# Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

Keita Kunikawa  Cooperative Faculty of Education, Utsunomiya University, 350
Mine-Machi, Utsunomiya, 321-8505, Japan<EMAIL_ADDRESS>and
Yohei Sakurai  Department of Mathematics, Saitama University, 255 Shimo-Okubo,
Sakura-ku, Saitama-City, Saitama, 338-8570, Japan
<EMAIL_ADDRESS>
(Date: January 28, 2022)

###### Abstract.

Bamler-Zhang have developed geometric analysis on Ricci flow with scalar
curvature bound. The aim of this paper is to extend their work to various
geometric flows. We generalize some of their results to super Ricci flow whose
Müller quantity is non-negative, and obtain Gaussian heat kernel estimates.

###### Key words and phrases:

Super Ricci flow; Müller quantity; Distance distortion estimate; Cutoff
function; Mean value inequality; Gaussian heat kernel estimate

###### 2020 Mathematics Subject Classification:

Primary 53C44; Secondly 58J35

##  1\. Introduction

###  1.1. Backgrounds

Let $(M,g(t))_{t\in[0,T)}$ be a solution to the Ricci flow

(1.1) $\partial_{t}g=-2\operatorname{Ric}$

on an $n$-dimensional compact smooth manifold with $T<+\infty$. One of the
common problems in the Ricci flow theory is to understand the blow-up behavior
of curvature when $t\nearrow T$. In a pioneering work by Hamilton [14], he has
proved that the supremum of the norm of the Riemannian curvature tensor blows
up at $T$; namely,

(1.2) $\lim_{t\nearrow T}\sup_{M}|\operatorname{Rm}|=+\infty.$

After that under the same setting, Šešum [29] has generalized (1.2) to the
Ricci curvature; namely, she has proved

$\limsup_{t\nearrow T}\sup_{M}|\operatorname{Ric}|=+\infty.$

It is natural to ask whether the same phenomenon occurs for the scalar
curvature; namely,

$\limsup_{t\nearrow T}\sup_{M}R=+\infty.$

This is known to occur in the case of $n=2,3$ or in the Kähler case (see [15],
[19], [37]), but it seems to be still open in the higher dimensional and non-
Kähler case. From this point of view, it is important to investigate the
structure of Ricci flow with scalar curvature bound.

Based on the background, Bamler-Zhang [7] have vastly developed geometric
analysis on Ricci flow with scalar curvature bound, and obtained useful tools
such as distance distortion estimates, a nice cutoff function, and mean value
inequalities. Furthermore, by using them, they have concluded Gaussian heat
kernel estimates, backward pseudolocality theorem, and strong
$\varepsilon$-regularity theorem. Recently, their estimates have yielded
numerous results in the sequel [8], and a series of works by Bamler [3], [4],
[5], [6] while being improved.

###  1.2. Main results

The purpose of this paper is to extend the results by Bamler-Zhang [7] to more
general geometric flows. We focus on a super solution to the Ricci flow (1.1).
An $n$-dimensional compact manifold $(M,g(t))_{t\in[0,T)}$ equipped with a
time-dependent metric is called super Ricci flow when

$\partial_{t}g\geq-2\operatorname{Ric}.$

McCann-Topping [25] have introduced this notion, and studied the relation
between the Ricci flow theory and optimal transport theory. Recently, this
notion has begun to be investigated not only from the viewpoint of the Ricci
flow theory but also from that of metric measure geometry (see e.g., [20],
[21], [22], [30]).

We examine super Ricci flow under the assumption that its Müller quantity is
non-negative. For a (time-dependent) vector field $V$, the Müller quantity is
defined by the following formula (see [26, Definition 1.3]):

$\mathcal{D}(V):=\partial_{t}H-\Delta
H-2|h|^{2}+4\operatorname{div}h(V)-2\langle\nabla
H,V\rangle+2\operatorname{Ric}(V,V)-2h(V,V),$

where

$h:=-\frac{1}{2}\partial_{t}g,\quad H:=\operatorname{tr}h.$

It is well-known that several results for Ricci flow can be extended to more
general geometric flow under the assumption that its Müller quantity is non-
negative; for instance, the monotonicity of $\mathcal{W}$-functional and the
reduced geometry (more precisely, see Subsections 2.2, 2.3). Also, there are
some examples of super Ricci flow whose Müller quantity is non-negative (see
[26, Section 2], [12, Section 7]):

  1. (1)

Ricci flow;

  2. (2)

Ricci flow coupled with the heat equation, called List flow ([23]);

  3. (3)

Ricci flow coupled with the harmonic map heat flow, called Müller flow ([27]);

  4. (4)

mean curvature flow for spacelike hypersurfaces in Lorentzian manifold of non-
negative sectional curvature;

  5. (5)

(scaled) twisted Kähler-Ricci flow.

For $x,y\in M$ and $s,t\in[0,T)$ with $s<t$, we denote by $G(x,t;y,s)$ the
heat kernel for the heat equation (more precisely, see Subsection 2.2). One of
our main results is the following Gaussian heat kernel estimate, which has
been formulated by Bamler-Zhang [7] for Ricci flow (see [7, Theorem 1.4]):

######  Theorem 1.1.

Let $(M,g(t))_{t\in[0,T)}$ be an $n$-dimensional compact super Ricci flow with
$T<+\infty$ satisfying $\mathcal{D}(V)\geq 0$ for all vector fields $V$. Then
for any $A>0$, there exist positive constants
$\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3},\mathcal{C}_{4},\mathcal{C}_{5},\mathcal{C}_{6}>0$
depending only on $n,T,g(0)$ and $A$ such that the following holds: We assume
$H\leq H_{1}$ for $H_{1}>0$. For $s,t\in[0,T)$ with $s<t$, we suppose $t-s\leq
AH_{1}^{-1}$ and $s\geq(t-s)/A$. Then we have

(1.3) $\displaystyle G(x,t;y,s)$
$\displaystyle\geq\frac{\mathcal{C}_{1}}{(t-s)^{n/2}}\exp\left({-\frac{\mathcal{C}_{2}\,d_{s}(x,y)^{2}}{t-s}}\right),$
(1.4) $\displaystyle G(x,t;y,s)$
$\displaystyle\leq\frac{\mathcal{C}_{3}}{(t-s)^{n/2}}\exp\left({-\frac{\mathcal{C}_{4}\,d_{s}(x,y)^{2}}{t-s}}\right),$
(1.5) $\displaystyle|\nabla_{x}G|(x,t;y,s)$
$\displaystyle\leq\frac{\mathcal{C}_{5}}{(t-s)^{(n+1)/2}}\exp\left({-\frac{\mathcal{C}_{6}\,d_{s}(x,y)^{2}}{t-s}}\right).$

######  Remark 1.2.

We give a remark for the dependence of the constants on $g(0)$. More
precisely, they depend on the volume and Sobolev constant of $M$ with respect
to $g(0)$, and the minimum of $H$ at $t=0$, which is derived from the
logarithmic Sobolev inequality of Fang-Zheng [12] (cf. [12, Theorem 1.1],
Subsection 2.2). This remark can be applied throughout this paper.

##  2\. Preliminaries

We recall some previous works for general geometric flow. Let
$(M,g(t))_{t\in[0,T)}$ denote an $n$-dimensional compact manifold with a time-
dependent metric with $T<+\infty$.

###  2.1. Notations and basics

In the present subsection, we introduce some notations, and elementary facts
that will be used frequently throughout this article.

We denote by $d_{t}$ and $m_{t}$ the Riemannian distance and Riemannian volume
measure with respect to $g(t)$, respectively. When $t$ is clear from the
context, we omit it, and write them as $d$ and $m$. Notice that the time
derivative of $dm$ is equal to $-H\,dm$. For $x_{0}\in M,r_{0}>0$ and
$t_{0}\in[0,T)$, we denote by $B(x_{0},r_{0},t_{0})$ the open ball of radius
$r_{0}$ centered at $x_{0}$ with respect to $g(t_{0})$. Let
$Q^{+}(x_{0},r_{0},t_{0})$ stand for the forward parabolic cube defined as

$Q^{+}(x_{0},r_{0},t_{0}):=\\{(x,t)\in M\times[0,T)\mid x\in
B(x_{0},r_{0},t),\,t\in[t_{0},t_{0}+r^{2}_{0}]\\}.$

We begin with the following elementary fact (see [12, Lemma 3.2]):

######  Proposition 2.1 ([12]).

Assume that $(M,g(t))_{t\in[0,T)}$ satisfies $\mathcal{D}(V)\geq 0$ for all
vector fields $V$. Then we have

$H\geq-\frac{n}{2t}.$

######  Remark 2.2.

We also mention the rescaling argument. For a positive $r_{0}>0$, we will
consider the parabolic rescaling

$\bar{g}(\tau):=r^{-2}_{0}\,g(r^{2}_{0}\tau)=r^{-2}_{0}g(t)$

with $\tau:=r^{-2}_{0}t$. We see that $\bar{h}=h,\bar{H}=r^{2}_{0}H$ and
$\overline{\mathcal{D}}(V)=r^{4}_{0}\mathcal{D}(r^{-2}_{0}V)$, where
$\bar{h},\bar{H},\overline{\mathcal{D}}(V)$ are corresponding objects for
$\bar{g}(\tau)$. In particular, the super Ricci flow and the non-negativity of
the Müller quantity are preserved under this rescaling.

###  2.2. Monotonicity of entropy

The monotonicity of Perelman’s $\mathcal{W}$-functional is one of the powerful
tools in the Ricci flow theory ([28]). It is well-known that such a
monotonicity can be extended to general geometric flow under the non-
negativity of the Müller quantity (see [18, Theorem 3.1], [13, Theorem 5.2],
[12, Lemma 3.1]). In this subsection, we collect some results that can be
derived from the monotonicity.

Fang-Zheng [12] obtained a logarithmic Sobolev inequality by using the
monotonicity of $\mathcal{W}$-functional, and derived the following Sobolev
inequality (see [12, Theorems 1.1, 1.3], and cf. [31], [35]):

######  Proposition 2.3 ([12]).

Assume that $(M,g(t))_{t\in[0,T)}$ satisfies $\mathcal{D}(V)\geq 0$ for all
vector fields $V$. Then for all $t\in[0,T)$ and $u\in W^{1,2}(M)$ we have

$\left(\int_{M}\,|u|^{\frac{2n}{n-2}}\,dm\right)^{\frac{n-2}{n}}\leq
C_{1}\int_{M}\,\left(|\nabla
u|^{2}+\frac{H}{4}u^{2}\right)\,dm+C_{2}\,\int_{M}\,u^{2}\,dm,$

where $C_{1},C_{2}>0$ are positive constants depending only on $n,T$ and
$g(0)$.

They further concluded that the following $\kappa$-noncollapsing estimate
holds (see [31, Theorem 6.1], [12, Theorem 5.1]):

######  Theorem 2.4 ([12]).

Assume that $(M,g(t))_{t\in[0,T)}$ satisfies $\mathcal{D}(V)\geq 0$ for all
vector fields $V$. For $x\in M,r\in(0,\sqrt{T})$ and $t\in[0,T)$, we assume
$H\leq r^{-2}$ on $B(x,r,t)$. Then we have

$m(B(x,r,t))\geq\kappa\,r^{n},$

where $\kappa>0$ is a positive constant depending only on $n,T$ and $g(0)$.

For $x,y\in M$ and $s,t\in[0,T)$ with $s<t$, we denote by $G(x,t;y,s)$ the
heat kernel for the heat equation; namely, for a fixed $(y,s)\in
M\times[0,T)$, it solves

$(\partial_{t}-\Delta_{x})G(\cdot,\cdot;y,s)=0,\quad\lim_{t\searrow
s}G(\cdot,t;y,s)=\delta_{y}.$

Notice that $G(x,t;\cdot,\cdot)$ is the kernel for the conjugate heat
equation; namely, for any $(x,t)\in M\times[0,T)$,

$(-\partial_{s}-\Delta_{y}+H)G(x,t;\cdot,\cdot)=0,\quad\lim_{s\nearrow
t}G(x,t;\cdot,s)=\delta_{x}.$

They also showed the following upper bound of Zhang type (see [36, (1.5)],
[12, Lemma 6.3]):

######  Proposition 2.5 ([12]).

Assume that $(M,g(t))_{t\in[0,T)}$ satisfies $\mathcal{D}(V)\geq 0$ for all
vector fields $V$. Then for all $x,y\in M$ and $s,t\in[0,T)$ with $s<t$, we
have

$G(x,t;y,s)\leq\frac{C}{(t-s)^{n/2}},$

where $C>0$ is a positive constant depending only on $n,T$ and $g(0)$.

###  2.3. Reduced geometry

Perelman’s reduced geometry is also a crucial concept in the Ricci flow theory
([28]). It is also well-known that the reduced geometry can be extended to
general geometric flow under the assumption that the Müller quantity is non-
negative (see e.g., [26], [18], [32], [33]). Here we present some results
concerning the reduced geometry.

Let us recall the notion of the reduced distance. Let $x,y\in M$ and
$s,t\in[0,T)$ with $s<t$. The $L$-distance $L_{(x,t)}(y,s)$ from a space-time
base point $(x,t)$ to $(y,s)$ is defined by

$L_{(x,t)}(y,s):=\inf_{\gamma}\int^{t}_{s}\sqrt{t-\xi}\left[H+\left|\frac{d\gamma}{d\xi}\right|^{2}\right]\,d\xi,$

where the infimum is taken over all curves $\gamma:[s,t]\to M$ with
$\gamma(s)=y$ and $\gamma(t)=x$. The reduced distance $\ell_{(x,t)}(y,s)$, and
its modified one $\overline{L}_{(x,t)}(y,s)$ are defined as

$\ell_{(x,t)}(y,s):=\frac{1}{2\sqrt{t-s}}L_{(x,t)}(y,s),\quad\overline{L}_{(x,t)}(y,s):=4(t-s)\,\ell_{(x,t)}(y,s).$

It is well-known that the following fundamental inequality holds under the
non-negativity of the Müller quantity (see [26, Lemmas 5.2, 5.3], and also
[18, Theorem 3.4], [32, Subsection 2.3], [33]):

######  Proposition 2.6 ([26], [18], [32], [33]).

Assume that $(M,g(t))_{t\in[0,T)}$ satisfies $\mathcal{D}(V)\geq 0$ for all
vector fields $V$. Then we have

(2.1) $(-\partial_{s}+\Delta)\overline{L}_{(x,t)}\leq 2n$

in the barrier sense.

For the Ricci flow, this estimate (2.1) together with the maximum principle
leads us to the uniform upper bound of the minimum of the reduced distance
(see [28], [10, Lemmas 7.48, 7.50]). By the same argument, Proposition 2.6
gives us the following:

######  Proposition 2.7.

Assume that $(M,g(t))_{t\in[0,T)}$ satisfies $\mathcal{D}(V)\geq 0$ for all
vector fields $V$. Then for all $x\in M$ and $s,t\in[0,T)$ with $s<t$ we have

$\min_{y\in M}\ell_{(x,t)}(y,s)\leq\frac{n}{2}.$

We also mention the following relation between the reduced distance and heat
kernel (see [9, Lemma 2.4]):

######  Proposition 2.8 ([9]).

Assume that $(M,g(t))_{t\in[0,T)}$ satisfies $\mathcal{D}(V)\geq 0$ for all
vector fields $V$. Then for all $x,y\in M$ and $s,t\in[0,T)$ with $s<t$ we
have

$G(x,t;y,s)\geq\frac{1}{(4\pi(t-s))^{n/2}}\exp\left(-\ell_{(x,t)}(y,s)\right).$

###  2.4. Zhang type gradient estimates

For the super Ricci flow, one can derive the following gradient estimate of
Zhang type (see [34, Theorem 3.1], [12, Lemma 6.5]):

######  Theorem 2.9 ([12]).

Let $(M,g(t))_{t\in[0,T)}$ be a super Ricci flow. For $t_{0}\in[0,T)$, let
$u\in C^{\infty}(M\times(t_{0},T))$ be a positive solution to the heat
equation. For $t_{1},t_{2}\in(t_{0},T)$ with $t_{1}<t_{2}$, we assume $u\leq
A$ for $A>0$ on $M\times[t_{1},t_{2}]$. Then for all $x\in M$ and
$t\in(t_{1},t_{2}]$ we have

$\frac{|\nabla
u|}{u}(x,t)\leq\frac{1}{\sqrt{t-t_{1}}}\sqrt{\log\frac{A}{u(x,t)}}.$

Due to Theorem 2.9, we have the following Harnack inequality (see [36], [12,
Lemma 6.5]):

######  Corollary 2.10 ([12]).

Let $(M,g(t))_{t\in[0,T)}$ be a super Ricci flow. For $t_{0}\in[0,T)$, let
$u\in C^{\infty}(M\times(t_{0},T))$ be a positive solution to the heat
equation. For $t_{1},t_{2}\in(t_{0},T)$ with $t_{1}<t_{2}$, we assume $u\leq
A$ for $A>0$ on $M\times[t_{1},t_{2}]$. Then for all $x,y\in M$ we have

$u(y,t_{2})\leq
A^{1/2}\,u(x,t_{2})^{1/2}\,\exp\left(\frac{d_{t_{2}}(x,y)^{2}}{4(t_{2}-t_{1})}\right).$

Using Propositions 2.5, 2.8 and Corollary 2.10, Fang-Zheng [12] have obtained
the following lower Gaussian heat kernel estimate (see [36], [12, Lemma 6.6]):

######  Theorem 2.11 ([12]).

Let $(M,g(t))_{t\in[0,T)}$ be a super Ricci flow satisfying
$\mathcal{D}(V)\geq 0$ for all vector fields $V$. For $s,t\in[0,T)$ with
$s<t$, and for $H_{1}>0$, we assume $H\leq H_{1}$ on $M\times[s,t]$. Then for
all $x,y\in M$ we have

$G(x,t;y,s)\geq\frac{C}{(t-s)^{n/2}}\exp\bigg{(}{-\frac{d_{t}(x,y)^{2}}{t-s}}\bigg{)},$

where $C>0$ is a positive constant depending only on $n,T,g(0)$ and
$(t-s)H_{1}$.

From Propositions 2.5, 2.8 and Corollary 2.10, they also concluded the
following $\kappa$-noninflating theorem of Zhang type (see [36, Theorem 1.1],
[12, Theorem 6.1]):

######  Theorem 2.12 ([12]).

Let $(M,g(t))_{t\in[0,T)}$ be a super Ricci flow satisfying
$\mathcal{D}(V)\geq 0$ for all vector fields $V$. For $x_{0}\in
M,t_{0}\in(0,T),r_{0}\in(0,\sqrt{t_{0}})$ and $A>0$, we assume $H\leq
A/(t_{0}-t)$ on $B(x_{0},r_{0},t_{0})\times[t_{0}-r^{2}_{0},t_{0}]$. Then we
have

$m(B(x_{0},r_{0},t_{0}))\leq\kappa\,r^{n}_{0},$

where $\kappa>0$ is a positive constant depending only on $n,T,g(0)$ and $A$.

###  2.5. Characterization of super Ricci flow

It is well-known that super Ricci flow can be characterized by a gradient
estimate, logarithmic Sobolev inequality, and Poincaré inequality for the heat
semigroup, which can be regarded as a time-dependent version of the
equivalence of a lower Ricci curvature bound established by Bakry-Émery [1],
Bakry-Ledoux [2] (see e.g., [17], [24], [16], [22], and also [25], [30] for
other characterizations). In this subsection, we present some results that can
be deduced from such a characterization.

Let $x,y\in M$, and let $s,t\in[0,T)$ with $s<t$. The heat kernel measure
$\nu_{(x,t)}(y,s)$ is defined by

$d\nu_{(x,t)}(y,s):=G(x,t;y,s)\,dm_{s}(y).$

We possess the following logarithmic Sobolev inequality (see [24, Theorem
1.1], [16, Theorem 1.5], [22, Theorem 1.3], and cf. [17, Theorem 1.10 (2)]):

######  Theorem 2.13 ([24], [16], [22]).

Let $(M,g(t))_{t\in[0,T)}$ be a super Ricci flow. Then for all $u\in
C^{\infty}(M)$ with $\int_{M}\,u^{2}(y)\,d\nu_{(x,t)}(y,s)=1$ we have

$\int_{M}\,u^{2}\log u^{2}(y)\,d\nu_{(x,t)}(y,s)\leq 4(t-s)\int_{M}\,|\nabla
u|^{2}(y)\,d\nu_{(x,t)}(y,s).$

Theorem 2.13 together with the method of the proof of [17, Theorem 1.13]
implies the following Gaussian concentration inequality (cf. [17, Theorem
1.13]):

######  Corollary 2.14.

Let $(M,g(t))_{t\in[0,T)}$ be a super Ricci flow. Let
$\nu:=\nu_{(x,t)}(\cdot,s)$. Then for all $\Omega_{1},\Omega_{2}\subset M$ we
have

$\nu(\Omega_{1})\nu(\Omega_{2})\leq\exp\left(-\frac{d_{s}(\Omega_{1},\Omega_{2})^{2}}{8(t-s)}\right).$

Corollary 2.14 together with the method of the proof of [17, (1.20)] tells us
the following average bound for the heat kernel (cf. [17, (1.20)]):

######  Proposition 2.15.

Let $(M,g(t))_{t\in[0,T)}$ be a super Ricci flow. Let $x,y\in M$, and let
$s,t\in[0,T)$ with $s<t$. For $c>0$ we assume
$m(B(y,c\sqrt{t-s},s))\geq\kappa(t-s)^{n/2}$. Then

$\displaystyle\quad\,\,\frac{1}{m(B(y,c\sqrt{t-s},s))}\int_{B(y,c\sqrt{t-s},s)}G(x,t;z,s)\,dm_{s}(z)$
$\displaystyle\leq\bigg{(}\int_{B(x,c\sqrt{t-s},s)}G(x,t;z,s)\,dm_{s}(z)\bigg{)}^{-1}\frac{\kappa^{-1}\,e^{c^{2}/2}}{(t-s)^{n/2}}\exp\left(-\frac{d_{s}(x,y)^{2}}{16(t-s)}\right).$

In [17], a similar inequality has been stated only for $c=1$. For the sake of
our argument, we present a slightly general form. The method is completely
same as that of [17, (1.20)].

##  3\. Distance distortion estimate

In what follows, we denote by $(M,g(t))_{t\in[0,T)}$ an $n$-dimensional
compact super Ricci flow with $T<+\infty$ satisfying $\mathcal{D}(V)\geq 0$
for all vector fields $V$. Unless otherwise stated, positive constants which
appear in the proof will depend only on $n,T$ and $g(0)$ (cf. Remark 1.2).

The aim of this section is to prove a distance distortion estimate. We start
with the following key lemma, which has been obtained by Bamler-Zhang [7] for
Ricci flow (see [7, Lemma 3.1]):

######  Lemma 3.1.

There exists a positive constant $C_{n}>0$ depending only on $n$ such that the
following holds: For $t_{0}\in[0,T)$, let $u\in C^{\infty}(M\times(t_{0},T))$
be a positive solution to the heat equation. For $t_{1},t_{2}\in(t_{0},T)$
with $t_{1}<t_{2}$, we assume $u\leq A$ for $A>0$ on $M\times[t_{1},t_{2}]$.
Then for all $x\in M$ and $t\in(t_{1},t_{2}]$ we have

$\left(|\Delta u|+\frac{|\nabla
u|^{2}}{u}-AH\right)(x,t)\leq\frac{C_{n}A}{t-t_{1}}.$

###### Proof.

We may assume $A=1$. We first estimate

$L_{1}:=-\Delta u+\frac{|\nabla u|^{2}}{u}-H.$

Using the Bochner formula, we deduce

(3.1) $\displaystyle(\partial_{t}-\Delta)\Delta u$
$\displaystyle=\partial_{t}\Delta
u-\Delta\partial_{t}u=2\langle\nabla^{2}u,h\rangle+2\operatorname{div}h(\nabla
u)-\langle\nabla H,\nabla u\rangle,$
$\displaystyle(\partial_{t}-\Delta)\frac{|\nabla u|^{2}}{u}$
$\displaystyle=-\frac{2}{u}\left(\left|\nabla^{2}u-\frac{du\otimes
du}{u}\right|^{2}+\operatorname{Ric}(\nabla u,\nabla u)-h(\nabla u,\nabla
u)\right),$ $\displaystyle(\partial_{t}-\Delta)H$
$\displaystyle=2|h|^{2}+\mathcal{D}(0).$

Therefore, by $u\leq 1,\operatorname{Ric}\geq h$ and $\mathcal{D}(V)\geq 0$ we
obtain

$\displaystyle(\partial_{t}-\Delta)L_{1}$
$\displaystyle=-\frac{2}{u}\left|\nabla^{2}u-\frac{du\otimes
du}{u}\right|^{2}-2\langle\nabla^{2}u,h\rangle-2|h|^{2}$
$\displaystyle\quad-2\operatorname{div}h(\nabla u)+\langle\nabla H,\nabla
u\rangle-\frac{2}{u}\left(\operatorname{Ric}(\nabla u,\nabla u)-h(\nabla
u,\nabla u)\right)-\mathcal{D}(0)$
$\displaystyle\leq-\left|\nabla^{2}u-\frac{du\otimes
du}{u}\right|^{2}-2\langle\nabla^{2}u,h\rangle-2|h|^{2}$
$\displaystyle\quad-2\operatorname{div}h(\nabla u)+\langle\nabla H,\nabla
u\rangle-(\operatorname{Ric}(\nabla u,\nabla u)-h(\nabla u,\nabla
u))-\mathcal{D}(0)$ $\displaystyle=-\left|\nabla^{2}u-\frac{du\otimes
du}{u}+h\right|^{2}-\left|\frac{du\otimes du}{u}+h\right|^{2}+\frac{|\nabla
u|^{4}}{u^{2}}-\frac{1}{2}(\mathcal{D}(\nabla u)+\mathcal{D}(0))$
$\displaystyle\leq-\frac{1}{n}\left(\Delta u-\frac{|\nabla
u|^{2}}{u}+H\right)^{2}+\frac{|\nabla
u|^{4}}{u^{2}}=-\frac{1}{n}L^{2}_{1}+\frac{|\nabla u|^{4}}{u^{2}}.$

Theorem 2.9 tells us that

(3.2) $\frac{|\nabla
u|^{2}}{u}\leq\frac{u}{t-t_{1}}\log\frac{1}{u}\leq\frac{1}{e(t-t_{1})}$

on $M\times(t_{1},t_{2}]$, and hence

$(\partial_{t}-\Delta)L_{1}\leq-\frac{1}{n}L^{2}_{1}+\frac{1}{e^{2}(t-t_{1})^{2}}.$

Let $C_{n,1}>0$ be a positive constant determined by

$\frac{C_{n,1}+e^{-2}}{C_{n,1}^{2}}=\frac{1}{n}$

depending only on $n$. Note that

$(\partial_{t}-\Delta)\left(\frac{C_{n,1}}{t-t_{1}}\right)=-\frac{1}{n}\left(\frac{C_{n,1}}{t-t_{1}}\right)^{2}+\frac{1}{e^{2}(t-t_{1})^{2}}.$

It follows that

$(\partial_{t}-\Delta)\left(L_{1}-\frac{C_{n,1}}{t-t_{1}}\right)\leq-\frac{1}{n}\left(L_{1}+\frac{C_{n,1}}{t-t_{1}}\right)\left(L_{1}-\frac{C_{n,1}}{t-t_{1}}\right).$

The maximum principle leads us to $L_{1}\leq C_{n,1}/(t-t_{1})$.

We next estimate

$L_{2}:=\Delta u+\frac{|\nabla u|^{2}}{u}-H.$

Using (3.1), $u\leq 1,\operatorname{Ric}\geq h$ and $\mathcal{D}(V)\geq 0$
again, one can calculate

$\displaystyle(\partial_{t}-\Delta)L_{2}$
$\displaystyle=-\frac{2}{u}\left|\nabla^{2}u-\frac{du\otimes
du}{u}\right|^{2}+2\langle\nabla^{2}u,h\rangle-2|h|^{2}$
$\displaystyle\quad+2\operatorname{div}h(\nabla u)-\langle\nabla H,\nabla
u\rangle-\frac{2}{u}\left(\operatorname{Ric}(\nabla u,\nabla u)-h(\nabla
u,\nabla u)\right)-\mathcal{D}(0)$
$\displaystyle\leq-\left|\nabla^{2}u-\frac{du\otimes
du}{u}\right|^{2}+2\langle\nabla^{2}u,h\rangle-2|h|^{2}$
$\displaystyle\quad-2\operatorname{div}h(-\nabla u)+\langle\nabla H,(-\nabla
u)\rangle-\left(\operatorname{Ric}(\nabla u,\nabla u)-h(\nabla u,\nabla
u)\right)-\mathcal{D}(0)$ $\displaystyle=-\left|\nabla^{2}u-\frac{du\otimes
du}{u}-h\right|^{2}-\left|\frac{du\otimes du}{u}-h\right|^{2}+\frac{|\nabla
u|^{4}}{u^{2}}-\frac{1}{2}\left(\mathcal{D}(-\nabla u)+\mathcal{D}(0)\right)$
$\displaystyle\leq-\frac{1}{n}\left(\Delta u-\frac{|\nabla
u|^{2}}{u}-H\right)^{2}+\frac{|\nabla u|^{4}}{u^{2}}$
$\displaystyle=-\frac{1}{2n}\left(L_{2}-4\frac{|\nabla
u|^{2}}{u}\right)^{2}-\frac{1}{2n}L^{2}_{2}+\left(1+\frac{4}{n}\right)\frac{|\nabla
u|^{4}}{u^{2}}\leq-\frac{1}{2n}L^{2}_{2}+\left(1+\frac{4}{n}\right)\frac{|\nabla
u|^{4}}{u^{2}}.$

The estimate (3.2) implies

$(\partial_{t}-\Delta)L_{2}\leq-\frac{1}{n}L^{2}_{2}+\left(1+\frac{4}{n}\right)\frac{1}{e^{2}(t-t_{1})^{2}}$

over $M\times(t_{1},t_{2}]$. Let $C_{n,2}>0$ be a constant determined by

$C^{-1}_{n,2}+\left(1+\frac{4}{n}\right)\frac{1}{e^{2}}C^{-2}_{n,2}=\frac{1}{2n},$

which depends only on $n$. We notice that

$(\partial_{t}-\Delta)\left(\frac{C_{n,2}}{t-t_{1}}\right)=-\frac{1}{2n}\left(\frac{C_{n,2}}{t-t_{1}}\right)^{2}+\left(1+\frac{4}{n}\right)\frac{1}{e^{2}(t-t_{1})^{2}};$

in particular,

$(\partial_{t}-\Delta)\left(L_{2}-\frac{C_{n,2}}{t-t_{1}}\right)\leq-\frac{1}{2n}\left(L_{2}+\frac{C_{n,2}}{t-t_{1}}\right)\left(L_{2}-\frac{C_{n,2}}{t-t_{1}}\right).$

Due to the maximum principle, we obtain $L_{2}\leq C_{n,2}/(t-t_{1})$. This
completes the proof. $\Box$

We are now in a position to prove the following distance distortion estimate,
which has been established by Bamler-Zhang [7] for Ricci flow (see [7, Theorem
1.1]):

######  Theorem 3.2.

There exists a positive constant $\alpha\in(0,1)$ depending only on $n,T$ and
$g(0)$ such that the following holds: For $t_{0}\in(0,T)$ and
$r_{0}\in(0,\sqrt{t_{0}}]$, we assume $H\leq r_{0}^{-2}$. Let $x_{0},y_{0}\in
M$ satisfy $d_{t_{0}}(x_{0},y_{0})\geq r_{0}$. Then for all $t\in[0,T)$ with
$|t-t_{0}|\leq\alpha r^{2}_{0}$,

$\alpha d_{t_{0}}(x_{0},y_{0})\leq
d_{t}(x_{0},y_{0})\leq\alpha^{-1}d_{t_{0}}(x_{0},y_{0}).$

###### Proof.

By the argument of [7, Theorem 1.1], it suffices to prove the upper bound for
$d_{t}(x_{0},y_{0})$ only in the case of $d_{t_{0}}(x_{0},y_{0})\leq 2r_{0}$.
By the parabolic rescaling, it is enough to show the claim when $r_{0}=1$ (see
Remark 2.2). Then we have $t_{0}\geq 1$ and $H\leq 1$. We further possess
$H\geq-n$ on $M\times[t_{0}-1/2,t_{0}+1/2]$ by virtue of Proposition 2.1. Let
$\gamma:[0,1]\to M$ be a minimal geodesic from $x_{0}$ to $y_{0}$ with respect
to $g(t_{0})$. By Proposition 2.7,

(3.3) $\ell_{(x_{0},t_{0})}\left(z,t_{0}-\frac{1}{2}\right)\leq\frac{n}{2}$

for some $z\in M$. We put $G(x,t):=G(x,t;z,t_{0}-1/2)$.

Due to Proposition 2.5, on $M\times[t_{0}-1/4,t_{0}+1/4]$ it holds that

(3.4) $G(x,t)\leq\frac{C_{1}}{(t-(t_{0}-1/2))^{n/2}}\leq C_{2}.$

Thanks to Proposition 2.8 and (3.3),

(3.5)
$G(x_{0},t_{0})\geq\frac{1}{(4\pi(t_{0}-(t_{0}-1/2))^{n/2}}\exp\left(-\ell_{(x_{0},t_{0})}(z,t_{0}-1/2)\right)\geq
C_{3}.$

By (3.4), Corollary 2.10, (3.5) and the assumption $d_{t_{0}}(x_{0},y_{0})\leq
2$, for all $x\in\gamma([0,1])$,

(3.6) $\displaystyle G(x,t_{0})$ $\displaystyle\geq
C^{-1}_{2}G(x_{0},t_{0})^{2}\,\exp\left(-\frac{d_{t_{0}}(x_{0},x)^{2}}{2(t_{0}-(t_{0}-1/4))}\right)$
$\displaystyle\geq
C^{-1}_{2}\,C^{2}_{3}\,\exp\left(-2d_{t_{0}}(x_{0},y_{0})^{2}\right)\geq
e^{-8}\,C^{-1}_{2}\,C^{2}_{3}=:C_{4}.$

For the dimensional constant $C_{n}>0$ obtained in Lemma 3.1, we set

$\alpha_{0}:=\min\left\\{\frac{1}{8},\frac{C_{4}}{4C_{2}(1+8C_{n})}\right\\}.$

Using (3.6), (3.4), Lemma 3.1, $H\leq 1$, for all $x\in\gamma([0,1])$ and
$t\in[t_{0}-\alpha_{0},t_{0}+\alpha_{0}]$, we see

(3.7) $\displaystyle G(x,t)$ $\displaystyle\geq
G(x,t_{0})-\int^{t_{0}+\alpha_{0}}_{t_{0}-\alpha_{0}}\,\left|\partial_{t}G\right|(x,t)\,dt=G(x,t_{0})-\int^{t_{0}+\alpha_{0}}_{t_{0}-\alpha_{0}}\,\left|\Delta
G\right|(x,t)\,dt$ $\displaystyle\geq
C_{4}-C_{2}\int^{t_{0}+\alpha_{0}}_{t_{0}-\alpha_{0}}\,\left(H(x,t)+\frac{C_{n}}{t-(t_{0}-1/4)}\right)\,dt$
$\displaystyle\geq
C_{4}-2\alpha_{0}\,C_{2}\left(1+\frac{C_{n}}{1/4-\alpha_{0}}\right)\geq
C_{4}-2\alpha_{0}\,C_{2}\left(1+8C_{n}\right)\geq\frac{C_{4}}{2}.$

By (3.4), Corollary 2.10, (3.7), for all
$s\in[0,1],t\in[t_{0}-\alpha_{0}/2,t_{0}+\alpha_{0}/2]$ and $x\in
B(\gamma(s),\sqrt{\alpha_{0}},t)$ we have

$G(x,t)\geq
C^{-1}_{2}G(\gamma(s),t)^{2}\,\exp\left(-\frac{d_{t}(x,\gamma(s))^{2}}{2(t-(t_{0}-\alpha_{0}))}\right)\geq\frac{C^{-1}_{2}C^{2}_{4}}{4}e^{-1}=:C_{5}.$

Let $\\{x_{i}\\}^{N}_{i=1}$ denote a maximal $(2\sqrt{\alpha_{0}})$-separated
set in $\gamma([0,1])$ with respect to $g(t)$. The argument in the proof of
[7, Theorem 1.1] tells us that $d_{t}(x_{0},y_{0})\leq 4N\sqrt{\alpha_{0}}$.
On $(t_{0}-1/2,t_{0}+1/2]$, we see

$\frac{d}{dt}\int_{M}\,G\,dm=\int_{M}\,\left(\Delta G-H\,G\right)\,dm\leq
n\int_{M}\,G\,dm$

since $H\geq-n$ on $M\times[t_{0}-1/2,t_{0}+1/2]$, and hence

(3.8) $\int_{M}\,G\,dm\leq\exp\left(n(t-(t_{0}-1/2))\right)\leq e^{n}.$

Combining (3.8) and Theorem 2.4, we see that if
$t\in[t_{0}-\alpha_{0}/2,t_{0}+\alpha_{0}/2]$, then

$e^{n}\geq\int_{M}\,G(\cdot,t)\,dm\geq\sum_{i=1}^{N}\int_{B(x_{i},\sqrt{\alpha_{0}},t)}\,G(\cdot,t)\,dm\geq
N(\kappa\alpha^{n/2}_{0})C_{5}.$

We conclude that for all $t\in[t_{0}-\alpha_{0}/2,t_{0}+\alpha_{0}/2]$,

$d_{t}(x_{0},y_{0})\leq\frac{4e^{n}}{\kappa\alpha^{(n-1)/2}_{0}C_{5}}<\frac{8e^{n}}{\kappa\alpha^{(n-1)/2}_{0}C_{5}}\,d_{t_{0}}(x_{0},y_{0}).$

Setting

$\alpha:=\min\left\\{\frac{\alpha_{0}}{2},\frac{\kappa\alpha^{(n-1)/2}_{0}C_{5}}{8e^{n}}\right\\},$

we arrive at the desired upper bound. Thus we complete the proof. $\Box$

##  4\. Cutoff function

Based on Theorem 3.2, we construct the following cutoff function, which has
been obtained by Bamler-Zhang [7] for Ricci flow (see [7, Theorem 1.3]):

######  Theorem 4.1.

There exists a constant $\rho\in(0,1)$ depending only on $n,T$ and $g(0)$ such
that the following holds: Let $t_{0}\in(0,T)$ and $r_{0}\in(0,\sqrt{t_{0}}]$.
We assume $H\leq r_{0}^{-2}$. Then for any $x_{0}\in M$ and
$\tau\in(0,\rho^{2}r_{0}^{2}]$, there is $\phi\in
C^{\infty}(M\times[t_{0}-\tau,t_{0}])$ satisfying the following properties:

  1. (1)

$0\leq\phi<1$;

  2. (2)

$\phi\geq\rho$ on $B(x_{0},\rho r_{0},t_{0})\times[t_{0}-\tau,t_{0}]$;

  3. (3)

$\phi=0$ on $(M\setminus B(x_{0},r_{0},t_{0}))\times[t_{0}-\tau,t_{0}]$;

  4. (4)

$|\nabla\phi|\leq r_{0}^{-1}$ and $|\partial_{t}\phi|+|\Delta\phi|\leq
r_{0}^{-2}$.

###### Proof.

By the rescaling, we may assume $r_{0}=1$ (see Remark 2.2). We possess
$t_{0}\geq 1$ and $H\leq 1$. Moreover, $H\geq-n$ on $M\times[t_{0}-1/2,t_{0}]$
by Proposition 2.1. Let $\theta\in(0,1/2),\tau\in(0,\theta/4)$, which will be
determined later.

Due to Proposition 2.7, there exists $z\in M$ such that

$\ell_{(x_{0},t_{0})}(z,t_{0}-\theta)\leq\frac{n}{2}.$

Set $G(x,t):=G(x,t;z,t_{0}-\theta)$. Thanks to Proposition 2.8,

(4.1) $\displaystyle G(x_{0},t_{0})$
$\displaystyle\geq\frac{1}{(4\pi(t_{0}-(t_{0}-\theta))^{n/2}}\exp\left(-\ell_{(x_{0},t_{0})}(z,t_{0}-\theta)\right)$
$\displaystyle\geq\frac{1}{(4\pi\theta)^{n/2}}e^{-n/2}=:C_{n}\theta^{-n/2},$

where $C_{n}>0$ depends only on $n$. We define

$\Omega:=\left\\{x\in M\mid G(x,t_{0})>C_{n}\theta^{-n/2}/2\right\\}.$

From (4.1) we derive $x_{0}\in\Omega$. Let $\Omega_{0}\subset\Omega$ stand for
the connected component of $\Omega$ containing $x_{0}$. We prove that for any
sufficiently small $\theta$, we have $\Omega_{0}\subset B(x_{0},1,t_{0})$ by
contradiction. Let us assume that $\Omega_{0}$ is not contained in
$B(x_{0},1,t_{0})$. Then we can take a curve $\gamma:[0,1]\to\Omega_{0}$ from
$x_{0}$ to a point in $\partial B(x_{0},1,t_{0})\cap\Omega_{0}$. Let
$\\{x_{i}\\}^{N}_{i=1}$ be a maximal $(2\sqrt{\theta})$-separated set in
$\gamma([0,1])$ with respect to $g(t_{0})$. The argument in the proof of [7,
Theorem 1.3] tells us that

(4.2) $N\geq\frac{1}{4\sqrt{\theta}}.$

Proposition 2.5 tells us that on $M\times[t_{0}-\theta/2,t_{0}]$ we see

(4.3) $G(x,t)\leq\frac{C_{1}}{(t-(t_{0}-\theta))^{n/2}}\leq
C_{2}\theta^{-n/2}.$

Due to (4.3), Corollary 2.10, for all $y\in\Omega_{0}$ and $z\in
B(y,\sqrt{\theta},t_{0})$ we have

(4.4) $G(z,t_{0})\geq
C^{-1}_{2}\,\theta^{n/2}\,G(y,t_{0})^{2}\,\exp\left(-\frac{d_{t_{0}}(y,z)^{2}}{2\theta}\right)\geq\frac{e^{-1/2}}{2}C^{-1}_{2}C_{n}\theta^{-n/2}=:C_{3}\theta^{-n/2}.$

On the other hand, on $(t_{0}-\theta,t_{0}]$ we see

$\frac{d}{dt}\int_{M}\,G\,dm=\int_{M}\,\left(\Delta G-H\,G\right)\,dm\leq
n\int_{M}\,G\,dm$

since $H\geq-n$ on $M\times[t_{0}-\theta,t_{0}]$, and hence

(4.5) $\int_{M}\,G\,dm\leq\exp\left(n(t-(t_{0}-\theta))\right)\leq
e^{n\theta}\leq e^{n/2}.$

Combining (4.2), (4.4), (4.5) and Theorem 2.4, we obtain

$e^{n/2}\geq\int_{M}G(\cdot,t_{0})dm\geq\sum_{i=1}^{N}\int_{B(x_{i},\sqrt{\theta},t_{0})}G(\cdot,t_{0})\,dm\geq
N(C_{3}\theta^{-n/2})(\kappa\theta^{n/2})=N\kappa C_{3}\geq\frac{\kappa
C_{3}}{4\sqrt{\theta}}.$

If $\theta<16^{-1}e^{-n}\kappa^{2}C^{2}_{3}$, then this is a contradiction.
Hereafter, we fix $\theta\in(0,16^{-1}e^{-n}\kappa^{2}C^{2}_{3})$ such that
$\Omega_{0}\subset B(x_{0},1,t_{0})$.

We now use (4.3) and Lemma 3.1. On $M\times[t_{0}-\tau,t_{0}]$ it holds that

(4.6) $\displaystyle|\partial_{t}G|(x,t)$ $\displaystyle\leq
C_{2}\theta^{-n/2}\left(H(x,t)+\frac{C_{4}}{t-(t_{0}-\theta/2)}\right)\leq
C_{2}\theta^{-n/2}(1+4C_{4}\theta^{-1})$ $\displaystyle\leq
C_{2}(1+4C_{4})\theta^{-n/2-1}=:C_{5}\theta^{-n/2-1}.$

Note that $C_{5}>C_{2}$. Let $\tau\in(0,0.1C_{n}C^{-1}_{5}\theta)$. Then on
$M\times[t_{0}-\tau,t_{0}]$,

$|G(x,t)-G(x,t_{0})|\leq\int^{t_{0}}_{t_{0}-\tau}\,|\partial_{t}G|(x,t)\,dt\leq
C_{5}\theta^{-n/2-1}\tau<0.1C_{n}\theta^{-n/2};$

in particular, on $\partial\Omega_{0}\times[t_{0}-\tau,t_{0}]$ we possess

$G(x,t)<G(x,t_{0})+0.1C_{n}\theta^{-n/2}=0.5C_{n}\theta^{-n/2}+0.1C_{n}\theta^{-n/2}=0.6C_{n}\theta^{-n/2}.$

Now, we can define a function $\psi\in C^{0}(M\times[t_{0}-\tau,t_{0}])$ by

$\hat{\psi}(x,t):=\begin{cases}\max\\{G(x,t)-0.6C_{n}\theta^{-n/2}\\}&\textrm{on}\
\Omega_{0}\times[t_{0}-\tau,t_{0}],\\\ 0&\textrm{on}\
(M\setminus\Omega_{0})\times[t_{0}-\tau,t_{0}],\end{cases}$

which is compactly supported on $\Omega_{0}\times[t_{0}-\tau,t_{0}]$. This
function enjoys the following properties:

  1. (1)

$0\leq\hat{\psi}<C_{5}\theta^{-n/2}$;

  2. (2)

$\hat{\psi}(x_{0},t_{0})\geq 0.4C_{n}\theta^{-n/2}$;

  3. (3)

$|\nabla\hat{\psi}|\leq 2C_{5}\theta^{-n/2-1/2}$;

  4. (4)

$|\partial_{t}\hat{\psi}|+|\Delta\hat{\psi}|\leq 2C_{5}\theta^{-n/2-1}$.

The first inequality follows from (4.3) and $C_{2}<C_{5}$. The second one is a
consequence of $x_{0}\in\Omega$. By (4.3) and Theorem 2.9, on
$M\times[t_{0}-\tau,t_{0}]$,

$|\nabla
G|(x,t)\leq\frac{1}{\sqrt{t-(t_{0}-\theta/2)}}\sqrt{\log\frac{C_{2}\theta^{-n/2}}{G(x,t)}}G(x,t)\leq
2C_{2}\theta^{-n/2-1/2}<2C_{5}\theta^{-n/2-1/2},$

and this implies the third one. The fourth one can be derived from (4.6). If
we further define $\tilde{\phi}:=32^{-1}C^{-1}_{5}\theta^{n/2+1}\hat{\psi}$,
then it satisfies the following properties:

  1. (1)

$0\leq\tilde{\phi}<1$;

  2. (2)

$\tilde{\phi}(x_{0},t_{0})\geq C_{6}\,\theta$;

  3. (3)

$|\nabla\tilde{\phi}|\leq 1/8$;

  4. (4)

$|\partial_{t}\tilde{\phi}|+|\Delta\tilde{\phi}|\leq 1/8$.

Once we obtain this function, we can conclude the desired assertion by the
same argument as in the proof of [7, Theorem 1.3]. We complete the proof.
$\Box$

##  5\. Mean value inequality

In the present section, we produce a mean value inequality for conjugate heat
equation. To do so, we first yield the following integral estimate based on
the Moser iteration argument, which has been obtained by Bamler-Zhang [7] for
Ricci flow (see [7, Lemma 4.1]):

######  Lemma 5.1.

Let $p\geq 2$. Then there are positive constants $\beta\in(0,1)$ and $C>0$
depending only on $n,T,g(0)$ and $p$ such that the following holds: For
$t_{0}\in(0,T)$ and $r_{0}\in(0,\sqrt{t_{0}}]$, we assume $H\leq r^{-2}_{0}$.
Let $u\in C^{\infty}(M\times[t_{0},t_{0}+r_{0}^{2}])$ denote a positive
solution to the conjugate heat equation. Then for all $x_{0}\in M$ we have

$\left(\int_{Q^{+}(x_{0},\beta
r_{0},t_{0})}u^{p}\,dm\,dt\right)^{1/p}\leq\frac{C}{r_{0}^{(n+2)(p-2)/2p}}\left(\int_{Q^{+}(x_{0},r_{0},t_{0})}u^{2}\,dm\,dt\right)^{1/2}.$

###### Proof.

By the rescaling, we may assume $r_{0}=1$ (see Remark 2.2). It holds that
$t_{0}\geq 1$ and $H\leq 1$. Moreover, $H\geq-n$ on
$M\times[t_{0}-1/2,t_{0}+1]$ in view of Proposition 2.1. Let
$\alpha,\rho\in(0,1)$ be the constants obtained in Theorems 3.2 and 4.1,
respectively. We define

$\theta:=\alpha\in(0,1).$

Note that $(\rho\theta)^{2}\in(0,\alpha)$. By Theorem 3.2 we have

(5.1) $B(x_{0},\theta,t_{1})\subset B(x_{0},1,t_{2})$

for all $t_{1},t_{2}\in[t_{0},t_{0}+(\rho\theta)^{2}]$. We further set

$\sigma:=\frac{\alpha\rho\theta}{\sqrt{2}}\in(0,\rho\theta).$

Notice that $2\sigma^{2}\in(0,\alpha(\rho\theta)^{2})$. Using Theorem 3.2
again, we obtain

(5.2) $B(x_{0},\sigma,t_{1})\subset B(x_{0},\rho\theta,t_{2})$

for all $t_{1},t_{2}\in[t_{0},t_{0}+2\sigma^{2}]$. With the help of Theorem
4.1, there is $\phi\in C^{\infty}(M\times[t_{0},t_{0}+2\sigma^{2}])$ such that
the following holds:

  1. (1)

$0\leq\phi<1$;

  2. (2)

$\phi\geq\rho$ on $B^{\prime}\times[t_{0},t_{0}+2\sigma^{2}]$;

  3. (3)

$\phi=0$ on $(M\setminus B)\times[t_{0},t_{0}+2\sigma^{2}]$;

  4. (4)

$|\nabla\phi|\leq\theta^{-1}$ and $|\partial_{t}\phi|\leq\theta^{-2}$,

where $B:=B(x_{0},\theta,t_{0}+2\sigma^{2})$ and
$B^{\prime}:=B(x_{0},\rho\theta,t_{0}+2\sigma^{2})$. By (5.1) and (5.2), we
possess

(5.3) $Q^{+}(x_{0},\sigma,t_{0})\subset
B^{\prime}\times[t_{0},t_{0}+2\sigma^{2}]\subset
B\times[t_{0},t_{0}+2\sigma^{2}]\subset Q^{+}(x_{0},1,t_{0}).$

Let $\eta\in C^{\infty}([t_{0},t_{0}+2\sigma^{2}])$ be a function satisfying
the following properties:

  1. (1)

$0\leq\eta\leq 1$;

  2. (2)

$\eta\equiv 1$ on $[t_{0},t_{0}+\sigma^{2}]$;

  3. (3)

$\eta(t_{0}+2\sigma^{2})=0$;

  4. (4)

$|\eta^{\prime}|\leq C_{1}$.

Then we define $\psi(x,t):=\eta(t)\phi(x,t)$. Note that $\psi\geq\rho$ on
$Q^{+}(x_{0},\sigma,t_{0})$.

Let $p\geq 2$. Since $u$ is a solution to the conjugate heat equation, we see

$\partial_{t}u^{p/2}+\Delta
u^{p/2}-\frac{p}{2}Hu^{p/2}=\frac{p}{2}\left(\frac{p}{2}-1\right)u^{\frac{p}{2}-2}|\nabla
u|^{2}\geq 0.$

On $[t_{0},t_{0}+2\sigma^{2}]$, it holds that

$\displaystyle\frac{d}{dt}\int_{M}\,u^{p}\psi^{2}\,dm$
$\displaystyle=2\int_{M}(\partial_{t}u^{p/2})u^{p/2}\psi^{2}\,dm+2\int_{M}u^{p}(\partial_{t}\psi)\psi\,dm-\int_{M}\,u^{p}\psi^{2}H\,dm$
$\displaystyle\geq 2\int_{M}\left(-\Delta
u^{p/2}+\frac{p}{2}Hu^{p/2}\right)u^{p/2}\psi^{2}dm-C_{2}\int_{B}u^{p}\,dm$
$\displaystyle\geq 2\int_{M}\,\langle\nabla
u^{p/2},\psi\,\nabla(u^{p/2}\psi)+(u^{p/2}\psi)\nabla\psi\rangle\,dm-
C_{3}\int_{B}u^{p}\,dm$
$\displaystyle=2\int_{M}\langle\nabla(u^{p/2}\psi)-u^{p/2}\nabla\psi,\nabla(u^{p/2}\psi)+u^{p/2}\nabla\psi\rangle\,dm-
C_{3}\int_{B}u^{p}\,dm$
$\displaystyle=2\int_{M}\big{|}\nabla(u^{p/2}\psi)\big{|}^{2}dm-2\int_{M}\,u^{p}|\nabla\psi|^{2}dm-
C_{3}\int_{B}u^{p}\,dm$ $\displaystyle\geq
2\int_{M}\big{|}\nabla(u^{p/2}\psi)\big{|}^{2}\,dm-C_{3}\int_{B}u^{p}\,dm.$

From Proposition 2.3, we conclude

$\frac{d}{dt}\int_{M}u^{p}\psi^{2}\,dm\geq
C_{4}\bigg{(}\int_{M}\,u^{\frac{pn}{n-2}}\psi^{\frac{2n}{n-2}}\,dm\bigg{)}^{\frac{n-2}{n}}-C_{5}\int_{B}\,u^{p}\,dm.$

Let us integrate this inequality from a fixed
$t_{1}\in[t_{0},t_{0}+2\sigma^{2}]$ to $t_{0}+2\sigma^{2}$. We deduce

$\displaystyle\quad\,\,-\int_{B^{\prime}}\,u^{p}(\cdot,t_{1})\psi^{2}(\cdot,t_{1})\,dm_{t_{1}}$
$\displaystyle\geq-\int_{M}u^{p}(\cdot,t_{1})\psi^{2}(\cdot,t_{1})\,dm_{t_{1}}$
$\displaystyle\geq
C_{4}\int_{t_{1}}^{t_{0}+2\sigma^{2}}\bigg{(}\int_{M}u^{\frac{pn}{n-2}}\psi^{\frac{2n}{n-2}}\,dm\bigg{)}^{\frac{n-2}{n}}dt-
C_{5}\int_{t_{1}}^{t_{0}+2\sigma^{2}}\int_{B}\,u^{p}\,dm\,dt.$

This leads us to

$\displaystyle\sup_{t\in[t_{0},t_{0}+2\sigma^{2}]}\int_{B^{\prime}}u^{p}(\cdot,t)\psi^{2}(\cdot,t)\,dm_{t}$
$\displaystyle\leq
C_{6}\,\int_{t_{0}}^{t_{0}+2\sigma^{2}}\int_{B}\,u^{p}\,dm\,dt,$
$\displaystyle\int_{t_{0}}^{t_{0}+2\sigma^{2}}\bigg{(}\int_{B^{\prime}}u^{\frac{pn}{n-2}}\psi^{\frac{2n}{n-2}}\,dm\bigg{)}^{\frac{n-2}{n}}dt$
$\displaystyle\leq
C_{7}\int_{t_{0}}^{t_{0}+2\sigma^{2}}\int_{B}\,u^{p}\,dm\,dt.$

By the Hölder inequality,

$\displaystyle\quad\,\,\int_{t_{0}}^{t_{0}+2\sigma^{2}}\,\int_{B^{\prime}}u^{p\left(1+\frac{2}{n}\right)}\psi^{2\left(1+\frac{2}{n}\right)}\,dm\,dt$
$\displaystyle=\int_{t_{0}}^{t_{0}+2\sigma^{2}}\,\int_{B^{\prime}}\,\left(u^{p}\psi^{2}\right)\left(u^{\frac{2p}{n}}\psi^{\frac{4}{n}}\right)\,dm\,dt$
$\displaystyle\leq\int_{t_{0}}^{t_{0}+2\sigma^{2}}\bigg{(}\int_{B^{\prime}}u^{\frac{pn}{n-2}}\psi^{\frac{2n}{n-2}}\,dm\bigg{)}^{\frac{n-2}{n}}\bigg{(}\int_{B^{\prime}}u^{p}\psi^{2}\,dm\bigg{)}^{\frac{2}{n}}\,dt$
$\displaystyle\leq\left[\int_{t_{0}}^{t_{0}+2\sigma^{2}}\bigg{(}\int_{B^{\prime}}u^{\frac{pn}{n-2}}\psi^{\frac{2n}{n-2}}\,dm\bigg{)}^{\frac{n-2}{n}}dt\right]\left[C_{6}\int_{t_{0}}^{t_{0}+2\sigma^{2}}\int_{B}u^{p}dm\,dt\right]^{\frac{2}{n}}$
$\displaystyle\leq\bigg{(}C_{8}\int_{t_{0}}^{t_{0}+2\sigma^{2}}\int_{B}u^{p}\,dm\,dt\bigg{)}^{1+\frac{2}{n}}.$

In view of (5.3) and $\psi\geq\rho$ on $Q^{+}(x_{0},\sigma,t_{0})$, we arrive
at

$\bigg{(}\int_{Q^{+}(x_{0},\sigma,t_{0})}u^{p\left(1+\frac{2}{n}\right)}\,dm\,dt\bigg{)}^{1/p\left(1+\frac{2}{n}\right)}\leq
C_{9}\bigg{(}\int_{Q^{+}(x_{0},1,t_{0})}u^{p}\,dm\,dt\bigg{)}^{1/p}.$

Once we obtain this estimate, we can conclude the desired one by the same
argument as in the proof of [7, Lemma 4.2]. $\Box$

We give a proof of the following mean value inequality, which has been
formulated by Bamler-Zhang [7] for Ricci flow (see [7, Lemma 4.2]):

######  Theorem 5.2.

There exist positive constants $\gamma\in(0,1)$ and $C>0$ depending only on
$n,T$ and $g(0)$ such that the following holds: For $t_{0}\in(0,T)$ and
$r_{0}\in(0,\sqrt{t_{0}}]$, we assume $H\leq r^{-2}_{0}$. Let $u\in
C^{\infty}(M\times[t_{0},t_{0}+r_{0}^{2}])$ be a positive solution to the
conjugate heat equation. Then for all $x_{0}\in M$ we have

$\sup_{Q^{+}(x_{0},\gamma
r_{0},t_{0})}u^{2}\leq\frac{C}{r_{0}^{n+2}}\,\int_{Q^{+}(x_{0},r_{0},t_{0})}u^{2}\,dm\,dt.$

###### Proof.

By the rescaling, we may assume $r_{0}=1$ (see Remark 2.2). We have $t_{0}\geq
1$ and $H\leq 1$. Furthermore, we have $H\geq-n$ on
$M\times[t_{0}-1/2,t_{0}+1]$ by Proposition 2.1. Let $\alpha,\rho\in(0,1)$ be
the constants obtained in Theorems 3.2 and 4.1, respectively. Let
$\beta\in(0,1)$ be the constant obtained in Lemma 5.1 for a fixed $p>n+2$. We
define

$\theta:=\alpha\beta\in(0,1).$

Note that $(\rho\theta)^{2}\in(0,\alpha\beta^{2})$. By Theorem 3.2, it holds
that

(5.4) $B(x_{0},\theta,t_{1})\subset B(x_{0},\beta,t_{2})$

for all $t_{1},t_{2}\in[t_{0},t_{0}+(\rho\theta)^{2}]$. We further define

$\gamma:=\frac{\alpha\rho\theta}{\sqrt{2}}\in(0,\rho\theta).$

Notice that $2\gamma^{2}\in(0,\alpha(\rho\theta)^{2})$. In virtue of Theorem
3.2, we possess

(5.5) $B(x_{0},\gamma,t_{1})\subset B(x_{0},\rho\theta,t_{2})$

for all $t_{1},t_{2}\in[t_{0},t_{0}+2\gamma^{2}]$. By Theorem 4.1, there is
$\phi\in C^{\infty}(M\times[t_{0},t_{0}+2\gamma^{2}])$ such that the following
holds:

  1. (1)

$0\leq\phi<1$;

  2. (2)

$\phi\geq\rho$ on $B^{\prime}\times[t_{0},t_{0}+2\gamma^{2}]$;

  3. (3)

$\phi=0$ on $(M\setminus B)\times[t_{0},t_{0}+2\gamma^{2}]$;

  4. (4)

$|\nabla\phi|\leq\theta^{-1}$ and
$|\partial_{t}\phi|+|\Delta\phi|\leq\theta^{-2}$,

where $B:=B(x_{0},\theta,t_{0}+2\gamma^{2})$ and
$B^{\prime}:=B(x_{0},\rho\,\theta,t_{0}+2\gamma^{2})$. From (5.4) and (5.5),
we deduce

(5.6) $\displaystyle Q^{+}(x_{0},\gamma,t_{0})\subset
B^{\prime}\times[t_{0},t_{0}+2\gamma^{2}]\subset
B\times[t_{0},t_{0}+2\gamma^{2}]\subset Q^{+}(x_{0},\beta,t_{0}).$

Let $\eta\in C^{\infty}([t_{0},t_{0}+2\gamma^{2}])$ be a function satisfying
the following properties:

  1. (1)

$0\leq\eta\leq 1$;

  2. (2)

$\eta\equiv 1$ on $[t_{0},t_{0}+\gamma^{2}]$;

  3. (3)

$\eta(t_{0}+2\gamma^{2})=0$;

  4. (4)

$|\eta^{\prime}|\leq C_{1}$.

Then we define $\psi(x,t):=\eta(t)\phi(x,t)$. In view of (5.6), we notice
$\psi\geq\rho$ on $Q^{+}(x_{0},\gamma,t_{0})$. For a fixed $(x,t)\in
Q^{+}(x_{0},\gamma,t_{0})$, we set $G(z,l):=G(z,l;x,t)$. Then

$(u\psi)(x,t)=-\int_{t}^{t_{0}+2\gamma^{2}}\int_{M}G\big{(}u\Delta\psi+u\partial_{l}\psi+2\langle\nabla
u,\nabla\psi\rangle\big{)}dm\,dl.$

Here we used

$\partial_{l}(u\psi)+\Delta(u\psi)-H(u\psi)=u\Delta\psi+u\partial_{l}\psi+2\langle\nabla
u,\nabla\psi\rangle,$

which is a consequence of the fact that $u$ is a solution to the conjugate
heat equation. Integration by parts yields

$(u\psi)(x,t)=-\int_{t}^{t_{0}+2\gamma^{2}}\int_{M}Gu(-\Delta\psi+\partial_{l}\psi)+2u\langle\nabla
G,\nabla\psi\rangle\,dm\,dl.$

From the properties of $\psi$ and (5.6), it follows that

(5.7) $(u\psi)(x,t)\leq
C_{2}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}\,\left(Gu+u|\nabla
G|\right)\,dm\,dl=:C_{2}(I_{1}+I_{2}).$

We first estimate $I_{1}$. By the Hölder inequality, (5.6) and Lemma 5.1,

(5.8) $\displaystyle I_{1}$
$\displaystyle\leq\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}G^{q}\,dm\,dl\bigg{)}^{1/q}\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}u^{p}\,dm\,dl\bigg{)}^{1/p}$
$\displaystyle\leq\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}G^{q}\,dm\,dl\bigg{)}^{1/q}\bigg{(}\int_{Q^{+}(x_{0},\beta,t_{0})}u^{p}\,dm\,dl\bigg{)}^{1/p}$
$\displaystyle\leq
C_{3}\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}G^{q}\,dm\,dl\bigg{)}^{1/q}\|u\|_{L^{2}(Q^{+}(x_{0},1,t_{0}))}$

for $q=p/(p-1)$. By Proposition 2.5, for all $l\in(t,t_{0}+2\gamma^{2}]$ we
have

(5.9) $G(z,l)\leq\frac{C_{4}}{(l-t)^{n/2}}.$

We also see

$\frac{d}{dl}\int_{M}\,G\,dm=-\int_{M}\,H\,G\,dm\leq n\int_{M}G\,dm$

since $H\geq-n$ on $M\times[t_{0}-1/2,t_{0}+1]$; in particular, that for all
$l\in(t,t_{0}+2\gamma^{2}]$,

(5.10) $\int_{M}G\,dm\leq\exp(n(l-t))\leq C_{5}.$

Combining (5.8), (5.9) and (5.10), we obtain

$\displaystyle I_{1}$ $\displaystyle\leq
C_{6}\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}G\frac{1}{(l-t)^{(q-1)n/2}}\,dm\,dl\bigg{)}^{1/q}\|u\|_{L^{2}(Q^{+}(x_{0},1,t_{0}))}$
$\displaystyle\leq
C_{7}\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\frac{1}{(l-t)^{(q-1)n/2}}\,dl\bigg{)}^{1/q}\|u\|_{L^{2}(Q^{+}(x_{0},1,t_{0}))}.$

Since $p>n+2$, we have $(q-1)n/2<1$; in particular,

(5.11) $I_{1}\leq C_{8}\|u\|_{L^{2}(Q^{+}(x_{0},1,t_{0}))}.$

We next derive a bound of $I_{2}$. The estimate (5.9) implies that for each
$l\in(t,t_{0}+2\gamma^{2}]$,

$A_{l}:=\sup_{M\times\left[\frac{l+t}{2},l\right]}G\leq\frac{C_{9}}{(l-t)^{n/2}}.$

Using Theorem 2.9, we possess

$\frac{|\nabla G|^{2}}{G^{2}}(z,l)\leq\frac{2}{l-t}\log\frac{A_{l}}{G(z,l)}.$

Since $G(z,l)/A_{l}\leq 1$, we have

$|\nabla G|^{q}\leq
G\frac{C_{10}A_{l}^{q-1}}{(l-t)^{q/2}}\bigg{(}\frac{G}{A_{l}}\bigg{)}^{q-1}\bigg{(}{\log\frac{A_{l}}{G}}\bigg{)}^{q/2}\leq
G\frac{C_{11}A_{l}^{q-1}}{(l-t)^{q/2}}\leq\frac{C_{12}}{(l-t)^{((n+1)q-n)/2}}G$

at $(z,l)$, where we used a fact that a function
$\varphi(\xi):=\xi^{q-1}(\log(1/\xi))^{q/2}$ takes its maximum value
$e^{-q/2}(q/2(q-1))^{q/2}$ at $\xi=e^{-q/2(q-1)}$ on $(0,1]$. Using the Hölder
inequality, this gradient bound, (5.10), Lemma 5.1 with (5.6), we conclude

$\displaystyle I_{2}$
$\displaystyle\leq\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}|\nabla
G|^{q}\,dm\,dl\bigg{)}^{1/q}\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}u^{p}\,dm\,dl\bigg{)}^{1/p}$
$\displaystyle\leq\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\int_{B}|\nabla
G|^{q}\,dm\,dl\bigg{)}^{1/q}\bigg{(}\int_{Q^{+}(x_{0},\beta,t_{0})}u^{p}\,dm\,dl\bigg{)}^{1/p}$
$\displaystyle\leq
C_{13}\,\bigg{(}\int_{t}^{t_{0}+2\gamma^{2}}\frac{1}{(l-t)^{((n+1)q-n)/2}}\,dl\bigg{)}^{1/q}\|u\|_{L^{2}(Q^{+}(x_{0},1,t_{0}))}.$

From $p>n+2$, we deduce $((n+1)q-n)/2<1$, and hence

(5.12) $I_{2}\leq C_{14}\|u\|_{L^{2}(Q^{+}(x_{0},1,t_{0}))}.$

Combining (5.7), (5.11), (5.12), we arrive at

$(u\psi)(x,t)\leq C_{15}\|u\|_{L^{2}(Q^{+}(x_{0},1,t_{0}))}.$

The desired assertion follows from $\psi(x,t)\geq\rho$. We complete the proof.
$\Box$

######  Remark 5.3.

Besides the mean value inequality for conjugate heat equation, Bamler-Zhang
[7] have also produced a mean value inequality for heat equation (see [7,
Lemma 4.3, Proposition 4.4]). Theorems 3.2 and 4.1 together with their
argument enable us to generalize such a mean value inequality in our setting.

##  6\. Gaussian heat kernel estimate

In this section, we give a proof of Theorem 1.1. First, we prove the lower
bound (1.3). To do so, we prepare the following lemma:

######  Lemma 6.1.

For any $A>0$, there is a constant $B>1$ depending only on $n,T,g(0)$ and $A$
such that the following holds: We assume $H\leq H_{1}$ for $H_{1}>0$. For
$s,t\in[0,T)$ with $s<t$, we suppose $t-s\leq A\,H_{1}^{-1}$ and
$s\geq(t-s)/A$. If $d_{t}(x,y)\geq B\sqrt{t-s}$, then $d_{s}(x,y)\geq
B^{-1}d_{t}(x,y)$.

###### Proof.

Let $\alpha\in(0,1)$ be the constant obtained in Theorem 3.2. We first
consider the case of $A\in(0,\alpha]$. We set

$\theta:=\alpha^{-1}A\in(0,1],\quad B:=\alpha^{-1}A^{-1/2},\quad
r_{0}:=A^{-1/2}\sqrt{t-s}.$

It holds that

$r_{0}\leq\sqrt{s}\leq\sqrt{t},\quad H\leq
H_{1}\leq\frac{A}{t-s}=r^{-2}_{0},\quad t-s\leq\theta^{-1}(t-s)\leq\alpha
r^{2}_{0},\quad d_{t}(x,y)\geq\alpha^{-1}r_{0}\geq r_{0},$

and hence one can apply Theorem 3.2. It follows that

$d_{s}(x,y)\geq\alpha d_{t}(x,y)\geq\alpha
A^{1/2}d_{t}(x,y)=B^{-1}d_{t}(x,y),$

which is a desired estimate.

We next consider the remaining case of $A\in(\alpha,\infty)$. We put

$\theta:=\alpha^{-1}A\in(1,\infty),\quad B:=\alpha^{-(\theta+2)},\quad
r_{0}:=A^{-1/2}\sqrt{t-s}.$

Let $N\in\mathbb{N}$ be the integer part of $\theta$, and let
$s=\tau_{N+1}\leq\tau_{N}<\dots<\tau_{1}<\tau_{0}=t$ be a division of $[s,t]$
such that $\tau_{i}-\tau_{i+1}=\theta^{-1}(t-s)$ for $i=0,\dots,N-1$. We see
$\tau_{N}-\tau_{N+1}\in[0,\theta^{-1}(t-s))$. We possess

$r_{0}\leq\sqrt{s}\leq\sqrt{\tau_{i}}\leq\sqrt{t},\quad H\leq
H_{1}\leq\frac{A}{t-s}=r^{-2}_{0},\quad\tau_{i}-\tau_{i+1}\leq\theta^{-1}(t-s)=\alpha
r^{2}_{0}$

for all $i=0,\dots,N$. In view of

$B\geq\alpha^{-\theta-1}A^{-1/2}\geq\alpha^{-N-1}A^{-1/2},$

we see

$d_{\tau_{0}}(x,y)=d_{t}(x,y)\geq
B\sqrt{t-s}\geq\alpha^{-N-1}A^{-1/2}\sqrt{t-s}=\alpha^{-N-1}r_{0}\geq r_{0},$

and hence Theorem 3.2 implies

$d_{\tau_{1}}(x,y)\geq\alpha\,d_{\tau_{0}}(x,y)\geq\alpha^{-N}r_{0}\geq
r_{0}.$

Using Theorem 3.2 again, we obtain

$d_{\tau_{2}}(x,y)\geq\alpha\,d_{\tau_{1}}(x,y)\geq\alpha^{-N+1}r_{0}\geq
r_{0}.$

By repeating this procedure, we arrive at

$d_{\tau_{i}}(x,y)\geq\alpha d_{\tau_{i-1}}(x,y)\geq\alpha^{-N+(i-1)}r_{0}\geq
r_{0}$

for all $i=1,\dots,N+1$. Therefore,

$\displaystyle d_{s}(x,y)$ $\displaystyle=d_{\tau_{N+1}}(x,y)$
$\displaystyle\geq\alpha
d_{\tau_{N}}(x,y)\geq\alpha^{2}d_{\tau_{N-1}}(x,y)\geq\cdots\geq\alpha^{N+1}d_{t}(x,y)$
$\displaystyle\geq\alpha^{\theta+2}d_{t}(x,y)\geq B^{-1}\sqrt{t-s}.$

This proves the lemma. $\Box$

Let us show the lower bound (1.3).

######  Proposition 6.2.

For any $A>0$, there exist positive constants
$\mathcal{C}_{1},\mathcal{C}_{2}>0$ depending only on $n,T,g(0)$ and $A$ such
that the following holds: We assume $H\leq H_{1}$ for $H_{1}>0$. For
$s,t\in[0,T)$ with $s<t$, we suppose $t-s\leq A\,H_{1}^{-1}$ and
$s\geq(t-s)/A$. Then we have

$G(x,t;y,s)\geq\frac{\mathcal{C}_{1}}{(t-s)^{n/2}}\exp\left({-\frac{\mathcal{C}_{2}\,d_{s}(x,y)^{2}}{t-s}}\right).$

###### Proof.

We possess

(6.1) $H\leq H_{1}\leq\frac{A}{t-s}.$

Furthermore, in view of Lemma 2.1,

(6.2) $-\frac{nA}{2(t-s)}\leq-\frac{n}{2s}\leq H$

on $M\times[s,t]$. Theorem 2.11 together with (6.1) tells us that

(6.3)
$G(x,t;y,s)\geq\frac{C}{(t-s)^{n/2}}\exp\left({-\frac{d_{t}(x,y)^{2}}{t-s}}\right).$

Let $B>1$ be the constant obtained in Lemma 6.1. If $d_{t}(x,y)\geq
B\sqrt{t-s}$, then $d_{s}(x,y)\geq B^{-1}\,d_{t}(x,y)$. From (6.3) it follows
that

$G(x,t;y,s)\geq\frac{C}{(t-s)^{n/2}}\exp\left({-\frac{d_{t}(x,y)^{2}}{t-s}}\right)\geq\frac{C}{(t-s)^{n/2}}\exp\left({-\frac{B^{2}d_{s}(x,y)^{2}}{t-s}}\right).$

On the other hand, if $d_{t}(x,y)<B\sqrt{t-s}$, then (6.3) also leads to

$G(x,t;y,s)\geq\frac{C}{(t-s)^{n/2}}\exp\left({-\frac{d_{t}(x,y)^{2}}{t-s}}\right)\geq\frac{Ce^{-B^{2}}}{(t-s)^{n/2}}\geq\frac{Ce^{-B^{2}}}{(t-s)^{n/2}}\exp\left({-\frac{d_{s}(x,y)^{2}}{t-s}}\right).$

Thus, we complete the proof. $\Box$

We next prove the upper bound (1.4). Similarly to Lemma 6.1, we prepare the
following:

######  Lemma 6.3.

For any $A>0$, there is a constant $B>1$ with $B^{-1}A^{-1/2}<1$ depending
only on $n,T,g(0)$ and $A$ such that the following holds: Assume $H\leq H_{1}$
for $H_{1}>0$. For $s,t\in[0,T)$ with $s<t$, we suppose $t-s\leq
A\,H_{1}^{-1}$ and $s\geq(t-s)/A$. Assume $d_{s}(x,y)\geq B\sqrt{t-s}$. Then
for all $l_{1},l_{2}\in[s,t]$ we have

$B^{-1}\,d_{l_{1}}(x,y)\leq d_{l_{2}}(x,y)\leq B\,d_{l_{1}}(x,y).$

###### Proof.

Let $\alpha\in(0,1)$ be the constant obtained in Theorem 3.2. We begin with
the case of $A\in(0,\alpha]$. Similarly to the proof of Lemma 6.1, we put

$\theta:=\alpha^{-1}A\in(0,1],\quad B:=\alpha^{-1}A^{-1/2},\quad
r_{0}:=A^{-1/2}\sqrt{t-s}.$

Note that $B>1$ and $B^{-1}A^{-1/2}<1$. For all $l_{1},l_{2}\in[s,t]$ we see

$\displaystyle r_{0}$ $\displaystyle\leq\sqrt{s}\leq\sqrt{t},\quad H\leq
H_{1}\leq\frac{A}{t-s}=r^{-2}_{0},$ $\displaystyle|l_{1}-l_{2}|$
$\displaystyle\leq\theta^{-1}(t-s)\leq\alpha r^{2}_{0},\quad
d_{s}(x,y)\geq\alpha^{-1}r_{0}\geq r_{0}.$

In virtue of Theorem 3.2, we obtain

$d_{l_{1}}(x,y)\geq\alpha d_{s}(x,y)\geq r_{0},\quad d_{l_{2}}(x,y)\geq\alpha
d_{l_{1}}(x,y)\geq\alpha A^{1/2}d_{l_{1}}(x,y)=B^{-1}d_{l_{1}}(x,y).$

By switching the roles of $l_{1}$ and $l_{2}$, we arrive at the desired
estimate.

Next, we study the case of $A\in(\alpha,\infty)$. As in the proof of Lemma
6.1, we define

$\theta:=\alpha^{-1}A\in(1,\infty),\quad B:=\alpha^{-(\theta+2)},\quad
r_{0}:=A^{-1/2}\sqrt{t-s},$

where $B>1$ and $B^{-1}A^{-1/2}<1$. Let $N\in\mathbb{N}$ be the integer part
of $\theta$, and let $s=\tau_{0}<\tau_{1}<\dots<\tau_{N}\leq\tau_{N+1}=t$ be a
division of $[s,t]$ such that $\tau_{i+1}-\tau_{i}=\theta^{-1}(t-s)$ for
$i=0,\dots,N-1$. We see $\tau_{N+1}-\tau_{N}\in[0,\theta^{-1}(t-s))$. It holds
that

$r_{0}\leq\sqrt{s}\leq\sqrt{\tau_{i}}\leq\sqrt{t},\quad H\leq
H_{1}\leq\frac{A}{t-s}=r^{-2}_{0},\quad\tau_{i+1}-\tau_{i}\leq\theta^{-1}(t-s)=\alpha
r^{2}_{0}$

for every $i=0,\dots,N$. Since

$d_{\tau_{0}}(x,y)=d_{s}(x,y)\geq
B\sqrt{t-s}\geq\alpha^{-N-1}A^{-1/2}\sqrt{t-s}=\alpha^{-N-1}r_{0}\geq r_{0},$

Theorem 3.2 tells us that

$d_{\tau_{1}}(x,y)\geq\alpha\,d_{\tau_{0}}(x,y)\geq\alpha^{-N}r_{0}\geq
r_{0}.$

In the same manner as in the proof of Lemma 6.1, we use Theorem 3.2
repeatedly. We obtain

(6.4) $d_{\tau_{i}}(x,y)\geq\alpha
d_{\tau_{i-1}}(x,y)\geq\alpha^{-N+(i-1)}r_{0}\geq r_{0}$

for all $i=1,\dots,N+1$. For $l_{1},l_{2}\in[s,t]$, let
$i_{0}\in\\{0,\dots,N\\}$ satisfy $l_{1}\in[\tau_{i_{0}},\tau_{i_{0}+1}]$.
Theorem 3.2 together with (6.4) leads us to

(6.5) $d_{l_{1}}(x,y)\geq\alpha
d_{\tau_{i_{0}}}(x,y)\geq\alpha^{-N+i_{0}}r_{0}\geq r_{0}.$

By using Theorem 3.2 together with (6.4), (6.5) at most $N+1$ times, we
conclude

$\displaystyle d_{l_{2}}(x,y)\geq\alpha^{N+1}d_{l_{1}}(x,y)\geq
B^{-1}d_{l_{1}}(x,y).$

Switching the roles of $l_{1}$ and $l_{2}$, we complete the proof. $\Box$

We now prove the upper bound (1.4).

######  Proposition 6.4.

For any $A>0$, there exist positive constants
$\mathcal{C}_{3},\mathcal{C}_{4}>0$ depending only on $n,T,g(0)$ and $A$ such
that the following holds: We assume $H\leq H_{1}$ for $H_{1}>0$. For
$s,t\in[0,T)$ with $s<t$, we also suppose $t-s\leq A\,H_{1}^{-1}$ and
$s\geq(t-s)/A$. Then we have

$G(x,t;y,s)\leq\frac{\mathcal{C}_{3}}{(t-s)^{n/2}}\exp\left({-\frac{\mathcal{C}_{4}\,d_{s}(x,y)^{2}}{t-s}}\right).$

###### Proof.

We again notice (6.1) and (6.2). By Proposition 2.5, for $l\in[s,t)$ we have

(6.6) $G(x,t;y,l)\leq\frac{C_{1}}{(t-l)^{n/2}}.$

Let $B>1$ be the constant in Lemma 6.3 satisfying $B^{-1}A^{-1/2}<1$. When
$d_{s}(x,y)<B\sqrt{t-s}$, (6.6) implies

$G(x,t;y,s)\leq\frac{C_{1}}{(t-s)^{n/2}}\leq\frac{C_{1}e^{B^{2}}}{(t-s)^{n/2}}\exp\left({-\frac{d_{s}(x,y)^{2}}{t-s}}\right).$

This is a desired one.

We consider the case of $d_{s}(x,y)\geq B\sqrt{t-s}$. We set

$c:=B^{-1}A^{-1/2}\in(0,1).$

For all $l\in[s,t)$ we see

$c\sqrt{t-l}<\sqrt{T},\quad
H\leq\frac{A}{t-s}\leq\frac{AB^{2}}{t-l}=\frac{1}{c^{2}(t-l)},$

and hence

(6.7) $m(B(x,c\sqrt{t-l},l))\geq\kappa_{1}(t-l)^{n/2}$

by Theorem 2.4. Proposition 2.15 together with (6.7) implies

(6.8)
$\displaystyle\quad\,\,\frac{1}{m(B(y,c\sqrt{t-l},l))}\int_{B(y,c\sqrt{t-l},l)}G(x,t;z,l)\,dm(z)$
$\displaystyle\leq\bigg{(}\int_{B(x,c\sqrt{t-l},l)}G(x,t;z,l)\,dm(z)\bigg{)}^{-1}\frac{C_{2}}{(t-l)^{n/2}}\exp\left(-\frac{d_{l}(x,y)^{2}}{16(t-l)}\right).$

By Proposition 6.2,

(6.9)
$G(x,t;z,l)\geq\frac{C_{3}}{(t-l)^{n/2}}\exp\left({-\frac{C_{4}d_{l}(x,z)^{2}}{t-l}}\right)\geq\frac{C_{5}}{(t-l)^{n/2}}$

for all $z\in B(x,c\sqrt{t-l},l)$. Combining (6.7), (6.8), (6.9), we conclude

(6.10)
$\frac{1}{m(B(y,c\sqrt{t-l},l))}\int_{B(y,c\sqrt{t-l},l)}G(x,t;z,l)\,dm(z)\leq\frac{C_{6}}{(t-l)^{n/2}}\exp\left(-\frac{d_{l}(x,y)^{2}}{16(t-l)}\right).$

Now, we possess

$\displaystyle c\sqrt{t-l}$ $\displaystyle<A^{-1/2}\sqrt{t-l}\leq
A^{-1/2}\sqrt{t-s}\leq\sqrt{s}\leq\sqrt{l},$ $\displaystyle H(\cdot,\tau)$
$\displaystyle\leq\frac{A}{t-s}\leq\frac{c^{2}(t-l)}{l-\tau}\frac{A}{t-s}\leq\frac{c^{2}A}{l-\tau}$

for $\tau\in[l-c^{2}(t-l),l]$, and hence Theorem 2.12 tells us that

(6.11) $m_{l}(B(y,c\sqrt{t-l},l))\leq\kappa_{2}(t-l)^{n/2}.$

From (6.10), (6.11) and Lemma 6.3, one can derive

$\int_{B(y,c\sqrt{t-l},l)}G(x,t;z,l)\,dm(z)\leq
C_{7}\exp\left({-\frac{d_{l}(x,y)^{2}}{16(t-l)}}\right)\leq
C_{7}\exp\left({-\frac{B^{-2}\,d_{s}(x,y)^{2}}{16(t-s)}}\right).$

This together with (6.6) leads us to

$\int_{B(y,c\sqrt{t-l},l)}G(x,t;z,l)^{2}\,dm(z)\leq\frac{C_{8}}{(t-l)^{n/2}}\exp\left({-\frac{B^{-2}\,d_{s}(x,y)^{2}}{16(t-s)}}\right).$

Integrating over $[s,(s+t)/2]$, we obtain

$\int^{\frac{s+t}{2}}_{s}\int_{B(y,c\sqrt{t-l},l)}\,G(x,t;z,l)^{2}\,dm(z)\,dl\leq\frac{C_{9}}{(t-s)^{n/2-1}}\exp\left({-\frac{B^{-2}\,d_{s}(x,y)^{2}}{16(t-s)}}\right).$

We now set

$r:=c\sqrt{\frac{t-s}{2}}.$

We notice that $t-l\geq(t-s)/2$ for $l\in[s,(s+t)/2]$, and hence

$Q^{+}(y,r,s)\subset\left\\{(z,l)\in M\times[0,T)~{}~{}\middle|~{}~{}z\in
B(y,c\sqrt{t-l},l),\,l\in\left[s,\frac{s+t}{2}\right]\right\\}.$

It follows that

$\int_{Q^{+}(y,r,s)}G(x,t;z,l)^{2}\,dm(z)\,dl\leq\frac{C_{9}}{(t-s)^{n/2-1}}\exp\left({-\frac{B^{-2}\,d_{s}(x,y)^{2}}{16(t-s)}}\right).$

By virtue of Theorem 5.2,

$G(x,t;y,s)^{2}\leq\frac{C_{10}}{r^{n+2}}\int_{Q^{+}(y,r,s)}G(x,t;z,l)^{2}\,dm(z)\,dl\leq\frac{C_{11}}{(t-s)^{n}}\exp\left({-\frac{B^{-2}\,d_{s}(x,y)^{2}}{16(t-s)}}\right).$

Thus, we complete the proof. $\Box$

We finally prove the gradient estimate (1.5).

######  Proposition 6.5.

For any $A>0$, there exist positive constants
$\mathcal{C}_{5},\mathcal{C}_{6}>0$ depending only on $n,T,g(0)$ and $A$ such
that the following holds: We assume $H\leq H_{1}$ for $H_{1}>0$. For
$s,t\in[0,T)$ with $s<t$, we also suppose $t-s\leq A\,H_{1}^{-1}$ and
$s\geq(t-s)/A$. Then we have

$|\nabla_{x}G|(x,t;y,s)\leq\frac{\mathcal{C}_{5}}{(t-s)^{(n+1)/2}}\exp\left({-\frac{\mathcal{C}_{6}\,d_{s}(x,y)^{2}}{t-s}}\right).$

###### Proof.

We set $G(x,t):=G(x,t;y,s)$. Due to Theorem 2.9,

$\frac{|\nabla
G|^{2}}{G^{2}}(x,t)\leq\frac{C_{1}}{t-s}\log\frac{\mathcal{A}}{G(x,t)},$

where

$\mathcal{A}:=\sup_{M\times\left[\frac{s+t}{2},t\right]}G.$

It follows that

$|\nabla
G|^{2}(x,t)\leq\frac{C_{1}}{t-s}\mathcal{A}G(x,t)\frac{G(x,t)}{\mathcal{A}}\log\frac{\mathcal{A}}{G(x,t)}\leq\frac{C_{1}}{t-s}\mathcal{A}G(x,t)$

since $G(x,t)/\mathcal{A}\leq 1$. By Propositions 2.5 and 6.4, we possess

$\mathcal{A}\leq\frac{C_{2}}{(t-s)^{n/2}},\quad
G(x,t)\leq\frac{C_{3}}{(t-s)^{n/2}}\exp\left({-\frac{C_{4}\,d_{s}(x,y)^{2}}{t-s}}\right).$

Combining them, we arrive at

$|\nabla
G|^{2}(x,t)\leq\frac{C_{5}}{(t-s)^{n+1}}\exp\left({-\frac{C_{4}\,d_{s}(x,y)^{2}}{t-s}}\right).$

This completes the proof. $\Box$

###### Proof of Theorem 1.1.

By Propositions 6.2, 6.4, 6.5, we complete the proof. $\Box$

### Acknowledgements

The first author was supported by JSPS KAKENHI (JP19K14521). The second author
was supported by JSPS KAKENHI (JP21K20315).

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