int_{M}e^{sb}\,dv_{t}=\int_{M}be^{sb}\,dv_{t}.$
(A.8)

Here, the differentiation under the integral sign is allowed by Theorem 4.16
and Remark 4.17. By the differential Harnack inequality [28, Theorem 21], we
calculate

$\displaystyle\int_{M}be^{sb}\,dv_{t}\leq$
$\displaystyle\int_{M}(\tau(-2\Delta b+|\nabla b|^{2}-R)+n)e^{sb}\,dv_{t}$
$\displaystyle=$ $\displaystyle\int_{M}(\tau((2s-1)|\nabla
b|^{2}-R)+n)e^{sb}\,dv_{t}\leq n\int_{M}e^{sb}\,dv_{t},$ (A.9)

where the integration by parts in the equality can be justified similarly as
in Remark 3.21. Combining (A.8) and (A.9), we obtain

$\displaystyle\int_{M}e^{sb}\,dv_{t}\leq e^{ns}.$

On the other hand, it follows from Theorem 4.4 that $b\geq-A$. Therefore, it
follows from (A.9) and the above inequality that

$\displaystyle\int_{M}\left\\{|b|+\tau(|\nabla
b|^{2}+R)\right\\}e^{sb}\,dv_{t}\leq C(n,A).$ (A.10)

Applying the differential Harnack inequality and the integration by parts
again, we obtain

$\displaystyle\int_{M}2\tau|\Delta b|e^{sb}\,dv_{t}\leq$
$\displaystyle\int_{M}\left\\{|u|+\tau(|\nabla
b|^{2}+R)+|b|+n\right\\}e^{sb}\,dv_{t}$ $\displaystyle=$
$\displaystyle\int_{M}\left\\{-u+\tau(|\nabla
b|^{2}+R)+|b|+n\right\\}e^{sb}\,dv_{t}$ $\displaystyle\leq$
$\displaystyle\int_{M}\left\\{2s\tau|\nabla
b|^{2}+2|b|+2n\right\\}e^{sb}\,dv_{t}\leq C(n,A),$

where $u=\tau(2\Delta b-|\nabla b|^{2}+R)+b-n\leq 0$. It is clear that (A.7)
follows from the combination of (A.10) and the above inequality. ∎

######  Lemma A.3.

There exists a constant $C=C(n)>1$ such that the following estimates hold for
any $t<t_{0}$ and $0\leq s\leq 1/4$.

$\displaystyle\int_{M}|\nabla b|^{4}e^{sb}\,dv_{t}\leq
C\int_{M}|\nabla^{2}b|^{2}e^{sb}\,dv_{t}.$ (A.11)

###### Proof.

In the proof, all constants $C>1$ depend only on $n$, which may be different
line by line.

We compute for $s\leq 1/4$ that

$\displaystyle\int_{M}|\nabla b|^{4}\phi^{r}e^{sb}\,dv_{t}$ $\displaystyle=$
$\displaystyle(4\pi\tau)^{-\frac{n}{2}}\int_{M}|\nabla
b|^{4}\phi^{r}e^{(s-1)b}\,dV_{t}$ $\displaystyle=$
$\displaystyle(4\pi\tau)^{-\frac{n}{2}}(s-1)^{-1}\int_{M}|\nabla
b|^{2}\langle\nabla b,\nabla e^{(s-1)b}\rangle\phi^{r}\,dV_{t}$
$\displaystyle=$
$\displaystyle(4\pi\tau)^{-\frac{n}{2}}(1-s)^{-1}\int_{M}\left(2\nabla^{2}b(\nabla
b,\nabla b)+|\nabla b|^{2}\Delta b\right)\phi^{r}e^{(s-1)b}\,dV_{t}+Z$
$\displaystyle\leq$ $\displaystyle
C(4\pi\tau)^{-\frac{n}{2}}(1-s)^{-1}\int_{M}|\nabla^{2}b||\nabla
b|^{2}\phi^{r}e^{(s-1)b}\,dV_{t}+Z$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\int_{M}|\nabla
b|^{4}\phi^{r}e^{sb}\,dv_{t}+C\int_{M}|\nabla^{2}b|^{2}\phi^{r}e^{sb}\,dv_{t}+Z,$
(A.12)

where the remainder

$\displaystyle Z:$ $\displaystyle=(1-s)^{-1}\int_{M}|\nabla
b|^{2}\langle\nabla b,\nabla\phi^{r}\rangle e^{sb}\,dv_{t}\leq
2\int_{M}|\nabla b|^{3}|\nabla\phi^{r}|e^{sb}\,dv_{t}$
$\displaystyle\leq\frac{1}{4}\int_{M}|\nabla
b|^{4}\phi^{r}e^{sb}\,dv_{t}+4\int_{M}|\nabla
b|^{2}|\nabla\phi^{r}|^{2}(\phi^{r})^{-1}e^{sb}\,dv_{t}.$ (A.13)

Applying Lemma A.2 and (2.12), we conclude from (A.12) and (A.13) that

$\displaystyle\int_{M}|\nabla b|^{4}e^{sb}\phi^{r}\,dv_{t}\leq
C\int_{M}|\nabla^{2}b|^{2}e^{sb}\phi^{r}\,dv_{t}+\epsilon(r)$ (A.14)

where $\epsilon(r)\to 0$ as $r\to\infty$. Thus we arrive at (A.11) by letting
$r\to\infty$ in the above inequality. ∎

The main result of this section is the following spacetime integral estimate.

######  Proposition A.4.

There exists a constant $C=C(n,A)>1$ and $\bar{s}=\bar{s}(n)<1$ such that the
following estimates hold for any $r>0$, $0<\theta<1/2$ and $s\leq\bar{s}$.

$\displaystyle\int_{t_{0}-\tau_{0}}^{t_{0}-\theta\tau_{0}}\int_{M}\tau(|Rc|^{2}+|\nabla^{2}b|^{2}+|\nabla
b|^{4})e^{sb}\,dv_{t}dt\leq C\log\theta^{-1}.$ (A.15)

###### Proof.

In the proof, all constants $C$ depend on $n$, and $C^{\prime}$ depend on $n$
and $A$. Moreover, we use $\epsilon(r)$ to denote a function independent of
$t$ such that $\epsilon(r)\to 0$ if $r\to\infty$. Those terms may be different
line by line. Without loss of generality, we assume $t_{0}=0$.

We set $u=\tau(2\Delta b-|\nabla b|^{2}+R)+b-n\leq 0$. Recall that from [34],
we have the celebrated identity

$\displaystyle\square^{*}(uw)=-2\tau\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}w.$

Moreover, we have

$\displaystyle\square b=-2\Delta b+|\nabla
b|^{2}-R+\frac{n}{2\tau}=\tau^{-1}(b-u-n/2).$

Direct computation shows that

$\displaystyle\partial_{t}\int_{M}uwe^{sb}\phi^{r}\,dV_{t}$ $\displaystyle=$
$\displaystyle\int_{M}\left\\{\square(e^{sb}\phi^{r})uw-e^{sb}\phi^{r}\square^{*}(uw)\right\\}dV_{t}$
$\displaystyle=$ $\displaystyle\int_{M}\left\\{\left((\square
e^{sb})\phi^{r}+e^{sb}(\square\phi^{r})-2\langle\nabla\phi^{r},\nabla
e^{sb}\rangle\right)uw+2\tau\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}we^{sb}\phi^{r}\right\\}dV_{t}$
$\displaystyle=$ $\displaystyle\int_{M}\left\\{\left((s\square b-s^{2}|\nabla
b|^{2})e^{sb}\phi^{r}+e^{sb}(\square\phi^{r})-2\langle\nabla\phi^{r},\nabla
e^{sb}\rangle\right)u+2\tau\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}e^{sb}\phi^{r}\right\\}wdV_{t},$
$\displaystyle=$
$\displaystyle\int_{M}\left\\{\left((s\tau^{-1}(b-u-n/2)-s^{2}|\nabla
b|^{2})\phi^{r}+\square\phi^{r}-2s\langle\nabla\phi^{r},\nabla
b\rangle\right)u+2\tau\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}\phi^{r}\right\\}e^{sb}dv_{t}.$

It follows that

$\displaystyle\partial_{t}\int_{M}uwe^{sb}\phi^{r}\,dV_{t}$
$\displaystyle\geq$
$\displaystyle\int_{M}\left\\{(s\tau^{-1}(b-u-n/2))u\phi^{r}+\left(\square\phi^{r}-2s\langle\nabla\phi^{r},\nabla
b\rangle\right)u+2\tau\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}\phi^{r}\right\\}e^{sb}dv_{t}$
$\displaystyle\geq$
$\displaystyle\int_{M}\left\\{-Cs\tau^{-1}(u^{2}+b^{2}+1)\phi^{r}+\left(\square\phi^{r}-2s\langle\nabla\phi^{r},\nabla
b\rangle\right)u+2\tau\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}\phi^{r}\right\\}e^{sb}dv_{t}$
$\displaystyle\geq$ $\displaystyle\int_{M}\left\\{-Cs\left(\tau((\Delta
b)^{2}+|\nabla
b|^{4}+R^{2})+\tau^{-1}(b^{2}+1)\right)+2\tau\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}\right\\}\phi^{r}e^{sb}dv_{t}+X_{t},$
(A.16)

where

$\displaystyle
X_{t}:=\int_{M}\left(\square\phi^{r}-2s\langle\nabla\phi^{r},\nabla
b\rangle\right)ue^{sb}dv_{t}.$

Define $X^{\prime}:=\int_{-\tau_{0}}^{-\theta\tau_{0}}|X_{t}|\,dt$. Then it
follows from Lemma A.2 and inequalities (2.12) to (2.15) that for any positive
$\delta$ we have

$\displaystyle|X^{\prime}|\leq$
$\displaystyle\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}(|\square\phi^{r}|+2s|\nabla\phi^{r}||\nabla
b|)|u|e^{sb}\,dv_{t}dt$ $\displaystyle\leq$
$\displaystyle\epsilon(r)+\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}\left(\delta^{-1}|\nabla\phi^{r}|^{2}(\phi^{r})^{-1}\tau|\nabla
b|^{2}+\delta\tau^{-1}u^{2}\phi^{r}\right)e^{sb}\,dv_{t}dt$
$\displaystyle\leq$
$\displaystyle\epsilon(r)+\delta\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}\tau^{-1}u^{2}\phi^{r}e^{sb}\,dv_{t}dt.$
(A.17)

It is clear from the definition of $u$ that $u^{2}\leq C\left(\tau^{2}(|\Delta
b|^{2}+|\nabla b|^{2}+R))+b^{2}+1\right)$. In addition, since $-A\leq
b\leq-\tau(2\Delta b-|\nabla b|^{2}+R)+n$, we have

$\displaystyle b^{2}\leq C^{\prime}\left(\tau^{2}(|\Delta b|^{2}+|\nabla
b|^{4}+R^{2})+1\right).$ (A.18)

Combining these facts with (A.14), we may choose $\delta$ in (A.17)
sufficiently small such that

$\displaystyle|X^{\prime}|\leq\epsilon(r)+\frac{1}{10}\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}\tau(|\nabla^{2}b|^{2}+|Rc|^{2})\phi^{r}e^{sb}\,dv_{t}dt+\log\theta^{-1}.$
(A.19)

Similarly, we compute

$\displaystyle\partial_{t}\int_{M}\tau Re^{sb}w\phi^{r}\,dV_{t}$
$\displaystyle=$ $\displaystyle\int_{M}\left\\{\square(\tau
Re^{sb})\phi^{r}w-\tau Re^{sb}\square^{*}(\phi^{r}w)\right\\}dV_{t}$
$\displaystyle=$ $\displaystyle\int_{M}\left\\{\left(\square(\tau
R)e^{sb}+\tau R\square e^{sb}-2\tau\langle\nabla R,\nabla
e^{sb}\rangle\right)\phi^{r}w+\tau
Re^{sb}\left((\Delta\phi^{r}+\phi^{r}_{t})w+2\langle\nabla
w,\nabla\phi^{r}\rangle\right)\right\\}dV_{t}$ $\displaystyle=$
$\displaystyle\int_{M}\left\\{\left(\square(\tau R)e^{sb}+\tau R\square
e^{sb}\right)\phi^{r}w+2\tau R\left(\Delta e^{sb}-\langle\nabla e^{sb},\nabla
b\rangle\right)\phi^{r}w\right\\}dV_{t}+Y_{t}$ $\displaystyle=$
$\displaystyle\int_{M}\left\\{2\tau|Rc|^{2}-R+\tau R\left(s\square
b+(s^{2}-2s)|\nabla b|^{2}+2s\Delta
b\right)\right\\}\phi^{r}e^{sb}dv_{t}+Y_{t}$ $\displaystyle\geq$
$\displaystyle\int_{M}\left\\{2\tau|Rc|^{2}-R-Cs\left(\tau(R^{2}+|\nabla
b|^{4}+(\Delta b)^{2})+R\right)\right\\}\phi^{r}e^{sb}dv_{t}+Y_{t},$ (A.20)

where

$\displaystyle Y_{t}:$ $\displaystyle=\int_{M}\tau
Re^{sb}\left((\Delta\phi^{r}+\phi^{r}_{t})w+2\langle\nabla
w,\nabla\phi^{r}\rangle+2s\langle\nabla
b,\nabla\phi^{r}\rangle\right)\,dV_{t}$ $\displaystyle=\int_{M}\tau
R\left(\Delta\phi^{r}+\phi^{r}_{t}+(2s-2)\langle\nabla
b,\nabla\phi^{r}\rangle\right)e^{sb}\,dv_{t}.$

We define similarly $Y:=\int_{-\tau_{0}}^{-\theta\tau_{0}}|Y_{t}|\,dt$. Then
it follows from Lemma A.2 and inequalities (2.12) to (2.15) that

$\displaystyle|Y^{\prime}|\leq$
$\displaystyle\epsilon(r)+C\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}\left(\delta^{-1}|\nabla\phi^{r}|^{2}(\phi^{r})^{-1}\tau|\nabla
b|^{2}+\delta\tau R^{2}\phi^{r}\right)e^{sb}\,dv_{t}dt$ $\displaystyle\leq$
$\displaystyle\epsilon(r)+C\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}\delta\tau
R^{2}\phi^{r}e^{sb}\,dv_{t}dt$ $\displaystyle\leq$
$\displaystyle\epsilon(r)+\frac{1}{10}\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}\tau(|\nabla^{2}b|^{2}+|Rc|^{2})\phi^{r}e^{sb}\,dv_{t}dt,$
(A.21)

for $\delta$ sufficiently small.

Combining (A.16) and (A.20), we obtain

$\displaystyle\partial_{t}\int_{M}(\tau R+u)e^{sb}\phi^{r}\,dv_{t}$
$\displaystyle\geq$ $\displaystyle\int_{M}\left\\{-Cs\left(\tau(|\nabla
b|^{4}+|\nabla^{2}b|^{2}+R^{2})+\tau^{-1}+R\right)+2\tau\left|Rc+\nabla^{2}b-\frac{g}{2\tau}\right|^{2}+2\tau|Rc|^{2}-R\right\\}\phi^{r}e^{sb}dv_{t}$
$\displaystyle+X_{t}+Y_{t}$ $\displaystyle\geq$
$\displaystyle\int_{M}\left\\{-Cs\left(\tau(|\nabla
b|^{4}+|\nabla^{2}b|^{2}+R^{2})\right)+\tau(|\nabla^{2}b|^{2}+|Rc|^{2})\right\\}\phi^{r}e^{sb}dv_{t}+X_{t}+Y_{t}-C^{\prime}\tau^{-1},$
(A.22)

where we have used Lemma A.2.

If $s$ is sufficiently small, it follows from (A.22) and (A.14) that

$\displaystyle\partial_{t}\int_{M}(\tau R+u)e^{sb}\phi^{r}\,dv_{t}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{M}\tau(|\nabla^{2}b|^{2}+|Rc|^{2})e^{sb}\phi^{r}\,dv_{t}+X_{t}+Y_{t}-C^{\prime}\tau^{-1}+\epsilon(r).$
(A.23)

By integration from $-\tau_{0}$ to $-\theta\tau_{0}$, we obtain from (A.23),
Lemma A.2, (A.19) and (A.21) that

$\displaystyle\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}\tau(|\nabla^{2}b|^{2}+|Rc|^{2})e^{sb}\phi^{r}\,dv_{t}dt\leq
C^{\prime}\log\theta^{-1}+\epsilon(r).$

Letting $r\to\infty$, we obtain

$\displaystyle\int_{-\tau_{0}}^{-\theta\tau_{0}}\int_{M}\tau(|\nabla^{2}b|^{2}+|Rc|^{2})e^{sb}\,dv_{t}dt\leq
C^{\prime}\log\theta^{-1}.$ (A.24)

Thus the inequality (A.15) follows from the combination of (A.24) and Lemma
A.3.

∎

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Yu Li, Institute of Geometry and Physics, University of Science and Technology
of China, No. 96 Jinzhai Road, Hefei, Anhui Province, 230026, China;
<EMAIL_ADDRESS>

Bing Wang, Institute of Geometry and Physics, School of Mathematical Sciences,
University of Science and Technology of China, No. 96 Jinzhai Road, Hefei,
Anhui Province, 230026, China<EMAIL_ADDRESS>