# A Liouville theorem of $VT$-harmonic map heat flow

Xiangzhi Cao School of Information Engineering, Nanjing Xiaozhuang University,
Nanjing 211171<EMAIL_ADDRESS>
###### Abstract

We proved an Liouville theorem for Backward $VT$-harmonic map heat flow from
evolution manifolds into generalized regular ball. Among others, we also
proved an Liouville theorem for $VT$-harmonic map heat flow from complete
manifolds into generalized regular ball.

Keywords and phrases: Dirichlet problem, Heat flow, $VT$-Harmonic map.

MSC 2010: 58E15, 58E20 , 53C27

##  1 Introduction

It is well known that harmonic map has a long history. Harmonic map represents
a kind of classical variation problem and nonlinear geometric analysis
problem. Harmonic map has many generalizations, such as V-harmonic map [4],
Hermitian harmonic map ([10]) which is the particular case of V-harmonic map,
affine harmonic map ([9]), Dirac-harmonic map([3]), etc. In this paper, we
consider another kind of generalized map which is introduced by Chen et al. in
2020:

######  Definition 1 (VT-harmonic map, cf. [5] ).

Let $(M,g)$ be a compact manifold with boundary, $(N,h)$ a compact Riemannian
manifold. A map $u:(M,g)\rightarrow(N,h)$ is called a $VT$-harmonic map iff
$u$ satisfies

$\tau_{V}{u}+Tr_{g}T(du,du)=0,$ (1.1)

where
$\tau_{V}{u}=\tau(u)+du(V),\tau(u)=Tr_{g}(Ddu),V\in\Gamma(TM),T\in\Gamma(\otimes^{1,2}TN)$.

######  Remark 1.

It is obvious that if $T\equiv 0$, $u$ is just $V$-harmonic map ([4]).
$V$-harmonic map is the special case of $VT$ harmonic map. Hermition harmonic
map is a particular case of $V$-harmonic map. When we take special $V$ and
$T$, $VT$\- harmonic map has wide application to other geometric probelm.

######  Remark 2.

The Liouville theorem of VT harmonic map is rare. In [2], we obtained a
Liouville type theorem for VT-harmonic map into horoball. The method in [5]
can’t be used directly to get the Liouville theorem. So, our main contribution
of this paper is to derive Liouville theorm for VT harmonic map into
generalized regualar ball(see definition 3).

Jost-Yau[10] investigated the existence of Hermitian harmonic maps from
Hermitian manifolds into compact nonpositively curved Riemannian manifolds by
using the heat flow method. The study of such kind of maps is more difficult
than that of harmonic map, since generally no variation structure can be
employed. The absence of variation structure often leads to the lack of
monotonicity inequality, which is a challenge for studying its blow-up
behaviour (e.g. energy identity) and proving the existence using heat flows.

In the first part of this paper, we will deal with evolving manifold. We
recall some definitions of backward Ricci flow. A smooth manifold
$(M,g(t)),t\in I$ with a time-dependent Riemannian metric is called Ricci flow
when

$\displaystyle\partial_{t}g=-2\operatorname{Ric}$ (1.2)

which was introduced by Hamilton.

A supersolution to (1.2) is called super Ricci flow. Namely, $(M,g(t)),t\in I$
is called super Ricci flow if

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

which has been introduced by McCann-Topping[22] from the viewpoint of optimal
transport theory. Harmonic Ricci flow ([23]) and List’s flow ([21]), mean
curvature flow for spacelike hypersurfaces in Lorentzian manifold of non-
negative sectional curvature, and (scaled) twisted K¨ahler-Ricci flow are the
examples of super Ricci flow include. One can refer to [1, 6, 7, 11, 12, 15,
20, 24, 25] the further study on super Ricci flow. When $t\leq 0$, super Ricci
flow is termed as ancient super Ricci flow.

For the function $u:(M,g)\times(\infty,0]\to R$,

$\displaystyle\frac{\partial}{\partial t}u=\Delta_{g}u,$

such equation is called ancient heat equation.

The evolving manifold $(M,g(\tau),\tau\geq 0$ is called backward super Ricci
flow if

$\partial_{\tau}g\leq 2\operatorname{Ric},$

For the function $u:(M,g)\times[0,\infty)\to\mathbf{R}$,

$\displaystyle\frac{\partial}{\partial\tau}u+\Delta_{g}u=0,$

such equation is called backward heat equation.

The ancient super Ricci flow and ancient heat equation are transformed into
backward super Ricci flow and backward heat equation through negative
transformation of time parameters. Wang [26] studied Liouville theorem of
ancient heat equation. The backward harmonic mapping heat flow is a
generalization of the backward heat equation. However, there are few studies
on the heat flow of backward harmonic mapping. Guo et al. [8] considered the
Liouville theorem of the backward heat equation heat flow along the backward
super Ricci flow.

Kunikawa et al. [13, 14] used the truncation function constructed by the
reduced distance function to obtain the gradient estimates of the backward
harmonic map along the backward super Ricci flow, and obtained the Liouville
theorem. In proving gradient estimates, this truncation function is still very
novel. In this paper, we hope to continue to use the skills of these two
papers [13, 14] and extend the corresponding results for harmonic map to
$VT$\- harmonic map. However, in this paper, unlike kunikawa et al[13, 14],
the image of our map is in the generalized regular ball in the target
manifold.

We generalized condition B in [18, 17, 16, 19] as follows

######  Definition 2 (Condtion C).

Let $N$ be a complete Riemannian manifold. Let $\Omega$ be a bounded open
subset of $N$. We say $\Omega$ satisfies condition $(C)$ if there exists a
positive function $f\in C^{2}(\Omega)$ satisfying the conditons:

$\begin{cases}-\nabla^{2}f-f(\frac{s_{0}-1}{s_{0}}\kappa+\frac{1}{4\varepsilon_{1}}\|\nabla
T\|_{L^{\infty}}^{2}+\frac{1}{\varepsilon_{2}}\|T\|_{L^{\infty}}^{2})h\geq
Qh\\\ 0<m_{1}(\Omega)\leq f(y)\leq m_{2}(\Omega)<\infty\\\ |\nabla f(y)|\leq
m_{3}(\Omega)<\infty\end{cases}$ (1.3)

for all $y\in\Omega$, where $\kappa(y)=\sup\\{K(y,\pi)|K(y,\pi)$ is the
sectional curvature of plane $\left.\pi\in
T_{y}N\right\\},Q>\frac{m_{3}^{2}}{2m_{1}}$， $\epsilon_{1},\epsilon_{2}$ are
two small positive constant, $s_{0}=\min\\{m,n\\}$, $m,m_{1},m_{2},m_{3}$ are
suitable positive constant.

######  Remark 3.

When $T=0$, it is just generalized regular ball defined in [18, 17, 16, 19].

######  Definition 3 (cf. [18, 17, 16, 19]).

If ${\Omega}$ satisfies condition (C) and there is a nonnegative convex
function ${f^{*}}$ on ${\Omega}$ such that
${\Omega=\left(f^{*}\right)^{-1}([0,r))}$, ${\Omega}$ is called a generalizad
regular ball.

######  Remark 4.

Regular ball is an exmaple of generalized regular ball. It is hard to get
Liouville theorem for $VT$-harmonic map into regular ball due to the extra
terms about $T$ and $V$. We can get get Liouville theorem for $VT$-harmonic
map into gerneralized regular ball due to the definition 3. This is our
motivation of this paper.

In this paper, we firstly consider the following backward $VT$-harmonic map
heat flow which is the generaliztion of backward harmonic map heat flow:

$\left\\{\begin{array}[]{ll}&\frac{\partial
u}{\partial\tau}+\tau_{V}{u}+Tr_{g}T(du,du)=0,\quad on\quad M.\\\
&u:M\times[0,\infty)\to\Omega\subset N.\\\ &\partial_{\tau}g\leq
2\operatorname{Ric}\end{array}\right.$ (1.4)

where $\Omega$ is the generalized regular ball in $N.$

Guo-Philipowski-Thalmaier[8] have approached this problem for backward
harmonic map heat flow from stochastic analytic viewpoint. Here we aim to
approach the problem from Perelman’s reduced geometric viewpoint.

We say that $(M,g(\tau)),\tau\in[0,\infty)$ is admissible(cf. [14, subsection
1.2]) if for every $\tau>0$ there is $c_{\tau}\geq 0$ depending only on $\tau$
such that $h\geq−c_{\tau}g$ on $[0,\tau]$. As said in [14], the admissibility
ensures that the $L$-distance is achieved by a minimal $L$-geodesic.

Now we are in a postiton to state our first main result

######  Theorem 1.

Let $(M^{m},g(\tau))_{\tau\in[0,\infty)}$ be an admissible complete backward
super Ricci flow. Let $(N^{n},h)$ be a complete Riemannian manifold with
$\sec\leq\kappa$ for $\kappa>0$. We denote by $h_{0}$ the funtion
$\frac{1}{2}\frac{\partial g(\tau)}{\partial\tau}$. We assume

$\mathcal{D}(V)\geq
0,\mathcal{H}(V)\geq-\frac{H}{\tau},Ric_{V}-h_{0}\geq-K,\left\|T\right\|_{\infty}<\frac{2Q}{m_{3}}-\frac{m_{3}}{m_{1}},$

for all vector fields $V$, $K\geq 0,H\geq 0$. Here, one can refer to (2.2) for
the definition of $\mathcal{D}(V),\mathcal{H}(V)$ , the constants
$Q,m_{1},m_{3}$ are the same as that in definition 2. Let
$Q_{R,\Lambda}:=\\{(x,\tau)|\mathfrak{d}(x,\tau)\leq R\\}$, here the function
$\mathfrak{d}$ refers to $\sqrt{4\tau\ell(x,\tau)}$, where the distance
function $\ell(x,\tau)$ is defined in (2.1). Let
$u:M\times[0,\infty)\rightarrow N$ be a solution to backward $VT$-harmonic map
heat flow (1.4) such that the image of $u$ is contained in $\Omega\subset N$,
Then on $Q_{R/2,\Lambda/4}$, for any $0\leq\Lambda<\infty$

$\displaystyle\sup_{Q_{R/2,\Lambda/4}}\frac{|du|^{2}}{f^{2}}(x,\tau)$ (1.5)
$\displaystyle\leq$ $\displaystyle\frac{C_{2}}{C_{1}-5\varepsilon-m_{3}^{2}}$
$\displaystyle+\frac{1}{C_{1}-5\varepsilon-
m_{3}^{2}}\bigg{(}\frac{C_{3/4}^{2}}{\varepsilon}\left(m^{2}+\frac{9}{4}\right)\frac{1}{R^{4}}+\frac{D^{2}}{4\varepsilon}\frac{1}{\Lambda^{2}}+\frac{C_{3/4}^{2}}{4\varepsilon}K^{2}$
$\displaystyle+\frac{9C_{3/4}^{4}}{\varepsilon}\frac{1}{R^{4}}+\frac{243C_{3/4}^{4}}{16}\frac{1}{m_{3}^{2}}\frac{1}{R^{4}}+\frac{1}{4\epsilon}\|V\|_{\infty}C^{2}_{1/2}\frac{1}{R^{2}}\bigg{)}^{\frac{1}{2}}$

Here the costant $C_{1}=2Qm_{1}-\|T\|_{L^{\infty}}m_{3}m_{1}$,
$C_{2}=2(K+\epsilon_{1})-\frac{8\epsilon_{3}-1}{4\epsilon_{3}}(\frac{m_{3}}{m_{1}})^{2},2\epsilon_{3}=1-\epsilon_{2}$.
The constants $D,C_{3/4}$ is defined in Lemma 3, the constant $\epsilon$ is
some small positive constant, the function $f$ is the defining function of the
domain $\Omega$, the constant $m_{3}$ is that defined in Definition 2. The
constants $\epsilon_{1},\epsilon_{2}$ are suitable small postive constants
defined in (3.3)

######  Remark 5.

Along admissible complete backward super Ricci flow, the condition
$Ric_{V}-h_{0}\geq-K$ is implied by the conditon $\frac{1}{2}L_{V}g\geq-K.$

Choosing some $\epsilon_{1},\epsilon_{2}$ , letting $R,\Lambda\to\infty$ in
Theorem 1 and seeing that the function $f$ is upper bounded, we get

######  Corollary 1.1.

In the situations of Theroem 1, let $u:M\times[0,\infty)\rightarrow N$ be a
solution to backward $VT$-harmonic map heat flow (1.4) such that the image of
$u$ is contained in $\Omega\subset N$, then the map $u$ is constant for some
choice of $\epsilon_{1},\epsilon_{2}$.

######  Remark 6.

Compared to the main theorems in [13, 14] , condition on the map $u$ near
infinity in Corollary 1.1 is not required, since the image of the map
cosidered in this paper is not in regular ball, but rather in generalized
regular ball. The method of gradient estimates in Theorem 1 was inspired by
[18], in order to get new estimates and nontrivial generalization.

In the second part of this paper, we also considered the following heat flow
combining method of [5] and [18, 17, 16]:

$\left\\{\begin{array}[]{ll}&\frac{\partial u}{\partial
t}=\tau_{V}{u}+Tr_{g}T(du,du),\quad on\quad M.\\\
&u=u_{0},~{}~{}~{}~{}~{}~{}~{}~{}~{}on~{}~{}~{}~{}~{}~{}~{}M\times\\{0\\}.\\\
&u:M\times[0,T_{max})\to\Omega.\end{array}\right.$ (1.6)

where $u_{0}:M\times[0,T_{max})\to\Omega$, here $\Omega$ is the generalized
regular ball.

About this problem (1.6), the second result in this paper is as follows:

######  Theorem 2.

Let $(M^{m},g)$ and $(N^{n},h)$ be two complete Riemannian manifolds. Let
$x_{o}\in M$ and $r(x)$ be the distance function from $x_{o}$, we use the
notations $B_{R}\left(x_{o}\right)=\\{x\in M|r(x)\leq$ R), in addition, we
assume that $Ric_{V}\geq-A,A\geq 0.$ Moreover, suppose that $\Omega\subset N$
satisfies condition $(B)$. Assume taht $\langle V,\nabla r\rangle\leq v(r)$
for some nondecreasing function $v(\cdot)$ and
$\left\|T\right\|_{\infty}<\frac{2Q}{m_{3}}$. If $u(x,t)$ is a solution of
equation (1.6) on
$B_{R}\left(x_{0}\right)\times\left[0,T_{1}\right),u\left(B_{R}\left(x_{0}\right)\times\left[0,T_{1}\right)\right)\subset\Omega$
and $B_{R}\left(x_{0}\right)\cap\partial M=\phi$ , then for $0<\Lambda<T_{1}$,
we have

$\displaystyle\sup_{B_{R/2}(x_{0})}|\nabla u(x,t)|\leq
m_{2}\left(\frac{C_{0}^{\frac{1}{2}}m_{3}}{K_{2}R}+C_{4}\frac{1}{\sqrt{K_{2}}}\left(\sqrt{K_{1}}+\sqrt{\frac{1}{R}}+\frac{1}{R}\right)+\frac{1}{\sqrt{K_{2}}\Lambda^{1/2}}\right),$
(1.7)

and

$\displaystyle\sup_{B_{R/2}(x_{0})}|\nabla
u(x,t)|\leq\frac{m_{2}}{m_{1}}\sup_{B_{R}\left(x_{0}\right)}|\nabla
u_{0}|+m_{2}\left(\frac{\frac{2m\sqrt{C_{0}}}{R}+\sqrt{\frac{2m\sqrt{C_{0}}}{R^{2}}+4K_{2}\left(K_{1}+\frac{C_{2}+2C_{0}}{R^{2}}+\frac{C_{3}}{R}\right)}}{2K_{2}}\right),$
(1.8)

for all $0<t<T_{1}$. Here
$K_{1}=2(A+\epsilon_{1})-\frac{3-4\epsilon_{2}}{2(1-\epsilon_{2})}(\frac{m_{3}}{m_{1}})^{2}$,
$K_{2}=2Qm_{1}-\|T\|_{L^{\infty}}{m_{3}}{m_{1}}$, $C_{0}>0$ is a universal
constant defined in (4.7),
$C_{1}=\sqrt{(m-1)A},C_{2}=C_{0}+\sqrt{C_{0}}(m-1),C_{3}=v(a)+C_{1},C_{4}=\operatorname{max}(C_{2}+2C_{0},C_{3}).$
The constants $\epsilon_{1},\epsilon_{2}$ are suitable small postive constants
defined in (3.3)

######  Corollary 1.2.

In the situations of Theroem 2, let $u:M\times[0,\infty)\rightarrow N$ be a
solution to $VT$-harmonic map heat flow (1.6) such that the image of $u$ is
contained in $\Omega\subset N$, then the map $u$ is constant for some choice
of $\epsilon_{1},\epsilon_{2}$.

As in Theroem 2, if we take the function $F$ directly instead of the function
$\lambda F$, slightly adapting the proof of Theroem 2, we will have

######  Corollary 1.3.

Let $(M^{m},g)$ be closed manifold. Let $N,\Omega,V$ be the same situations in
Theorem 2. If $u(x,t)$ is a solution of equation (1.6) on
$M\times\left[0,T_{1}\right),u\left(M\times\left[0,T_{1}\right)\right)\subset\Omega$
, then for $0<t<T_{1}$, we have

${|\nabla u(x,t)|\leq
m_{2}\left(C_{4}\frac{1}{\sqrt{K_{2}}}\sqrt{K_{1}}+\frac{1}{\sqrt{K_{2}}t^{1/2}}\right),}$

and

$\displaystyle|\nabla
u(x,t)|\leq\frac{m_{2}}{m_{1}}\sup_{B_{R}\left(x_{0}\right)}|\nabla
u(x,0)|+m_{2}\sqrt{\frac{K_{1}}{K_{2}}},$ (1.9)

for all $0<t<T_{1}$. Here
$K_{1}=2(A+\epsilon_{1})-\frac{3-4\epsilon_{2}}{2(1-\epsilon_{2})}(\frac{m_{3}}{m_{1}})^{2}$,
$K_{2}=2Qm_{1}-\|T\|_{L^{\infty}}{m_{3}}{m_{1}}$, $C_{0}>0$ is a universal
constant defined in (4.7),
$C_{1}=\sqrt{(m-1)A},C_{2}=C_{0}+\sqrt{C_{0}}(m-1),C_{3}=v(a)+C_{1},C_{4}=\operatorname{max}(C_{2}+2C_{0},C_{3}).$
The constants $\epsilon_{1},\epsilon_{2}$ are suitable small postive constants
defined in (3.3)

For $VT$-harmonic map, from Theroem 2 and Corollary 1.3, we get

######  Corollary 1.4.

Let $M,N,\Omega,V$ be the same situations in Theorem 2. If $u(x)$ is
$VT$-harmonic map from $B_{R}(x_{0})$ into $\Omega$, then for
$0<\Lambda<T_{1}$, we have

${\sup_{B_{R/2}(x_{0})}|\nabla u|\leq
m_{2}\left(\frac{C_{0}^{\frac{1}{2}}m_{3}}{K_{2}R}+C_{4}\frac{1}{\sqrt{K_{2}}}\left(\sqrt{K_{1}}+\sqrt{\frac{1}{R}}+\frac{1}{R}\right)\right),}$

If $u(x)$ is $VT$-harmonic map from $M$ into $\Omega$ ,then we have

$\displaystyle\sup_{B_{R/2}(x_{0})}|\nabla u|\leq
m_{2}\sqrt{\frac{K_{1}}{K_{2}}},$ (1.10)

Here
$K_{1}=2(A+\epsilon_{1})-\frac{3-4\epsilon_{2}}{2(1-\epsilon_{2})}(\frac{m_{3}}{m_{1}})^{2}$,
$K_{2}=2Qm_{1}-\|T\|_{L^{\infty}}{m_{3}}{m_{1}}$, $C_{0}>0$ is a universal
constant defined in (4.7),
$C_{1}=\sqrt{(m-1)A},C_{2}=C_{0}+\sqrt{C_{0}}(m-1),C_{3}=v(a)+C_{1},C_{4}=\operatorname{max}(C_{2}+2C_{0},C_{3}).$
The constants $\epsilon_{1},\epsilon_{2}$ are suitable small postive constant
defined in (3.3)

By a similar method as in Theorem 2, we can get

######  Corollary 1.5.

Let ${M}$ be a complete Riemannian manifold with Ricci curvature bounded below
by

$\operatorname{Ric_{V}}\geq A\geq 0,$

Let $N,\Omega,V$ be in the same situations in Theorem 2. If $u(x)$ is a
$VT$-harmonic map from ${M}$ into ${\Omega}$ and
$A\geq\frac{\epsilon_{1}}{2}-\frac{3-4\epsilon_{2}}{4(1-\epsilon_{2})}\left(\frac{m_{3}}{m_{1}}\right)^{2},$
then ${u}$ is constront.

Next we slightly mention the case when the domain manifold is compact manifold
with boundary. Concretely speaking, we consider the following initial boundary
problem

$\left\\{\begin{array}[]{ll}&\frac{\partial u}{\partial
t}=\tau_{V}{u}+Tr_{g}T(du,du),\quad on\quad M.\\\
&u=u_{0},~{}~{}~{}~{}~{}~{}~{}~{}on~{}~{}~{}~{}~{}~{}~{}\partial
M\times[0,T_{max});\\\
&u=u_{0},~{}~{}~{}~{}~{}~{}~{}~{}~{}on~{}~{}~{}~{}~{}~{}~{}M\times\\{0\\}.\\\
&u:M\times[0,T_{max})\to\Omega.\end{array}\right.$ (1.11)

where $u_{0}:M\times[0,T_{max})\to\Omega$, here $\Omega$ is the generalized
regular ball.

We have the following result without giving its proof.

######  Corollary 1.6.

Let $(M^{m},g)$ be compact manifold with boundary. Let $N,\Omega,V$ be the
same situations in Theorem 2. If $u(x,t)$ is a solution of equation (1.11) on
$M\times\left[0,T_{1}\right),u\left(M\times\left[0,T_{1}\right)\right)\subset\Omega$
, then for $0<t<T_{1}$, we have

$\displaystyle|\nabla u(x,t)|\leq\frac{m_{2}}{m_{1}}\sup_{M}|\nabla
u_{0}|+m_{2}\sqrt{\frac{K_{1}}{K_{2}}},$ (1.12)

Here
$K_{1}=2(A+\epsilon_{1})-\frac{3-4\epsilon_{2}}{2(1-\epsilon_{2})}(\frac{m_{3}}{m_{1}})^{2}$,
$K_{2}=2Qm_{1}-\|T\|_{L^{\infty}}{m_{3}}{m_{1}}$, $C_{0}>0$ is a universal
constant defined in (4.7),
$C_{1}=\sqrt{(m-1)A},C_{2}=C_{0}+\sqrt{C_{0}}(m-1),C_{3}=v(a)+C_{1},C_{4}=\operatorname{max}(C_{2}+2C_{0},C_{3}).$
The constants $\epsilon_{1},\epsilon_{2}$ are suitable small postive constant
defined in (3.3)

This paper is orgalized as follows: In section 2, we recalled some backgound
of backward Ricci flow and gives some lemmas used in the subsequent section.
In section 3, we proved Theroem 1. In section 4, we prove Theorem 2.

##  2 Preliminary

Let $(M,g(\tau))_{\tau\in[0,\infty)}$ be an $m$-dimensional, complete time-
dependent Riemannian manifold. For a curve
$\gamma:\left[\tau_{1},\tau_{2}\right]\rightarrow M$, its $\mathcal{L}$-length
is defined as

$\mathcal{L}(\gamma):=\int_{\tau_{1}}^{\tau_{2}}\sqrt{\tau}\left(H+\left\|\frac{d\gamma}{d\tau}\right\|^{2}\right)d\tau.$

It is well-known that its critical point over all curves with fixed endpoints
is characterized by the following $\mathcal{L}$-geodesic equation:

$X:=\frac{d\gamma}{d\tau},\quad\nabla_{X}X-\frac{1}{2}\nabla
H+\frac{1}{2\tau}X+2h(X)=0.$

For $(x,\tau)\in M\times(0,\infty)$, the $L$-distance $L(x,\tau)$ and reduced
distance $\ell(x,\tau)$ from a space-time base point $\left(x_{0},0\right)$
are defined by

$\displaystyle
L(x,\tau):=\inf_{\gamma}\mathcal{L}(\gamma),\ell(x,\tau):=\frac{1}{2\sqrt{\tau}}L(x,\tau),$
(2.1)

Where the infimum is taken over all curves $\gamma:[0,\tau]\rightarrow M$ with
$\gamma(0)=x_{0}$ and $\gamma(\tau)=x$. A curve is called minimal
$\mathcal{L}$-geodesic from $\left(x_{0},0\right)$ to $(x,\tau)$ if it attains
the infimum of (2.1). Hereafter, we use the following notations:

$\displaystyle\begin{cases}\bar{L}(x,\tau):=4\tau\ell(x,\tau),\\\
\mathcal{D}(V):=-\partial_{\tau}H-\Delta
H-2\|h\|^{2}+4\operatorname{div}h(V)-2g(\nabla
H,V)+2\operatorname{Ric}(V,V)-2h(V,V),\\\
\mathcal{H}(V):=-\partial_{\tau}H-\frac{H}{\tau}-2g(\nabla H,V)+2h(V,V),\\\
R(V):=\operatorname{Ric}(V,V)-h(V,V).\end{cases}$ (2.2)

The function $\mathcal{D}(V)$ is refered to as M$\ddot{u}$ller qunantity. We
now assume that $(M,g(\tau))_{\tau\in[0,\infty)}$ is admissible (see
Subsection 1.2). In this case, for every $(x,\tau)\in M\times(0,\infty)$,
there exists at least one minimal $\mathcal{L}$-geodesic. Also, the functions
$L(\cdot,\tau)$ and $L(x,\cdot)$ are locally Lipschitz in $(M,g(\tau))$ and
$(0,\infty)$, respectively; in particular, they are differentiable almost
everywhere. Assume that $\ell$ is smooth at $(\bar{x},\bar{\tau})\in
M\times(0,\infty)$. We have

######  Lemma 1 ([cf. [13, 14]).

] Let $K\geq 0$. We assume

$\mathcal{D}(V)\geq-2K\left(H+\|V\|^{2}\right),H\geq 0,$

for all vector fields $V$. Then at $(\bar{x},\bar{\tau})$ we have

$\left(\Delta+\partial_{\tau}\right)\bar{L}\leq 2m+2K\bar{L}.$

######  Lemma 2 (cf. [13, 14]).

We assume

$\mathcal{H}(V)\geq-\frac{H}{\tau},H\geq 0$

for all vector fields $V$. Then at $(\bar{x},\bar{\tau})$ we have

$\|\nabla\mathfrak{d}\|^{2}\leq 3.$

Here, the function $\mathfrak{d}$ refers to $\sqrt{4\tau\ell(x,\tau)}$ .

######  Lemma 3 (cf. Lemma 4.4 in [14] ).

Let $R,\Lambda>0,\alpha\in(0,1)$. Then there is a smooth function
$\psi:[0,\infty)\times[0,\infty)\rightarrow$ $[0,1]$ which is supported on
$[0,R]\times[0,\lambda]$, and a constant $C_{\alpha}>0$ depending only on
$\alpha$ such that the following hold:

(1) $\psi\equiv 1$ on $[0,R/2]\times[0,\Lambda/4]$;

(2) $\partial_{r}\psi\leq 0$ on $[0,\infty)\times[0,\infty)$, and
$\partial_{r}\psi\equiv 0$ on $[0,R/2]\times[0,\infty)$;

(3) we have

$\frac{\left|\partial_{r}\psi\right|}{\psi^{\alpha}}\leq\frac{C_{\alpha}}{R},\frac{\left|\partial_{r}^{2}\psi\right|}{\psi^{\alpha}}\leq\frac{C_{\alpha}}{R^{2}},\frac{\left|\partial_{\tau}\psi\right|}{\psi^{1/2}}\leq\frac{D}{\Lambda},$

where $C>0$ is a universal constant,

In the sequel, these constants
$\epsilon,\epsilon_{1},\epsilon_{2},\epsilon_{3}$ etc. are used to denote
small positive constant. The constants $C,C_{1},C_{3/4}$ etc. which occured in
the inequalities may be different at defferent lines. The energy density
$e(u)=|du|^{2}$. We hope that the reader will not find these notations
confusing.

##  3 Proof of Theorem 1

We first fix the notation. We denote by $h_{0}$ the funtion
$\frac{1}{2}\frac{\partial g(\tau)}{\partial\tau}$ in order to make
differences with the metric $h$ of the manifold $N$. The index $\alpha,\beta$
etc. ranges from 1 to $m$. Let $e_{1},e_{2},\cdots,e_{m}$ be a focal
orthonormal frame field of the domain manifold $M$ . In this section, the
constant such as $C_{1/2},C_{3/4}$ are defined in Lemma 3.

Since $u$ is the solution of backward $VT$-harmonic heat flow,

$\frac{\partial u}{\partial\tau}+\tau_{V}{u}+Tr_{g}T(du,du)=0,$

Computing directly, we have

$\displaystyle\frac{\partial}{\partial\tau}|du|^{2}=-\langle
du(h_{0}(e_{i})),du(e_{i})\rangle+\langle\nabla_{e_{\alpha}}(\frac{\partial
u}{\partial\tau})),du(e_{\alpha})\rangle,$ (3.1)

We can also deduce Bochner type formula for the backward $VT$-harmonic map
heat flow,

$\displaystyle\frac{1}{2}(\triangle+\frac{\partial}{\partial\tau})|du|^{2}=$
$\displaystyle|\nabla du|^{2}-\langle
R^{N}(du(e_{\alpha}),du(e_{\beta}))du(e_{\alpha}),du(e_{\beta})\rangle$
$\displaystyle+\langle
du(Ric^{M}(e_{\alpha})),du(e_{\alpha})\rangle+\langle\nabla_{e_{\alpha}}(\tau(u)),du(e_{\alpha})\rangle$
$\displaystyle-\langle
du(h(e_{\alpha})),du(e_{\alpha})\rangle+\langle\nabla_{e_{\alpha}}(\frac{\partial
u}{\partial\tau})),du(e_{\alpha})\rangle.$

However,

$\displaystyle\langle\nabla_{e_{\alpha}}(\tau(u)),du(e_{\alpha})\rangle+\langle\nabla_{e_{\alpha}}(\frac{\partial
u}{\partial\tau})),du(e_{\alpha})\rangle$ (3.2)
$\displaystyle=-\langle\nabla_{e_{\alpha}}(du(V)+Tr_{g}T(du,du),du(e_{\alpha})\rangle$
$\displaystyle=-\frac{1}{2}V|du|^{2}-\frac{1}{2}\langle
du(L_{V}g(e_{i})),du(e_{i})\rangle-\langle\nabla_{e_{\alpha}}Tr_{g}T(du,du),du(e_{\alpha})\rangle,$

and

$\displaystyle\langle\nabla_{e_{\alpha}}Tr_{g}T(du,du),du(e_{\alpha})\rangle=\sum_{\alpha,\beta=1}^{m}\left\langle\left(\nabla_{e_{\alpha}}T\right)\left(du\left(e_{\beta}\right),du\left(e_{\beta}\right)\right),du\left(e_{\alpha}\right)\right\rangle$
$\displaystyle-\sum_{\alpha,\beta=1}^{m}\left\langle
2T\left(\left(\nabla_{e_{\alpha}}du\right)\left(e_{\beta}\right),du\left(e_{\beta}\right)\right),du\left(e_{\alpha}\right)\right\rangle,$

Taking the above formula into count, we have

$\displaystyle\frac{1}{2}(\triangle_{V}+\frac{\partial}{\partial\tau}|du|^{2}$
$\displaystyle=|\nabla du|^{2}-\langle
R^{N}(du(e_{\alpha}),du(e_{\beta}))du(e_{\alpha}),du(e_{\beta})\rangle$
$\displaystyle+\langle
du((Ric_{V}^{M}-h_{0})(e_{\alpha})),du(e_{\alpha})\rangle-\sum_{\alpha,\beta=1}^{m}\left\langle\left(\nabla_{e_{\alpha}}T\right)\left(du\left(e_{\beta}\right),du\left(e_{\beta}\right)\right),du\left(e_{\alpha}\right)\right\rangle$
$\displaystyle+\sum_{\alpha,\beta=1}^{m}\left\langle
2T\left(\left(\nabla_{e_{\alpha}}du\right)\left(e_{\beta}\right),du\left(e_{\beta}\right)\right),du\left(e_{\alpha}\right)\right\rangle,$

By the formula in [5],

$\displaystyle\begin{cases}\sum_{\alpha,\beta}R^{N}\left(du\left(e_{\alpha}\right),du\left(e_{\beta}\right),du\left(e_{\alpha}\right),du\left(e_{\beta}\right)\right)\leq\frac{s_{0}-1}{s_{0}}\kappa|du|^{4},\\\
\left|\left\langle\left(\nabla_{e_{\alpha}}T\right)\left(du\left(e_{\beta}\right),du\left(e_{\beta}\right)\right),du\left(e_{\alpha}\right)\right\rangle\right|\leq\varepsilon_{1}e(u)+\frac{1}{4\varepsilon_{1}}\|\nabla
T\|_{L^{\infty}}^{2}e(u)^{2},\\\ \\\ \left|\left\langle
2T\left(\left(\nabla_{e_{\alpha}}du\right)\left(e_{\beta}\right),du\left(e_{\beta}\right)\right),du\left(e_{\alpha}\right)\right\rangle\right|\leq\varepsilon_{2}|\nabla
du|^{2}+\frac{1}{\varepsilon_{2}}\|T\|_{L^{\infty}}^{2}e(u)^{2}.\end{cases}$
(3.3)

Here $\epsilon_{1},\epsilon_{2}$ are suitable small positive constant,
${s_{0}:=\min\\{m,n\\}}$ .

Using the assumption $Ric_{V}-h_{0}\geq-K$ and the above estimates, we have

$\displaystyle\frac{1}{2}(\Delta_{V}+\frac{\partial}{\partial\tau})|du|^{2}$
(3.4) $\displaystyle\geq(1-\epsilon_{2})|\nabla
du|^{2}-(K+\epsilon_{1})e(u)-(\frac{s_{0}-1}{s_{0}}\kappa+\frac{1}{4\varepsilon_{1}}\|\nabla
T\|_{L^{\infty}}^{2}+\frac{1}{\varepsilon_{2}}\|T\|_{L^{\infty}}^{2})e(u)^{2}.$

Let the function $f$ be the function in Definition 2 and
$\omega=\frac{|du|^{2}}{f^{2}}$, a routine computation will give

$\displaystyle\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)\omega=$
$\displaystyle\frac{\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)|du|^{2}}{f^{2}}-2\frac{\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)f(u(x,\tau))|du|^{2}}{f^{3}}$
(3.5) $\displaystyle-4\frac{\nabla f\nabla|du|^{2}}{f^{3}}+6\frac{|\nabla
f|^{2}|du|^{2}}{f^{4}}.$

Computing directly , one has

$\displaystyle\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)f(u(x,t))=\nabla^{2}(f)(du,du)-\langle
Tr_{g}T(du,du),\nabla f\rangle.$ (3.6)

Substituting (3.6) and (3.4) into (3.5), we may have

$\displaystyle\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)\omega$
$\displaystyle\geq$ $\displaystyle-2(K+\epsilon_{1})\frac{{|\nabla
u|^{2}}}{f^{2}}+2(1-\epsilon_{2})\frac{|\nabla du|^{2}}{f^{2}}$
$\displaystyle-2\dfrac{(\frac{s_{0}-1}{s_{0}}\kappa+\frac{1}{4\varepsilon_{1}}\|\nabla
T\|_{L^{\infty}}^{2}+\frac{1}{\varepsilon_{2}}\|T\|_{L^{\infty}}^{2})e(u)^{2}}{f^{2}}-2\frac{\left(\Delta+\frac{\partial}{\partial\tau}\right)f(u(x,\tau))|du|^{2}}{f^{3}}$
$\displaystyle-2\frac{\nabla f\cdot\nabla|du|^{2}}{f^{3}}+2\frac{|\nabla
f|^{2}|du|^{2}}{f^{4}}-2\nabla\omega\cdot\frac{\nabla f}{f}.$

But, since the domain $\Omega$ satisfies conditon (B),

$\displaystyle-2\dfrac{(\frac{s_{0}-1}{s_{0}}\kappa+\frac{1}{4\varepsilon_{1}}\|\nabla
T\|_{L^{\infty}}^{2}+\frac{1}{\varepsilon_{2}}\|T\|_{L^{\infty}}^{2})e(u)^{2}}{f^{2}}-2\frac{\left(\Delta+\frac{\partial}{\partial\tau}\right)f(u(x,\tau))|du|^{2}}{f^{3}}+2Q\frac{|du|^{4}}{f^{3}}$
$\displaystyle\geq\frac{\langle Tr_{g}T(du,du),\nabla
f\rangle}{f^{3}}+2Q\frac{|du|^{4}}{f^{3}}$
$\displaystyle\geq-\|T\|_{L^{\infty}}\frac{\|\nabla
f\|\|du\|^{4}}{f^{3}}+2Q\frac{|du|^{4}}{f^{3}}.$

The Hölder’s inequality implies

${4\frac{|\nabla du||du||\nabla f|}{f^{3}}\leq 4\epsilon_{3}\frac{|\nabla
du|^{2}}{f^{2}}+\frac{1}{4\epsilon_{3}}\frac{|\nabla f|^{2}|du|^{2}}{f^{4}},}$

and it is trivial to see

${\left.|\nabla|du\right|^{2}|\leq 2|\nabla du||du|}$

Taking $2\epsilon_{3}=1-\epsilon_{2}$ and substituting the last two
inequalities into (3), we have

$\left(\Delta+\frac{\partial}{\partial\tau}\right)\omega\geq
C_{1}\omega^{2}-2\nabla\omega\cdot\frac{\nabla f}{f}-C_{2}\omega$

where
$C_{1}=2Qm_{1}-\|T\|_{L^{\infty}}m_{3}m_{1},C_{2}=2(K+\epsilon_{1})-\frac{8\epsilon_{3}-1}{4\epsilon_{3}}(\frac{m_{3}}{m_{1}})^{2}.$

We choose the function $\psi$ which is defined in Lemma 3, then we get

$\displaystyle\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)(\psi\omega)-2\frac{\langle\nabla(\psi\omega),\nabla\psi\rangle}{\psi}+\frac{\langle\nabla(\psi\omega),\nabla
f\rangle}{f}$ $\displaystyle=$
$\displaystyle\psi\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)(\omega)+\omega\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)(\psi)-2|\nabla\psi|^{2}\frac{\omega}{\psi}+\langle\psi\nabla\omega,\nabla\operatorname{log}f\rangle+\langle\psi\nabla\omega,\nabla\operatorname{log}f\rangle$
$\displaystyle\geq$ $\displaystyle
C_{1}\psi\omega^{2}-2\psi\nabla\omega\cdot\frac{\nabla
f}{f}-C_{2}\psi\omega+\omega\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)\psi-2|\nabla\psi|^{2}\frac{\omega}{\psi}$
$\displaystyle+2\langle\psi\nabla\omega,\nabla\operatorname{log}f\rangle+2\langle\omega\nabla\psi,\nabla\operatorname{log}f\rangle$
$\displaystyle=$ $\displaystyle
C_{1}\psi\omega^{2}-C_{2}\psi\omega+\omega\left(\Delta_{V}+\frac{\partial}{\partial\tau}\right)\psi-2\frac{|\nabla\psi|^{2}}{\psi}\omega+2\langle\omega\nabla\psi,\nabla\operatorname{log}f\rangle.$
(3.7)

Now we can estimate the last three terms on the right hand of (3). By the
estimates in [13, 14], we have

$\displaystyle\omega\left(\Delta+\frac{\partial}{\partial\tau}\right)(\psi)\leq
4\varepsilon\psi\omega^{2}+\frac{C_{3/4}^{2}}{\varepsilon}\left(m^{2}+\frac{9}{4}\right)\frac{1}{R^{4}}+\frac{D^{2}}{4\varepsilon}\frac{1}{\Lambda^{2}}+\frac{C_{3/4}^{2}}{4\varepsilon}K^{2}.$
(3.8)

Here and in the sequel $\epsilon$ denotes a small positive constant. This
formula is derived using Lemma 1 and Lemma 2. One can refer to [13, 14] for
details.

Next, by Young’s inequality, we get

$\displaystyle\omega\langle V,d\psi\rangle=\omega\sqrt{\psi}\langle
V,\frac{d\psi}{\sqrt{\psi}}\rangle\leq\epsilon\psi
w^{2}+\frac{1}{4\epsilon}\|V\|_{\infty}\frac{|d\psi|^{2}}{\psi}$
$\displaystyle\leq\epsilon\psi
w^{2}+\frac{1}{4\epsilon}\|V\|_{\infty}C^{2}_{1/2}\frac{1}{R^{2}}$

In addition,

$\displaystyle\frac{2w|\nabla\psi|^{2}}{\psi}\leq\varepsilon\psi
w^{2}+\frac{|\nabla\psi|^{4}}{\varepsilon\psi^{3}}\leq\varepsilon\psi
w^{2}+\frac{9C_{3/4}^{4}}{\varepsilon}\frac{1}{R^{4}},$ (3.9)

and

$\displaystyle-2\frac{wg\left(\nabla\psi,\nabla\left(f\circ
u\right)\right)}{f\circ u}$
$\displaystyle\leq\frac{2w|\nabla\psi|\left|\nabla\left(f\circ
u\right)\right|}{f\circ u}\leq 2m_{3}w^{3/2}|\nabla\psi|$ $\displaystyle\leq$
$\displaystyle m_{3}^{2}\psi
w^{2}+\frac{27}{16}\frac{1}{m_{3}^{2}}\frac{|\nabla\psi|^{4}}{\psi^{3}}\leq
m_{3}^{2}\psi
w^{2}+\frac{243C_{3/4}^{4}}{16}\frac{1}{m_{3}^{2}}\frac{1}{R^{4}},$ (3.10)

where we have used the bound $f\leq m_{3}$ and the definition of $\omega$, the
constants $C_{3/4},D$ are the same as that in Lemma 3.

$\displaystyle C_{1}\psi\omega^{2}-C_{2}\psi\omega\leq$
$\displaystyle\left(\Delta+\frac{\partial}{\partial\tau}\right)(\psi\omega)-2\frac{\langle\nabla(\psi\omega),\nabla\psi\rangle}{\psi}+\frac{\langle\nabla(\psi\omega),\nabla
f\rangle}{f}$ (3.11)
$\displaystyle+\left(4\varepsilon\psi\omega^{2}+\frac{C_{3/4}^{2}}{\varepsilon}\left(m^{2}+\frac{9}{4}\right)\frac{1}{R^{4}}+\frac{D^{2}}{4\varepsilon}\frac{1}{\Lambda^{2}}+\frac{C_{3/4}^{2}}{4\varepsilon}K^{2}.\right)$
$\displaystyle+\epsilon\psi
w^{2}+\frac{1}{4\epsilon}\|V\|_{\infty}C^{2}_{1/2}\frac{1}{R^{2}}$
$\displaystyle+\left(\varepsilon\psi
w^{2}+\frac{9C_{3/4}^{4}}{\varepsilon}\frac{1}{R^{4}}\right)+\left(m_{3}^{2}\psi
w^{2}+\frac{243C_{3/4}^{4}}{16}\frac{1}{m_{3}^{2}}\frac{1}{R^{4}}\right),$

where the constants $C_{3/4},D$ are as above.

We can suppose the reduced distance is smooth at maximal point
($\bar{x},\bar{\tau}$) of $\psi\omega$, thus at the point
($\bar{x},\bar{\tau}$), we have

$\displaystyle\Delta(\psi\omega)\leq 0,\partial_{\tau}(\psi\omega)\leq
0,\nabla(\psi\omega)=0,$ (3.12)

Hence,

$\displaystyle\left(C_{1}-6\varepsilon-
m_{3}^{2}\right)\psi^{2}\omega^{2}-C_{2}\psi\omega$ (3.13) $\displaystyle\leq$
$\displaystyle\left(\frac{C_{3/4}^{2}}{\varepsilon}\left(m^{2}+\frac{9}{4}\right)\frac{1}{R^{4}}+\frac{D^{2}}{4\varepsilon}\frac{1}{T^{2}}+\frac{C_{3/4}^{2}}{4\varepsilon}K^{2}.\right)$
$\displaystyle+\left(\frac{9C_{3/4}^{4}}{\varepsilon}\frac{1}{R^{4}}\right)+\frac{243C_{3/4}^{4}}{16}\frac{1}{m_{3}^{2}}\frac{1}{R^{4}}+\frac{1}{4\epsilon}\|V\|_{\infty}C^{2}_{1/2}\frac{1}{R^{2}},$

where
$C_{1}=2Qm_{1}-\|T\|_{L^{\infty}}m_{3}m_{1},C_{2}=2(K+\epsilon_{1})-\frac{8\epsilon_{3}-1}{4\epsilon_{3}}(\frac{m_{3}}{m_{1}})^{2},\varepsilon_{3}=4(1-\varepsilon_{2})$,
the $\epsilon,\epsilon_{1}$ is same that in (3.3),(3.8),(3.9),(3). It is easy
to see that we can choose special $\epsilon$ such that $C_{1}-5\varepsilon-
m_{3}^{2}>0$,

Let $Q_{R,\Lambda}:=\\{(x,\tau)|\mathfrak{d}(x,\tau)\leq R\\}$. Since
$\psi=1,$ on $Q_{R/2,\Lambda/4,\theta}:=\\{(x,\tau)\in
Q_{R/4,\Lambda/2}|\tau\in[\theta,\Lambda/4]\\}\\}$, The quadratic formula
immplies that

$\displaystyle\omega(x,\tau)\leq$
$\displaystyle\frac{C_{2}}{C_{1}-5\varepsilon-m_{3}^{2}}$ (3.14)
$\displaystyle+\frac{1}{C_{1}-5\varepsilon-
m_{3}^{2}}\bigg{(}\frac{C_{3/4}^{2}}{\varepsilon}\left(m^{2}+\frac{9}{4}\right)\frac{1}{R^{4}}+\frac{D^{2}}{4\varepsilon}\frac{1}{\Lambda^{2}}+\frac{C_{3/4}^{2}}{4\varepsilon}K^{2}$
$\displaystyle+\frac{9C_{3/4}^{4}}{\varepsilon}\frac{1}{R^{4}}+\frac{243C_{3/4}^{4}}{16}\frac{1}{m_{3}^{2}}\frac{1}{R^{4}}+\frac{1}{4\epsilon}\|V\|_{\infty}C^{2}_{1/2}\frac{1}{R^{2}}\bigg{)}^{\frac{1}{2}}$

Letting $\theta\to 0$, the proof is complete.

##  4 Proof of Theorem 2

In this section, some constants are different from the previous section, e.g,
$C_{1},C_{2},\varepsilon_{1},$ etc. We hope that this the readers will not
find confusing.

###### Proof.

Let $\omega(x,t)=\frac{|\nabla u(x,t)|^{2}}{f^{2}(u(x,t))}.$ The first step is
to estimate $\left(\Delta_{V}-\frac{\partial}{\partial t}\right)\omega$. A
calculation shows that

$\displaystyle\nabla\omega=\frac{\nabla|\nabla u|^{2}}{f^{2}}-2\frac{\nabla
f|\nabla u|^{2}}{f^{3}},$ (4.1) $\displaystyle\Delta\omega=\frac{\Delta|\nabla
u|^{2}}{f^{2}}-4\frac{\nabla f\nabla|\nabla u|^{2}}{f^{3}}-2\frac{\Delta
f|\nabla u|^{2}}{f^{3}}+6\frac{|\nabla f|^{2}|\nabla u|^{2}}{f^{4}},$ (4.2)

and

$\frac{\partial\omega}{\partial t}=\frac{\frac{\partial}{\partial t}|\nabla
u|^{2}}{f^{2}}-2\frac{\frac{\partial f}{\partial t}|\nabla u|^{2}}{f^{3}}.$

Combing all together gives

$\displaystyle\left(\Delta-\frac{\partial}{\partial t}\right)\omega=$
$\displaystyle\frac{\left(\Delta-\frac{\partial}{\partial t}\right)|\nabla
u|^{2}}{f^{2}}-2\frac{\left(\Delta-\frac{\partial}{\partial
t}\right)f(u(x,t))|\nabla u|^{2}}{f^{3}}$ (4.3) $\displaystyle-4\nabla
f\frac{\nabla|\nabla u|^{2}}{f^{3}}+6\frac{|\nabla f|^{2}|\nabla
u|^{2}}{f^{4}},$

We recall Weitzenböck type formula for $VT$ harmonic map given in [5],

$\displaystyle\frac{1}{2}(\Delta_{V}-\frac{\partial}{\partial t})|du|^{2}=$
$\displaystyle|\nabla du|^{2}+\sum_{\alpha=1}^{m}\left\langle
du\left(\operatorname{Ric}_{V}\left(e_{\alpha}\right)\right),du\left(e_{\alpha}\right)\right\rangle$
$\displaystyle-\sum_{\alpha,\beta=1}^{m}R^{N}\left(du\left(e_{\alpha}\right),du\left(e_{\beta}\right),du\left(e_{\alpha}\right),du\left(e_{\beta}\right)\right)$
$\displaystyle-\sum_{\alpha,\beta=1}^{m}\left\langle\left(\nabla_{e_{\alpha}}T\right)\left(du\left(e_{\beta}\right),du\left(e_{\beta}\right)\right),du\left(e_{\alpha}\right)\right\rangle$
$\displaystyle-\sum_{\alpha,\beta=1}^{m}\left\langle
2T\left(\left(\nabla_{e_{\alpha}}du\right)\left(e_{\beta}\right),du\left(e_{\beta}\right)\right),du\left(e_{\alpha}\right)\right\rangle,$

where $e_{1},e_{2},\cdots,e_{m}$ is a focal orthonormal frame field of the
domain manifold.

Noticing that $Ric_{V}\geq-A$ and (3.3), we further have

$\displaystyle\frac{1}{2}(\Delta_{V}-\frac{\partial}{\partial t})|du|^{2}$
(4.4) $\displaystyle\geq(1-\epsilon_{2})|\nabla
du|^{2}-(A+\epsilon_{1})e(u)-(\frac{s_{0}-1}{s_{0}}\kappa+\frac{1}{4\varepsilon_{1}}\|\nabla
T\|_{L^{\infty}}^{2}+\frac{1}{\varepsilon_{2}}\|T\|_{L^{\infty}}^{2})e(u)^{2},$

where the constants $s_{0},\epsilon_{1},\epsilon_{2}$ are the same as that in
(3.3). Plugging the above formula into (4.3), we get

$\displaystyle\left(\Delta_{V}-\frac{\partial}{\partial t}\right)\omega\geq$
$\displaystyle-2(A+\epsilon_{1})\frac{{|\nabla
u|^{2}}}{f^{2}}+2(1-\epsilon_{2})\frac{|\nabla du|^{2}}{f^{2}}$ (4.5)
$\displaystyle-2\dfrac{(\frac{s_{0}-1}{s_{0}}\kappa+\frac{1}{4\varepsilon_{1}}\|\nabla
T\|_{L^{\infty}}^{2}+\frac{1}{\varepsilon_{2}}\|T\|_{L^{\infty}}^{2})e(u)^{2}}{f^{2}}-2\frac{\left(\Delta-\frac{\partial}{\partial
t}\right)f(u(x,t))|\nabla u|^{2}}{f^{3}}$ $\displaystyle-2\frac{\nabla
f\cdot\nabla|\nabla u|^{2}}{f^{3}}+2\frac{|\nabla f|^{2}|\nabla
u|^{2}}{f^{4}}-2\nabla\omega\cdot\frac{\nabla f}{f}.$

The chain rule gives

$\left(\Delta_{V}-\frac{\partial}{\partial
t}\right)f(u(x,t))=\nabla^{2}f(\nabla u,\nabla u)+\langle
Tr_{g}T(du,du),\nabla f\rangle.$

Since ${\Omega}$ satisfies condition (B),

$\displaystyle-2\dfrac{(\frac{s_{0}-1}{s_{0}}\kappa+\frac{1}{4\varepsilon_{1}}\|\nabla
T\|_{L^{\infty}}^{2}+\frac{1}{\varepsilon_{2}}\|T\|_{L^{\infty}}^{2})e(u)^{2}}{f^{2}}-2\frac{\left(\Delta-\frac{\partial}{\partial
t}\right)f(u(x,t))|\nabla u|^{2}}{f^{3}}$ $\displaystyle\geq-\frac{\langle
Tr_{g}T(du,du),\nabla f\rangle}{f^{3}}++2Q\frac{|du|^{4}}{f^{3}}$
$\displaystyle\geq-\|T\|_{L^{\infty}}\frac{\|\nabla
f\|\|du\|^{4}}{f^{3}}+2Q\frac{|du|^{4}}{f^{3}}.$

By Hölder’s inequality, we deduce that

${4\epsilon_{3}\frac{|\nabla
du|^{2}}{f^{2}}+\frac{1}{4\epsilon_{3}}\frac{|\nabla f|^{2}|\nabla
u|^{2}}{f^{4}}\geq 4\frac{|\nabla du||\nabla u||\nabla f|}{f^{3}},}$

and

${\left.|\nabla|\nabla u\right|^{2}|\leq 2|\nabla du||\nabla u|,}$

It follows that

$\displaystyle\left(\Delta_{V}-\frac{\partial}{\partial t}\right)\omega\geq$
$\displaystyle-2(A+\epsilon_{1})\frac{{|\nabla
u|^{2}}}{f^{2}}+2(1-\epsilon_{2})\frac{|\nabla du|^{2}}{f^{2}}$
$\displaystyle-\|T\|_{L^{\infty}}\frac{\|\nabla
f\|\|du\|^{4}}{f^{3}}+2Q\frac{|du|^{4}}{f^{3}}$
$\displaystyle-4\epsilon_{3}\frac{|\nabla
du|^{2}}{f^{2}}-\frac{1}{4\epsilon_{3}}\frac{|\nabla f|^{2}|\nabla
u|^{2}}{f^{4}}+2\frac{|\nabla f|^{2}|\nabla
u|^{2}}{f^{4}}-2\nabla\omega\cdot\frac{\nabla f}{f}.$

Taking $2\epsilon_{3}=1-\epsilon_{2}$ and using the bound of $f$, we can get
the following estimates,

${\left(\Delta_{V}-\frac{\partial}{\partial t}\right)\omega\geq
K_{2}\omega^{2}-2\nabla\omega\cdot\frac{\nabla f}{f}-K_{1}\omega,}$

where
$K_{1}=2(A+\epsilon_{1})-\frac{3-4\epsilon_{2}}{2(1-\epsilon_{2})}(\frac{m_{3}}{m_{1}})^{2},K_{2}=2Qm_{1}-\|T\|_{L^{\infty}}{m_{3}}{m_{1}}.$
Henceforth,

$\displaystyle\left(\Delta_{V}-\frac{\partial}{\partial t}\right)F\geq
K_{2}\frac{F^{2}}{t}-2\nabla F\cdot\frac{\nabla
f}{f}-\left(K_{1}+\frac{1}{t}\right)F,$ (4.6)

which is the key formula to derive Theorem 2.

$\displaystyle\psi(r)=\left\\{\begin{array}[]{ll}1,\quad if\quad
r\in[0,1/2]\\\ 0,\quad if\quad r\in(1,\infty)\\\ 0\leq\psi(r)\leq 1,\\\
\psi{{}^{\prime}}(r)\leq 0,\\\ \psi{{}^{\prime\prime}}(r)\geq-C_{0}\\\
{\frac{\left|\psi{{}^{\prime}}(r)\right|^{2}}{\psi(r)}\leq
C_{0}},\end{array}\right.$ (4.7)

where ${C_{0}}$ is an absolute constant.

Let ${\lambda(x)=\psi\left(\frac{r(x)}{R}\right)}$, let ${F(x,t)=}$
${t\omega(x,t)}$. Assume that ${\left(x_{1},t_{1}\right)}$ is the point where
${\lambda F}$ achieves its maxisum in
${B_{R}\left(x_{o}\right)\times[0,\Lambda]\left(0<\Lambda<T_{1}\right)}$. It
is well known that we can assume ${\lambda(x)}$ to be smooth at ${x_{1}}$. And
we may also assume ${(\lambda F)\left(x_{1},t_{1}\right)>0}$. At
${\left(x_{1},t_{1}\right)}$, we have

$\displaystyle\nabla(\lambda F)=0,\frac{\partial}{\partial t}(\lambda F)\geq
0,\Delta(\lambda F)\leq 0.$ (4.8)

Hence at ${\left(x_{1},t_{1}\right)}$,

$\displaystyle\left(\Delta_{V}-\frac{\partial}{\partial t}\right)(\lambda
F)\leq 0.$ (4.9)

As ${\operatorname{Ric}^{M}-\frac{1}{2}L_{V}g\geq-A}$, by the ${V}$-Laplacian
comparison Theorem (cf. [4, Theroem 3]), we have

${\Delta_{V}r\leq\sqrt{(m-1)A}\operatorname{coth}\sqrt{\frac{A}{m-1}}r+v(r).}$

Here $v(\cdot)$ is the function defined in Theorem 2.

Noticing that $kr\coth kr\leq 1+kr$, there exists a constant ${C_{1}>0}$
depending on ${A}$ such that

${r\Delta_{V}r\leq(v(a)+C_{1})r+m-1.}$

where $C_{1}=\sqrt{(m-1)A}.$ It is clear that

$\displaystyle\nabla\lambda=\psi^{\prime}\frac{r}{R}$

Noticing that $\psi^{\prime}\leq 0$, we deduce that for ${x\in B_{R}(x_{0})}$,

$\displaystyle\Delta_{V}\lambda=$
$\displaystyle\psi^{\prime\prime}\left(\frac{1}{R}\right)^{2}\left(r^{\prime}\right)^{2}+\psi^{\prime}\frac{1}{R}\Delta_{V}r$
$\displaystyle\geq\frac{-C_{0}}{R^{2}}+\frac{\psi^{\prime}}{R}\left(v(a)+C_{1}+\frac{m-1}{r}\right)$
$\displaystyle=\frac{-C_{0}}{R^{2}}+\frac{\psi^{\prime}}{R}\left(v(a)+C_{1}\right)+\frac{\psi^{\prime}}{R}\frac{m-1}{R}$
$\displaystyle=\frac{\psi^{\prime}(m-1)-C_{0}}{R^{2}}+\frac{\psi^{\prime}}{R}{\left(v(a)+C_{1}\right)}$
$\displaystyle\geq\frac{{-\sqrt{C_{0}}(m-1)-C_{0}}{}}{R^{2}}-\frac{\left(v(a)+C_{1}\right)}{R}\sqrt{C_{0}}.$

By the difinition of $\psi$, We can conclude from the above estimates that

$\displaystyle\frac{|\nabla\lambda|^{2}}{\lambda}\leq\frac{C_{0}}{R^{2}},\,\quad\Delta_{V}\lambda\geq-\frac{C_{2}}{R^{2}}-\frac{C_{3}}{R}.$
(4.10)

where $C_{2}=C_{0}+\sqrt{C_{0}}(m-1),C_{3}=v(a)+C_{1}$.

In the sequel of this section, we continue the proof at the point
$(x_{1},t_{1})$ where (4.8) holds. Then (4.9) and (4.10) gives

${0\geq-\left(\frac{C_{2}}{R^{2}}+\frac{C_{3}}{R}\right)F+2\nabla\lambda\nabla
F+\lambda\left(\Delta_{V}-\frac{\partial}{\partial t}\right)F,}$

According to (4.8) and (4.6) and noticing that $|\nabla f(u(x,t)|\leq
m_{3}|\nabla u|$, we get

$\displaystyle
0\geq-\left(\frac{C_{2}}{R^{2}}+\frac{C_{3}}{R}\right)F-2\frac{|\nabla\lambda|^{2}}{\lambda}F+\lambda\left(\Delta-\frac{\partial}{\partial
t}\right)F$
$\displaystyle\geq-\left(\frac{C_{2}+2C_{0}}{R^{2}}+\frac{C_{3}}{R}\right)F+K_{2}\frac{1}{t_{1}}\lambda
F^{2}-\left(K_{1}+\frac{1}{t_{1}}\right)\lambda
F-m_{3}\frac{|\nabla\lambda|}{\lambda^{1/2}t_{1}^{1/2}}\left(\lambda
F\right)^{1/2}F,$ (4.11)

where we have used the definition of $F$ and $\omega.$

By (4) and the first inequality in (4.10), we have

${0\geq K_{2}\frac{1}{t_{1}}(\lambda
F)^{2}-\left(\frac{C_{2}+2C_{0}}{R^{2}}+\frac{C_{3}}{R}+K_{1}+\frac{1}{t_{1}}\right)\lambda
F-C_{0}^{\frac{1}{2}}m_{3}\frac{1}{R}\left(\frac{\lambda
F}{t_{1}}\right)^{1/2}\lambda F.}$

The quadratic formula immplies that

$\displaystyle\left(\frac{\lambda F}{t_{1}}\right)^{1/2}\leq$
$\displaystyle\frac{C_{0}^{\frac{1}{2}}m_{3}}{K_{2}R}+\sqrt{\frac{1}{K_{2}}\left(\frac{C_{2}+2C_{0}}{R^{2}}+\frac{C_{3}}{R}+K_{1}+\frac{1}{t_{1}}\right)}$
(4.12) $\displaystyle\leq$
$\displaystyle\frac{C_{0}^{\frac{1}{2}}m_{3}}{K_{2}R}+C_{4}\frac{1}{\sqrt{K_{2}}}\left(\sqrt{K_{1}}+\sqrt{\frac{1}{R}}+\frac{1}{R}\right)+\frac{1}{\sqrt{t_{1}}}\frac{1}{\sqrt{K_{2}}}$

where $C_{4}=\operatorname{max}(C_{2}+2C_{0},C_{3}).$ Noticing that $0\leq
t_{1}\leq\Lambda$

${\begin{array}[]{c}(\lambda
F)^{1/2}\left(x_{1},t_{1}\right)\leq\frac{C_{0}^{\frac{1}{2}}m_{3}}{K_{2}R}\sqrt{\Lambda}+C_{4}\frac{1}{\sqrt{K_{2}}}\left(\sqrt{K_{1}}+\sqrt{\frac{1}{R}}+\frac{1}{R}\right)\sqrt{\Lambda}+\frac{1}{\sqrt{K_{2}}}.\end{array}}$

So

$\displaystyle\sup\left\\{\left|t^{1/2}\right|\nabla u(x,t)||(x,t)\in
B_{R/2}\left(x_{0}\right)\times[0,\Lambda]\right\\}$ $\displaystyle\leq$
$\displaystyle
m_{2}\left(\frac{C_{0}^{\frac{1}{2}}m_{3}}{K_{2}R}\sqrt{\Lambda}+C_{4}\frac{1}{\sqrt{K_{2}}}\left(\sqrt{K_{1}}+\sqrt{\frac{1}{R}}+\frac{1}{R}\right)\sqrt{\Lambda}+\frac{1}{\sqrt{K_{2}}}\right)$

Hence

${\sup_{B_{R/2}(x_{0})}\Lambda^{1/2}|\nabla u(x,t)|\leq
m_{2}\left(\frac{C_{0}^{\frac{1}{2}}m_{3}}{K_{2}R}\sqrt{\Lambda}+C_{4}\frac{1}{\sqrt{K_{2}}}\left(\sqrt{K_{1}}+\sqrt{\frac{1}{R}}+\frac{1}{R}\right)\sqrt{\Lambda}+\frac{1}{\sqrt{K_{2}}}\right).}$

This proves (1.7).

To prove (1.8), we set ${F(x,t)=\omega(x,t)}$. If $\lambda F$ achieves its
maximum in ${B_{R/2}\left(x_{0}\right)\times[0,\Lambda]}$ for
${0<\Lambda<T_{1}}$ at ${\left(x_{1},0\right)}$, then we have

$\displaystyle{\sup_{B_{R/2}\left(x_{0}\right)}|\nabla
u(x,t)|\leq\frac{m_{2}}{m_{1}}\sup_{B_{R}\left(x_{0}\right)}|\nabla u(x,0)|}$
(4.13)

If $gF$ achieves its maximum at
${\left(x_{1},t_{1}\right)\left(t_{1}>0\right)}$, then at
${\left(x_{1},t_{1}\right)}$,

$\displaystyle\nabla(\lambda\omega)=0,\frac{\partial}{\partial
t}(\lambda\omega)\geq 0,\Delta(\lambda\omega)\leq 0.$ (4.14)

Thus, we get

$\displaystyle 0\geq$
$\displaystyle-\left(\frac{C_{2}}{R^{2}}+\frac{C_{3}}{R}\right)\omega-2\frac{|\nabla\lambda|^{2}}{\lambda}\omega+\lambda\left(\Delta-\frac{\partial}{\partial
t}\right)\omega$ $\displaystyle\geq$
$\displaystyle-\left(\frac{C_{2}}{R^{2}}+\frac{C_{3}}{R}\right)\omega-2\frac{|\nabla\lambda|^{2}}{\lambda}\omega+\lambda\left(K_{2}\omega^{2}-2\nabla\omega\cdot\frac{\nabla
f}{f}-K_{1}\omega\right)$ $\displaystyle\geq$ $\displaystyle
K_{2}\lambda\omega^{2}-\left(\frac{C_{2}}{R^{2}}+\frac{C_{3}}{R}\right)\omega-2\frac{|\nabla\lambda|^{2}}{\lambda}\omega-2\nabla\lambda\cdot\frac{\nabla
f}{f}\omega-K_{1}\lambda\omega$ $\displaystyle\geq$ $\displaystyle
K_{2}\lambda\omega^{2}-\left(\frac{C_{2}+2C_{0}}{R^{2}}+\frac{C_{3}}{R}\right)\omega-\frac{2m\sqrt{C_{0}}}{R}\sqrt{\lambda\omega}-K_{1}\lambda\omega$
$\displaystyle\geq$ $\displaystyle
K_{2}g\omega^{2}-\left(\frac{C_{2}}{R^{2}}+\frac{C_{3}}{R}\right)\omega-2\frac{|\nabla\lambda|^{2}}{\lambda}\omega-2\nabla\lambda\cdot\frac{\nabla
f}{f}\omega-K_{1}\lambda\omega$ $\displaystyle\geq$ $\displaystyle
K_{2}\lambda\omega^{2}-\left(K_{1}+\frac{C_{2}+2C_{0}}{R^{2}}+\frac{C_{3}}{R}\right)\omega-\frac{2m\sqrt{C_{0}}}{R}\sqrt{\lambda\omega}\omega.$

By the quadratic formula, one obtains

$\displaystyle\sqrt{\lambda\omega}(x_{1},t_{1})\leq\frac{\frac{2m\sqrt{C_{0}}}{R}+\sqrt{\frac{2m\sqrt{C_{0}}}{R^{2}}+4K_{2}\left(K_{1}+\frac{C_{2}+2C_{0}}{R^{2}}+\frac{C_{3}}{R}\right)}}{2K_{2}},$
(4.15)

Using the definiton of $\omega$ and $\lambda$,

$\displaystyle{\sup_{B_{R/2}\left(x_{o}\right)}|\nabla u(x,t)|\leq
m_{2}\left(\frac{\frac{2m\sqrt{C_{0}}}{R}+\sqrt{\frac{2m\sqrt{C_{0}}}{R^{2}}+4K_{2}\left(K_{1}+\frac{C_{2}+2C_{0}}{R^{2}}+\frac{C_{3}}{R}\right)}}{2K_{2}}\right).}$
(4.16)

Then, (1.8) follows from (4.16) and (4.13).

∎

## References

  * [1] Abolarinwa, A., and Azami, S. Variation of the first eigenvalue of Witten Laplacian and consequences under super Perelman-Ricci flow.  J. Elliptic Parabol. Equ. 9, 1 (2023), 231–245.
  * [2] Cao, X. Liouville theorem of vt harmonic map from complete noncompact manifold into horoball.  arXiv:2303.07686 (2023).
  * [3] Chen, Q., Jost, J., Li, J., and Wang, G. Dirac-harmonic maps.  Math. Z. 254, 2 (2006), 409–432.
  * [4] Chen, Q., Jost, J., and Qiu, H. Existence and Liouville theorems for $V$-harmonic maps from complete manifolds.  Ann. Global Anal. Geom. 42, 4 (2012), 565–584.
  * [5] Chen, Q., Jost, J., and Qiu, H. On $VT$-harmonic maps.  Ann. Global Anal. Geom. 57, 1 (2020), 71–94.
  * [6] Dung, H. T., Manh, N. T., and Tuyen, N. D. Liouville type theorems and gradient estimates for nonlinear heat equations along ancient $K$-super Ricci flow via reduced geometry.  J. Math. Anal. Appl. 519, 2 (2023), Paper No. 126836, 17.
  * [7] Erbar, M., and Kopfer, E. Super Ricci flows for weighted graphs.  J. Funct. Anal. 279, 6 (2020), 108607, 51.
  * [8] Guo, H., Philipowski, R., and Thalmaier, A. An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions.  Potential Anal. 42, 2 (2015), 483–497.
  * [9] Jost, J., and Şimşir, F. M. Affine harmonic maps.  Analysis (Munich) 29, 2 (2009), 185–197.
  * [10] Jost, J., and Yau, S.-T. A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry.  Acta Math. 170, 2 (1993), 221–254.
  * [11] Kopfer, E. Super-Ricci flows and improved gradient and transport estimates.  Probab. Theory Related Fields 175, 3-4 (2019), 897–936.
  * [12] Kopfer, E., and Sturm, K.-T. Heat flow on time-dependent metric measure spaces and super-Ricci flows.  Comm. Pure Appl. Math. 71, 12 (2018), 2500–2608.
  * [13] Kunikawa, K., and Sakurai, Y. Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry.  J. Geom. Anal. 31, 12 (2021), 11899–11930.
  * [14] Kunikawa, K., and Sakurai, Y. Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry.  Calc. Var. Partial Differential Equations 60, 5 (2021), Paper No. 199, 24.
  * [15] Lakzian, S., and Munn, M. On weak super Ricci flow through neckpinch.  Anal. Geom. Metr. Spaces 9, 1 (2021), 120–159.
  * [16] Li, J. The heat flows and harmonic maps of complete noncompact Riemannian manifolds.  Math. Z. 212, 2 (1993), 161–173.
  * [17] Li, J. Heat flows and harmonic maps with a free boundary.  Math. Z. 217, 3 (1994), 487–495.
  * [18] Li, J. The heat flows and harmonic maps from complete manifolds into generalized regular balls.  preprint (1994).
  * [19] Li, J., and Wang, S. The heat flows and harmonic maps from complete manifolds into regular balls.  Bull. Austral. Math. Soc. 58, 2 (1998), 177–187.
  * [20] Li, S., and Li, X.-D. $W$-entropy, super Perelman Ricci flows, and $(K,m)$-Ricci solitons.  J. Geom. Anal. 30, 3 (2020), 3149–3180.
  * [21] List, B. Evolution of an extended Ricci flow system.  Comm. Anal. Geom. 16, 5 (2008), 1007–1048.
  * [22] McCann, R. J., and Topping, P. M. Ricci flow, entropy and optimal transportation.  Am. J. Math. 132, 3 (2010), 711–730.
  * [23] Müller, R. Ricci flow coupled with harmonic map flow.  Ann. Sci. Éc. Norm. Supér. (4) 45, 1 (2012), 101–142.
  * [24] Sturm, K.-T. Super-Ricci flows for metric measure spaces.  J. Funct. Anal. 275, 12 (2018), 3504–3569.
  * [25] Taheri, A. Gradient Estimates for a Weighted $\Gamma$-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications.  Potential Anal. 59, 1 (2023), 311–335.
  * [26] Wang, M. Liouville theorems for the ancient solution of heat flows.  Proc. Am. Math. Soc. 139, 10 (2011), 3491–3496.