# Heat flow for $VT$ harmonic map from compact manifold

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

In this paper, we obtain the existence of Dirichlet problem for VT harmonic
map from compact Riemannian manifold with or without boundary into compact
manifold via the heat flow method. We also obtain the existence of $VT$
geodesics uncer certain conditions on $T$.

###### Contents

  1. 1 Introduction
  2. 2 Domain manifold is closed manifold
  3. 3 Domain manifold is compact manifold with boundary
  4. 4 Existence and convergencve of heat flow of $VT$ geodeisc

Keywords: Heat flow, $VT$ Harmonic map, $VT$ geodesic.

MSC 2010: 58E15, 58E20

##  1 Introduction

Heat flow method was used in [7] to obtain the existence of harmonic map
requiring that the target manifold has nonpositive sectional curvature.
Hamilton [8] consider the existence problem of harmonic map from manifolds
with boundary via the heat flow method.

Harmonic map with a two form and potential is related to the prescribed mean
curvature equation of surface in $\mathbb{R}^{3}$. Harmonic map with two form
and potential is the natural generalization of harmonic map with potential and
harmonic map with two form . Branding [1] considered the existence of the heat
flow for the harmonic map with a potential and two form from closed surface
under some curvature conditions of target manifold. In [4], we considered the
Dirichlet problem for such kind of map using the method of Hamilton [8] .

When domain manifold is $S^{1}$, Koh [10] obtained the exsitence of the heat
flow of magnetic geodesics. Branding [2] further investigated the covergence
of the heat flow of magnetic geodesics.

In 2020, Chen et al.[6] introduced the notion of $VT$-harmonic map which is
defined as

$\tau(u)+du(V)+\operatorname{Tr}_{g}T(du,du)=0$

where $V\in\Gamma(TM),T\in\Gamma(\otimes^{1,2}(TN)).$ They obtained the
existence of VT harmonic map from compact manifold with boundary into geodesic
ball in Riemannian manifold using heat flow method and maximum principle. In
[5], we improved theorem 3 in [6] using Leray-Schauder degree theory.

In general, $VT$ harmonic map doesn’t neccesarily have variation structure,
which cause much difficulties. At present, there is a gap on the existence of
general VT harmonic map heat flow when domanin manifold is closed manifold.
This is one of our motivations of this paper.

In this paper, we first consider the following problem on closed manifold:

$\displaystyle\left\\{\begin{array}[]{ll}&\frac{\partial u}{\partial
t}=\tau_{V}(u)+\operatorname{Tr}_{g}T(du,du),\quad on\quad
M\times[0,\infty);\\\
&u=u_{0},~{}~{}~{}~{}~{}on~{}~{}~{}~{}~{}~{}M\times\\{0\\}.\end{array}\right.$
(1.3)

Where $u_{0}$ is independent of $t$. We consider the following functional

$\begin{split}E(u)=\int_{M}\left(\frac{1}{2}|du|^{2}+\left(\operatorname{Tr}_{g}u^{*}\Phi\right)\right)f\mathrm{d}v_{g},\end{split}$

where $f\in C^{1}(S^{1}),f>0$ , the tensor $T$ is defined by

$\begin{split}\left\langle
T(Y,Z),X\right\rangle=\left(L_{X}\Phi\right)(Y,Z),T\in\Gamma(\otimes^{1,2}N),\Phi\in\Gamma(\otimes^{1,2}N).\end{split}$
(1.4)

Its critical point is a special case of VT harmonic map. In this case,
$V=\nabla\log f.$

In this paper, our first main result is as follows:

######  Theorem 1.1 (see Theorem 2.4).

Let $M$ be a closed Riemannian manifold , $N$ is a Riemannian manifold. The
tensor $T$ is defined by (1.4). Furthermore, the curvature tensor $R^{N}$ and
$T$ satisfy (2.1). Let $V=\nabla\log f$, $\|\Phi\|_{L^{\infty}}<\frac{1}{2}$.
For a given $C^{2+\alpha}$ map $u_{0}\in C^{2+\alpha}(M,N)$, there exists a
solution $u\in C^{2+\alpha,1+\alpha/2}(M\times[0,+\infty),N)\cap
C^{\infty}(M\times(0,+\infty),N)$ which solves

$\left\\{\begin{array}[]{l}\tau_{V}\left(u\right)(x)=\operatorname{Tr}_{g}T(du,du)+\frac{\partial
u_{t}}{\partial t}(x),(x,t)\in M\times(0,\infty),\\\
u(x,0)=u_{0}(x).\end{array}\right.$

Moreover, $u(x,t)$ subconverges in $C^{2}\left(M,N\right)$ to a VT harmonic
map $u_{\infty}$.

Secondly, when $M$ is compact manifold with boundary, we consider the heat
flow of Neumann problem

$\displaystyle\left\\{\begin{array}[]{l}\frac{\partial u}{\partial
t}=\tau(u)+\operatorname{Tr}_{g}\left(u^{*}T\right),\quad in\quad M\\\
u=u_{0}(\cdot),\partial M\times[0,\infty)\\\ \frac{\partial u}{\partial
n}|_{\partial M}=0\\\ u(\cdot,0)=u_{0}(\cdot)\end{array}\right.$ (1.9)

We get

######  Theorem 1.2 (see Theorem 3.3).

Let $M$ be a compact Riemann manifold with nonempty boundary and $N$ be a
Riemannian manifold. Furthermore, let $T$ be defined by (1.4), the curvature
tensor $R^{N}$ and $T$ satisfy (2.1), $V=\nabla\log f$,
$\|\Phi\|_{L^{\infty}}<\frac{1}{2}$. For a given $C^{2+\alpha}$ map $u_{0}\in
C^{2+\alpha}(M,N)$, there exist a solution $u\in
C^{2+\alpha,1+\alpha/2}(M\times[0,+\infty),N)\cap
C^{\infty}(M\times(0,+\infty),N)$ of (1.9) holds. Moreover, $u(x,t)$
subconverges in $C^{2}\left(M,N\right)$ to VT harmonic map $u_{\infty}$ which
sloves the Neumann boundary problem

$\displaystyle\left\\{\begin{array}[]{l}\tau(u)+\operatorname{Tr}_{g}\left(u^{*}T\right)=0,\quad
in\quad M\\\ u=u_{0}(\cdot),\partial M\times[0,\infty)\\\ \frac{\partial
u}{\partial n}|_{\partial M}=0\\\ u(\cdot,0)=u_{0}(\cdot)\end{array}\right.$
(1.14)

When domain manifold has boundary, the conditions in our theorem and the
method of proof are also different from that in [6] .

When domain manifold is one dimensinal, Ottarsson [13] studied the closed
geodecis on Riemannianfolds via the heat flow. Koh [10] obtained the exsitence
of the heat flow of magnetic geodesics. Branding [2] further investigate the
covergence of the heat flow of magnetic geodesics. One can refer to [10, 2]
and reference therein for more information about magnetic geodesic. Inspired
by Koh [10] and Branding [2], it is natural to investigate the heat flow for
VT geodesic, including existence and convergence .

In this paper, we also consider the following problem:

$\displaystyle\left\\{\begin{array}[]{ll}&\frac{\partial u}{\partial
t}=\tau_{V}(u)+\operatorname{Tr}_{g}T(du,du),\quad on\quad
S^{1}\times[0,\infty);\\\
&u=u_{0},~{}~{}~{}~{}~{}on~{}~{}~{}~{}~{}~{}S^{1}\times\\{0\\}.\end{array}\right.$
(1.17)

where $T\in\Gamma\left(\otimes^{1,2}M\right),\tau_{V}(u)=\tau(u)+du(V).$

When $VT$ harmonic map has variation structure, we get

######  Theorem 1.3 (see Theorem 4.6).

If the tensor $T$ is befined by (1.4), $\|\Phi\|_{L^{\infty}}<\frac{1}{2}$,
the tensor $T$ satisfies the condition (4.2), then the solution $u$ of (1.17)
exists globally and subconverges to a VT geodesic $u_{\infty}$.

######  Remark 1.1.

The curvature condition of N is weaker than that in Theorem 2.4 and Theorem
3.3 in the case when the domain manifold is one dimensional.

This paper is orgalized as follows: in section 2, we use the method of heat
flow to solve (1.3). In section 3, we use the method of Hamilton [8] to solve
(1.9) when the compact target manifold has convex boundary. In section 4, we
solve (1.17) obtained the existence of VT geodesics in the homotopy class of
initial map.

##  2 Domain manifold is closed manifold

######  Theorem 2.1 (Short time existence).

Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds. For a given
$C^{2+\alpha}$ map $u_{0}\in C^{2+\alpha}(M,N)$ there exist a positive number
$T_{max}>0$ and a map $u\in C^{2+\alpha,1+\alpha/2}(M\times[0,T_{max}],N)\cap
C^{\infty}(M\times(0,T_{max}),N)$ such that

$\left\\{\begin{array}[]{l}\tau_{V}\left(u\right)(x)=\operatorname{Tr}_{g}T(du,du)+\frac{\partial
u_{t}}{\partial t}(x),(x,t)\in M\times(0,T_{max}),\\\
u(x,0)=u_{0}(x)\end{array}\right.$

holds. Here, $T_{max}$ is a constant depending only on $N,M,T,f$ and $\alpha$
.

###### Proof.

The term $du(V)$ and $\operatorname{Tr}_{g}T(du,du)$ don’t affect the argument
of contraction map principle in this case . We omit the details for the
shortness of this paper. One can refer to the proof of the short time
existence of harmonic map heat flow from closed manifold (cf. [11, chapter 5]
or [1, Lemma 4.2]) and adjust it. ∎

######  Lemma 2.1.

Let $M$ be a compact Riemannian manifold, N is a Riemannian manifold. Let
$u:M\times[0,T_{max})\rightarrow N$ be a smooth solution of (1.3) or (1.9).
The curvature tensor $R^{N}$ and $T$ satisfy

$\langle
R^{N}(X,Y)X,Y\rangle+C_{0}|T(X^{\sharp},Y)|^{2}-\frac{1}{2}\langle(\nabla
T)(X^{\sharp},Y,Z),X\rangle\leq 0;$ (2.1)

where $X,Y,Z\in\Gamma(TN)$, constant $C_{0}>1$, $X^{\sharp}$ denotes the dual
1-form of $X$. Then for $t\in[0,T_{Max})$, we have

$\displaystyle\frac{1}{2}\bigg{(}\frac{\partial}{\partial
t}-\Delta_{V}\bigg{)}|du|^{2}\leq-(1+\frac{1}{4C_{0}})|\nabla
du|^{2}+C|du|^{2};$ (2.2)

and

$\begin{split}&\frac{1}{2}\bigg{(}\frac{\partial}{\partial
t}-\Delta_{V}\bigg{)}|\frac{du}{dt}|^{2}\leq-(1+\frac{1}{4C_{0}})|\nabla\frac{du}{dt}|^{2}.\end{split}$
(2.3)

Here $C$ is an constant depending only on the bounds on the Ricci curvature of
$M$ and $V$.

###### Proof.

We modify the argument in [4]. By the standard Bochner formula, we have for
all $t\in[0,T_{Max})$

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

Let $\\{e_{\alpha}\\}$ be a local orthonormal frame of $M$ at the considered
point.

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

Therefore, by the proof of Lemma 6.3 in [6], we get

$\displaystyle\frac{1}{2}\triangle_{V}|du|^{2}-\frac{1}{2}\frac{\partial}{\partial
t}|du|^{2}=$ $\displaystyle|\nabla
du|^{2}+\langle\nabla_{e_{\beta}}\big{(}T(du(e_{\alpha})),du(e_{\alpha}))\big{)},du(e_{\beta})\rangle$
$\displaystyle-\langle
R^{N}(du(e_{\alpha}),du(\beta))du(e_{\alpha}),du(e_{\beta})\rangle+\langle
du(Ric_{V}^{M}(e_{\alpha})),du(e_{\alpha})\rangle.$

We choose the normal coordinates at the considered point of $N$,

$\displaystyle-\langle\nabla_{e_{\beta}}\big{(}T(du(e_{\alpha})),du(e_{\alpha}))\big{)},du(e_{\beta})\rangle$
$\displaystyle=\langle\nabla_{e_{\beta}}\big{(}T(u_{\alpha}^{i}\partial_{i}\wedge
u_{\alpha}^{j}\partial_{j}\big{)},u_{\beta}^{r}\partial_{r}\rangle=\langle\nabla_{e_{\beta}}\big{(}T_{ij}^{k}u_{\alpha}^{i}u_{\alpha}^{j}\partial_{k}\big{)},u_{\beta}^{r}\partial_{r}\rangle$
$\displaystyle=-\langle
T_{ij,l}^{k}u_{\beta}^{l}u_{\alpha}^{i}u_{\alpha}^{j}\partial_{k},u_{\beta}^{r}\partial_{r}\rangle-\langle
T_{ij}^{k}u_{\alpha\beta}^{i}u_{\alpha}^{j}\partial_{k},u_{\beta}^{r}\partial_{r}\rangle-\langle
T_{ij}^{k}u_{\alpha}^{i}u_{\alpha\beta}^{j}\partial_{k},u_{\beta}^{r}\partial_{r}\rangle.$
(2.4)

The formula (2.5)(2.6)(2.7) can be found in [3] . Then the Young inequality
implies that

$\begin{split}&-\langle
T_{ij}^{k}u_{\alpha\beta}^{i}u_{\alpha}^{j}\partial_{k},u_{\beta}^{r}\partial_{r}\rangle=-\sum_{i,j,k=1}^{n}T_{ij}^{r}u_{\alpha\beta}^{i}u_{\alpha}^{j}u_{\beta}^{r}=-\sum_{i=1}^{n}\bigg{(}\sum_{j,k=1}^{n}T_{ij}^{k}u_{\alpha}^{j}u_{\beta}^{k}\bigg{)}u_{\alpha\beta}^{i}\\\
&\leq\sum_{i=1}^{n}\frac{1}{4\epsilon}\bigg{(}\sum_{j,k=1}^{n}T_{ij}^{k}u_{\alpha}^{j}u_{\beta}^{k}\bigg{)}^{2}+\epsilon\sum_{i=1}^{n}(u_{\alpha\beta}^{i})^{2}\\\
&=\sum_{i=1}^{n}\frac{1}{4\epsilon}\bigg{(}\sum_{j,k=1}^{n}T_{ij}^{k}u_{\alpha}^{j}u_{\beta}^{k}\sum_{s,t=\alpha}^{n}T_{is}^{t}u_{\alpha}^{s}u_{\beta}^{t}\bigg{)}+\epsilon\sum_{i=1}^{n}(u_{\alpha\beta}^{i})^{2}\\\
&=\frac{1}{4\epsilon}\sum_{j,k,t,s=1}^{n}\bigg{(}\sum_{i=1}^{n}(T_{ij}^{k}T_{is}^{t})u_{\alpha}^{j}u_{\beta}^{k}u_{\alpha}^{s}u_{\beta}^{t}\bigg{)}+\epsilon\sum_{i=1}^{n}(u_{\alpha\beta}^{i})^{2}.\end{split}$
(2.5)

The third term on the right of (2) can also be estimated as follows:

$\begin{split}&-\langle
T_{ij}^{k}u_{\alpha}^{i}u_{\alpha\beta}^{j}\partial_{k},u_{\beta}^{r}\partial_{r}\rangle=-\sum_{i,j,k=1}^{n}T_{ij}^{k}u_{\alpha}^{i}u_{\alpha\beta}^{j}u_{\beta}^{k}=-\sum_{j=1}^{n}\bigg{(}\sum_{i,k=1}^{n}T_{ij}^{k}u_{\alpha}^{i}u_{\beta}^{k}\bigg{)}u_{\alpha\beta}^{j}\\\
&\leq\frac{1}{4\epsilon}\sum_{j=1}^{n}\bigg{(}\sum_{i,k=1}^{n}T_{ij}^{k}u_{\alpha}^{i}u_{\beta}^{k}\bigg{)}^{2}+\epsilon\sum_{j=1}^{n}(u_{\alpha\beta}^{j})^{2}\\\
&=\frac{1}{4\epsilon}\sum_{j=1}^{n}\bigg{(}\sum_{i,k=1}^{n}T_{ij}^{k}u_{\alpha}^{i}u_{\beta}^{k}\sum_{s,t=1}^{n}T_{sj}^{t}u_{\alpha}^{s}u_{\beta}^{t}\bigg{)}+\epsilon\sum_{j=\alpha}^{n}(u_{\alpha\beta}^{j})^{2}\\\
&=\frac{1}{4\epsilon}\sum_{i,k,s,t=1}^{n}\bigg{(}\sum_{j=1}^{n}(T_{ij}^{k}T_{sj}^{t})u_{\alpha}^{i}u_{\beta}^{k}u_{\alpha}^{s}u_{\beta}^{t}\bigg{)}+\epsilon\sum_{j=1}^{n}(u_{\alpha\beta}^{j})^{2}\\\
&=\frac{1}{4\epsilon}\sum_{j,k,s,t=1}^{n}\bigg{(}\sum_{i=1}^{n}(T_{ij}^{k}T_{is}^{t})u_{\alpha}^{j}u_{\beta}^{k}u_{\alpha}^{s}u_{\beta}^{t}\bigg{)}+\epsilon\sum_{i=1}^{n}(u_{\alpha\beta}^{j})^{2}.\\\
\end{split}$ (2.6)

Combining the above inequalities,

$\begin{split}&-\sum_{\beta,\alpha}\langle\nabla_{e_{\beta}}\big{(}T(du(e_{\alpha}),du(e_{\alpha}))\big{)},du(e_{\beta})\rangle\\\
&\leq-\sum_{\beta,\alpha}\langle
T_{ij,l}^{k}u_{\beta}^{l}u_{\alpha}^{i}u_{\alpha}^{j}\partial_{k},u_{\beta}^{r}\partial_{r}\rangle+\frac{1}{4\epsilon}\sum_{\beta,\alpha}\sum_{j,k,t,s=1}^{n}\bigg{(}\sum_{i=1}^{n}(T_{ij}^{k}T_{is}^{t})u_{\alpha}^{j}u_{\beta}^{k}u_{\alpha}^{s}u_{\beta}^{t}\bigg{)}\\\
&+\epsilon\sum_{\beta,\alpha}\sum_{i=\alpha}^{n}(u_{\alpha\beta}^{i})^{2}+\frac{1}{4\epsilon}\sum_{\beta,\alpha}\sum_{j,k,s,t=1}^{n}\bigg{(}\sum_{i=1}^{n}(T_{ij}^{k}T_{is}^{t})u_{\alpha}^{j}u_{\beta}^{k}u_{\alpha}^{s}u_{\beta}^{t}\bigg{)}+\epsilon\sum_{\beta,\alpha}\sum_{j=1}^{n}(u_{\alpha\beta}^{j})^{2}\\\
&=\sum_{\beta,\alpha}\langle
T_{ij,l}^{k}u_{\beta}^{l}u_{\alpha}^{i}u_{\alpha}^{j}\partial_{k},u_{\beta}^{r}\partial_{r}\rangle\\\
&+\sum_{\beta,\gamma}\frac{1}{2\epsilon}\sum_{j,k,s,t=1}^{n}\bigg{(}\sum_{i=1}^{n}(T_{ij}^{k}T_{is}^{t})u_{\gamma}^{j}u_{\beta}^{k}u_{\gamma}^{s}u_{\beta}^{t}\bigg{)}+\epsilon\sum_{j=1}^{n}\sum_{\beta,\alpha}(u_{\alpha\beta}^{j})^{2}\\\
&=\frac{1}{2\epsilon}\sum_{\beta,\gamma}|T(du(e_{\beta})^{\sharp},du(e_{\gamma}))|^{2}+\langle(\nabla
T)(du(e_{\beta})^{\sharp},du(e_{\alpha}),du(e_{\alpha})),du(e_{\beta})\rangle\\\
&+\epsilon|\nabla du|^{2}.\end{split}$ (2.7)

Therefore, we get

$\displaystyle\frac{1}{2}(\triangle_{V}|du|^{2}-\frac{\partial}{\partial
t}|du|^{2})$ $\displaystyle\geq(1-\epsilon)|\nabla
du|^{2}-\sum_{\beta,\beta=\alpha}\langle
R^{N}(du(e_{\beta}),du(\beta))du(e_{\beta}),du(e_{\beta})\rangle+\sum_{\beta=\alpha}\langle
du(Ric^{M}(e_{\beta})),du(e_{\beta})\rangle$
$\displaystyle-\frac{1}{4\epsilon}\sum_{\beta,\gamma}|T(du(e_{\beta})^{\sharp},du(e_{\gamma}))|^{\alpha}-\sum_{\beta,\alpha}\langle(\nabla
T)(du(e_{\beta})^{\sharp},du(e_{\alpha}),du(e_{\alpha})),du(e_{\beta})\rangle$
$\displaystyle=(1-\epsilon)|\nabla du|^{2}-\sum_{\beta,\alpha}\bigg{(}\langle
R^{N}(du(e_{\beta}),du(e_{\alpha}))du(e_{\beta}),du(e_{\alpha})\rangle+\frac{1}{2\epsilon}|T(du(e_{\beta})^{\sharp},du(e_{\alpha}))|^{2}$
$\displaystyle+\langle(\nabla
T)(du(e_{\beta})^{\sharp},du(e_{\alpha}),du(e_{\alpha})),du(e_{\beta})\rangle\bigg{)}+\sum_{\beta}\langle
du(Ric_{V}^{M}(e_{\beta})),du(e_{\beta})\rangle.$

Therefore, we choose $\epsilon=\frac{1}{4C_{0}}$,

$\frac{1}{2}(\Delta_{V}|du|^{2}-\frac{\partial}{\partial
t}|du|^{2})\geq-(1-\frac{1}{4C_{0}})|\nabla du|^{2}+C_{1}|du|^{2}.$ (2.8)

where the constant $C_{1}$ depends only on $Ric_{V}^{M}.$

Similarly, we can treat the index $e_{\beta}$ as the index $\partial_{t}$ in
the previous proof. Recall Bochner formula of the kinetic energy,

$\displaystyle\frac{1}{2}\frac{\partial}{\partial t}|\frac{\partial
u}{\partial t}|^{2}$ $\displaystyle=\frac{1}{2}\triangle|\frac{\partial
u}{\partial t}|^{2}-|\nabla\frac{\partial u}{\partial t}|^{2}+\langle
R^{N}(du(e_{\alpha}),\frac{\partial u}{\partial
t})du(e_{\alpha}),\frac{\partial u}{\partial t}\rangle$
$\displaystyle-\langle\frac{\nabla}{\partial
t}\big{(}T(du(e_{\alpha}),du(e_{\alpha})+du(V))\big{)},\frac{\partial
u}{\partial t}\rangle.$

Therefore choosing $\epsilon=\frac{1}{4C_{0}}$,

$\begin{split}\frac{1}{2}(\Delta_{V}-\frac{\partial}{\partial
t})|\frac{du}{dt}|^{2}\geq&(1-\epsilon)|\nabla_{\partial_{t}}du|^{2}\\\
&-\bigg{(}\sum_{\gamma}\big{(}\langle
R^{N}(du(\partial_{t}),du(\gamma))du(\partial_{t}),du(e_{\gamma})\rangle+\frac{1}{2\epsilon}|T(du(\partial_{t})^{\sharp},du(e_{\gamma}))|^{2}\\\
&+\frac{1}{2}\langle(\nabla
T)(du(\partial_{t})^{\sharp},du(e_{\alpha}),du(e_{\alpha})),du(\partial_{t})\rangle\bigg{)}\\\
\geq&(1-\frac{1}{4C_{0}})|\nabla\frac{du}{dt}|^{2}.\end{split}$ (2.9)

∎

######  Theorem 2.2 (Energy estimates).

Let $u\in C^{2,1}\left(M\times[0,T_{max}),N\right)\cap
C^{\infty}\left(M\times(0,T_{max}),N\right)$ be a solution to the (1.3). The
curvature tensor $R^{N}$ and $T$ satisfy (2.1). Then the following statements
hold:

(1) If $|T|_{L^{\infty}}<\infty$, then for $(x,t)\in M\times[0,T_{max})$,

$e\left(u\right)(x,t)\leq e^{CT_{max}}\sup_{x\in M}e(u_{0})(x).$

(2) For all $(x,t)\in M\times[0,T_{max})$,

$\left|\frac{\partial u}{\partial t}(x,t)\right|\leq\sup_{x\in
M}\left|\frac{\partial u}{\partial t}(x,0)\right|.$

###### Proof.

We modify the proof of Lemma 1 in [9] and (15)(16) in [9], since holomorphic
laplacian operator is special case of $V$-laplacian operator (cf. section 2 in
[12]). One can also refer to Koh [10] although the dimension of the domain
manifold may be different. By preceeding lemma, we have

$\displaystyle\frac{1}{2}(\frac{\partial}{\partial
t}|du|^{2}-\Delta_{V}|du|^{2})\leq-(1+\frac{1}{4C_{0}})|\nabla
du|^{2}+C|du|^{2};$ (2.10)

and

$\displaystyle\frac{1}{2}(\frac{\partial}{\partial
t}|\frac{du}{dt}|^{2}-\Delta_{V}|\frac{du}{dt}|^{2})\leq-(1+\frac{1}{4C_{0}})|\nabla\frac{du}{dt}|^{2}$
(2.11)

Then the theorem follows from maximum principle of parabolic equation. ∎

######  Theorem 2.3 (Long time existence and convergence ).

Let $(M,g)$ be a closed manifold and $(N,h)$ be a Riemannian manifold.
Furthermore, the curvature tensor $R^{N}$ and $T$ satisfy (2.1) . For a given
$C^{2+\alpha}$ map $u_{0}\in C^{2+\alpha}(M,N)$, there exists a global map
$u\in C^{2+\alpha,1+\alpha/2}(M\times[0,\infty),N)\cap
C^{\infty}(M\times(0,\infty),N)$ , which solves

$\left\\{\begin{array}[]{l}\tau_{V}\left(u\right)(x)=\operatorname{Tr}_{g}T(du,du)+\frac{\partial
u}{\partial t}(x),(x,t)\in M\times(0,+\infty),\\\
u(x,0)=u_{0}(x)\end{array}\right.$

Furthermore, if we have uniform bounds

$\int_{M}e(u)ds\leq C,\int_{0}^{\infty}\int_{M}k(u)dsdt\leq C$

for all $t\in[0,\infty)$, then the solution of the evolution equation (1.9)
subconverges in $C^{2}\left(M,N\right)$ to a $VT$ harmonic map $u_{\infty}$.

###### Proof.

It follows from Theorem 2.2. The proof of the first part is standard, we omit
the details. We begin to prove the second part. By the assumption and Moser
iteration, we have

$\begin{split}e(u)\leq C,k(u)\leq C.\end{split}$

Applying the Schauder estimates, we have

$\begin{split}|u|_{C^{2,\alpha}}\leq C,|\partial_{t}u|_{C^{2,\alpha}}\leq
C\end{split}$

However,

$\begin{split}\int_{0}^{\infty}\int_{M}k(u)dsdt\leq C\end{split}$

implies that there exists a sequence $t_{k}\to\infty$ such that

$\begin{split}|\partial_{t}u(\cdot,t_{k})|_{L^{2}}\to 0\end{split}$

We can draw conclusion from here.

∎

In order to apply Theorem 2.3, we consider the energy functional

$\begin{split}E(u)=\int_{M}\left(\frac{1}{2}f|du|^{2}+f\left(\operatorname{Tr}_{g}u^{*}\Phi\right)\right)dv_{g}\end{split}$

where $\left\langle T(Y,Z),X\right\rangle=\left(L_{X}\Phi\right)(Y,Z),f\in
C^{1}(M),f>0.$ Let $V=\nabla\log f$, its critical point of $E(u)$ is
$\tau_{f}(u)+\operatorname{Tr}_{g}T(du,du)=0,$ where
$\tau_{f}(u)=\tau(u)+du(\nabla\log f)$.

######  Lemma 2.2 (Monotonicity formula).

Let $M$ be a closed Riemannian manifold and $N$ be a Riemannian manifold, $u$
is the solution of (1.17) where $V=\nabla\log f$ and the tensor $T$ is befined
by (1.4). Then the energy functional $E(u)$ satisfies that

$\frac{d}{dt}E(u)=-\int_{M}f|\frac{\partial u}{\partial t}|^{2}\leq 0.$ (2.12)

In particular,

$\frac{1}{2}\int_{M}f|du|^{2}+f\operatorname{Tr}_{g}\left(u^{*}\Phi\right)\mathrm{d}v_{g}+\int_{0}^{t}\int_{M}f|\frac{\partial
u}{\partial t}|^{2}\mathrm{d}v_{g}\mathrm{d}s\leq E(u_{0})<\infty.$ (2.13)

###### Proof.

$\begin{split}\frac{d}{dt}E(u)=&-\int_{M}f\langle\frac{\partial u}{\partial
t},\tau_{f}(u)\rangle+f(u^{*}(L_{du(\partial_{t})}\Phi)(e_{i},e_{i})\mathrm{d}v_{g}\\\
=&-\int_{M}f\langle\tau_{f}(u)+T(du(e_{i}),du(e_{i})),\frac{\partial
u}{\partial
t}\rangle\mathrm{d}v_{g}=-\int_{M}f|\frac{du}{dt}|^{2}\mathrm{d}v_{g}.\\\
\end{split}$

where $\tau_{f}(u)=\tau(u)+du(\nabla\log f)$. ∎

######  Theorem 2.4 ( Long time existence and convergence ).

Let $M$ be a closed Riemannian manifold , $N$ is a Riemannian manifold. The
tensor $T$ is befined by (1.4), $V=\nabla\log f,$
$\|\Phi\|_{L^{\infty}}<\frac{1}{2}$. Furthermore, the curvature tensor $R^{N}$
and $T$ satisfy (2.1). For a given $C^{2+\alpha}$ map $u_{0}\in
C^{2+\alpha}(M,N)$, there exists a solution $u\in
C^{2+\alpha,1+\alpha/2}(M\times[0,T),N)\cap C^{\infty}(M\times(0,T),N)$ which
solves (1.9). Moreover, $u(x,t)$ subconverges in $C^{2}\left(M,N\right)$ to a
VT harmonic map $u_{\infty}$.

###### Proof.

By Lemma 2.2, the conditions in Theroem 2.3 are satisfied. ∎

##  3 Domain manifold is compact manifold with boundary

In this section, we adapt the method of [4] to study the problem (1.9). The
extra term $\operatorname{Tr}_{g}T(du,du)$ doesn’t affect the argument of
contraction map principle in this case. We omit the details for simplicity.
One can refer to the proof of the short time existence of harmonic map heat
flow from closed manifold with boundary(cf. [8]) and adjust it. We can assume
the existence of $u:M\times[0,T_{max})\rightarrow N$.

As in in [4], we choose the frame { $\pi,\nu$ } for $M$, and $\nu$ is the
normal vector field of $\partial M$.

######  Lemma 3.1.

Let $u:M\times[0,T_{max})\rightarrow N$ be a smooth solution of (1.9) , $H$ is
the mean curvature of $\partial M$, the following differential equality holds
on $\partial M\times[0,T_{max})$,

$\displaystyle\nabla_{\nu}e(u)-2He(u)=a\nabla^{2}u+b\nabla u;$ (3.1)

where $a,b$ only depends on $u_{0}$, $e(u)$ is the energy density
$\frac{1}{2}|du|^{2}$.

###### Proof.

We adjust the argument in [4]. According to corollary in page 132 of part V in
[8], $\tau_{M}u=\tau_{\partial M}u+\nabla_{\nu}\nabla_{\nu}u-H\nabla_{\nu}u$.
Recall that $\frac{\partial u}{\partial
t}=\tau(u)+\operatorname{Tr}_{g}T(du,du)$. Since $u|_{\partial M\times[0,t)}$
is independent of $t$, we conclude that on $\partial M$

$\tau(u)+\operatorname{Tr}_{g}T(du,du)=0.$

It follows that

$\operatorname{Tr}_{g}T(du,du)-\tau_{\partial
M}u+H\nabla_{\nu}u=\nabla_{\nu}\nabla_{\nu}u;$

and

$\displaystyle\nabla_{\nu}e(u)=$
$\displaystyle\langle\nabla_{\nu}\nabla_{\pi}u,\nabla_{\pi}u\rangle+\langle\nabla_{\nu}\nabla_{\nu}u,\nabla_{\nu}u\rangle$
$\displaystyle=$
$\displaystyle\langle\nabla_{\nu}\nabla_{\pi}u,\nabla_{\pi}u\rangle+\langle\operatorname{Tr}_{g}T(du,du)-\tau_{\partial
M}u,\nabla_{\nu}u\rangle+\langle H\nabla_{\nu}u,\nabla_{\nu}u\rangle$
$\displaystyle=$
$\displaystyle\langle\nabla_{\nu}\nabla_{\pi}u,\nabla_{\pi}u\rangle+\langle\operatorname{Tr}_{g}T(du,du),\nabla_{\nu}u\rangle$
$\displaystyle-\langle\tau_{\partial
M}u,\nabla_{\nu}u\rangle+2He(u)-H\langle\nabla_{\pi}u,\nabla_{\pi}u\rangle.$
$\displaystyle=$
$\displaystyle\langle\nabla_{\nu}\nabla_{\pi}u,\nabla_{\pi}u\rangle+\langle\operatorname{Tr}_{g}T(du,du),\nabla_{\nu}u\rangle$
$\displaystyle-\langle\tau_{\partial
M}u,\nabla_{\nu}u\rangle+2He(u)-H\langle\nabla_{\pi}u,\nabla_{\pi}u\rangle.$

Then (3.1) follows as in [8, pp. 132-133] , noticing that $\nabla_{\nu}u=0$.

∎

######  Theorem 3.1.

Let $u:M\times[0,T_{max})\rightarrow N$ be a smooth solution of (1.9). If we
have uniform bounds

$\int_{M}e(u)\mathrm{d}s\leq
C,\int_{0}^{\infty}\int_{M}k(u)\mathrm{d}s\mathrm{d}t\leq C$

for all $t\in[0,\infty)$. Assume that the curvature tensor $R^{N}$ and $T$
satisfy (2.1), and $|\Phi|_{L^{\infty}(N)}<\frac{1}{2}$. Then there is a
constant $C$ which may depend on $\delta$ and is independent of $t$, such that
for any $p<\infty$,

$\displaystyle\|u\|^{2}_{W_{2}^{2}(M\times[t,t+\delta])}$ $\displaystyle\leq
C(\int_{\partial M\times[t,t+\delta]}|\nabla^{2}u|+1).$ (3.2)
$\displaystyle||u||^{2}_{W_{2}^{2}(M\times[t,t+\delta])}$ $\displaystyle\leq
C.$ (3.3) $\displaystyle||u||_{W^{p}_{2}(M\times[t,t+\delta])}$
$\displaystyle\leq C.$ (3.4)

for $0\leq t<T_{max}-\delta$ (or $0\leq t<\infty$ for $T_{max}=\infty$). For
any $p<\infty,n<\infty$,

$||u||_{W^{p}_{n}(M\times[t,t+\delta])}\leq C.$ (3.5)

for $\delta\leq t<T_{max}-\delta$ (or $1\leq t<\infty$ if $T_{max}=\infty$).

###### Proof.

We adjust the argument in [4]. By the assumption,

$\int_{M}e(u)\mathrm{d}s\leq
C,\int_{0}^{\infty}\int_{M}k(u)\mathrm{d}s\mathrm{d}t\leq C$

for all $t\in[0,\infty)$. By Lemma (2.1), we have

$\displaystyle\left(\frac{\partial}{\partial t}-\Delta\right)|\frac{\partial
u}{\partial t}|^{2}\leq-2(1+\frac{1}{4C_{0}})\bigg{|}\nabla|\frac{\partial
u}{\partial t}|\bigg{|}^{2}.$ (3.6)

since $|\nabla\frac{\partial u}{\partial t}|^{2}\geq|\nabla|\frac{\partial
u}{\partial t}||^{2}$. Integrating the above equations, we infer that

$\begin{split}\int_{M}k(u)\mathrm{d}v_{g}\leq C.\end{split}$

Using Moser iteration, we conclude that

$\begin{split}\|\frac{\partial u}{\partial
t}\|_{L^{\infty}(M\times[0,T_{max}))}&\leq C\end{split}$

Noticing the Neumann boundary condition and using the same proof as in
Hamilton [8] (here we omit the details for simplicity), we can finish the
proof. ∎

######  Theorem 3.2 (Long time existence and convergence).

Let $M$ be a closed Riemannian manifold with nonempty boundary and $N$ be a
Riemannian manifold. Assume that the curvature tensor $R^{N}$ and $T$ satisfy
(2.1) and $|\Phi|_{L^{\infty}(N)}<\frac{1}{2}$. Assume that

$\int_{M}e(u)\mathrm{d}s\leq
C,\int_{0}^{\infty}\int_{M}k(u)\mathrm{d}s\mathrm{d}t\leq C$

for all $t\in[0,\infty)$. For a given $C^{2+\alpha}$ map $u_{0}\in
C^{2+\alpha}(M,N)$, there exist a global map $u\in
C^{2+\alpha,1+\alpha/2}(M\times[0,+\infty),N)\cap
C^{\infty}(M\times(0,+\infty),N),$ which solves (1.9). Furthermore, the
solution $u(x,t)$ of the evolution equation (1.9) subconverges in
$C^{2}\left(M,N\right)$ to a VT harmonic map $u_{\infty}$.

In order to apply Theorem 3.2, we consider the energy functional

$\begin{split}E(u)=\int_{M}\left(\frac{1}{2}|du|^{2}+\left(\operatorname{Tr}_{g}u^{*}\Phi\right)\right)dv_{g}\end{split}$

where $\left\langle T(Y,Z),X\right\rangle=\left(L_{X}\Phi\right)(Y,Z).$ The
corresponding heat flow of Neumann problem is

$\displaystyle\left\\{\begin{array}[]{l}\frac{\partial u}{\partial
t}=\tau(u)+\operatorname{Tr}_{g}u^{*}T,\quad in\quad M\\\ u=u_{0},\partial
M\times[0,\infty)\\\ \frac{\partial u}{\partial n}|_{\partial M}=0\\\
u(\cdot,0)=u_{0}(\cdot)\end{array}\right.$ (3.11)

######  Lemma 3.2 (Monotonicity formula).

Let $M$ be a compact Riemann manifold with boundary and $N$ be a Riemannian
manifold, $u$ is the solution of (1.17). Then the energy functional $E(u)$
satisfies that

$\frac{d}{dt}E(u)=-\int_{M}|\frac{\partial u}{\partial t}|^{2}\leq 0.$ (3.12)

In particular,

$\int_{M}\frac{1}{2}|du|^{2}+\operatorname{Tr}_{g}\left(u^{*}\Phi\right)\mathrm{d}v_{g}+\int_{0}^{t}\int_{M}|\frac{\partial
u}{\partial t}|^{2}\mathrm{d}v_{g}\mathrm{d}s\leq E(u_{0})<\infty.$ (3.13)

###### Proof.

The proof is almost the same as that in Lemma 2.2 .

∎

By Theorem 3.1 and Lemma 3.2, it is easy to get

######  Theorem 3.3 ( Long time existence and convergence ).

Let $M$ be a compact Riemann manifold with non-empty boundary and $N$ be a
Riemannian manifold. Furthermore, the tensor $T$ is befined by (1.4). Assume
that the curvature tensor $R^{N}$ and $T$ satisfy (2.1) and
$|\Phi|_{L^{\infty}(N)}<\frac{1}{2}$. For a given $C^{2+\alpha}$ map $u_{0}\in
C^{2+\alpha}(M,N)$, there exist a solution $u\in
C^{2+\alpha,1+\alpha/2}(M\times[0,+\infty),N)\cap
C^{\infty}(M\times(0,+\infty),N)$ of (3.11) such that

$\displaystyle\left\\{\begin{array}[]{l}\frac{\partial u}{\partial
t}=\tau(u)+\operatorname{Tr}_{g}\left(u^{*}T\right),\quad in\quad M\\\
u=u_{0},\partial M\times[0,\infty)\\\ \frac{\partial u}{\partial n}|_{\partial
M}=0\\\ u(\cdot,0)=u_{0}(\cdot)\end{array}\right.$ (3.18)

holds. Moreover, $u(x,t)$ subconverges in $C^{2}\left(M,N\right)$ to VT
harmonic map $u_{\infty}$which sloves the Neumann boundary problem

$\displaystyle\left\\{\begin{array}[]{l}\tau(u)+\operatorname{Tr}_{g}\left(u^{*}T\right)=0,\quad
in\quad M\\\ u=u_{0}(\cdot),\partial M\times[0,\infty)\\\ \frac{\partial
u}{\partial n}|_{\partial M}=0\\\ u(\cdot,0)=u_{0}(\cdot)\end{array}\right.$
(3.23)

##  4 Existence and convergencve of heat flow of $VT$ geodeisc

In this section, we mainly consider the case when domanin manifold is $S^{1}.$

######  Lemma 4.1.

Suppose that $N$ is isometrically embedded into $\mathbb{R}^{q}$. Then (1.17)
acquires the form

$\begin{split}\frac{\partial u}{\partial
t}=\Delta_{V}u-\mathbb{I}(du,du)-\tilde{T}\left(du\left(e_{1}\right),du\left(e_{1}\right)\right),\end{split}$
(4.1)

where $u:S^{1}\times[0,T)\rightarrow\mathbb{R}^{q},\tilde{T}$ denote the
extension of T to $\mathbb{R}^{q}$, $\mathbb{I}(du,du)$ is the second
fundermental form of $N$.

###### Proof.

The tensor $T$ can be extended to the ambient space by projecting to a tubular
neighborhood of N. The solution of (4.1) is equivalent to that of (1.17). ∎

By theorem 2.1, we immediately get

######  Theorem 4.1 (Short time existence).

Let $(N,h)$ be a Riemannian manifold. For a given $C^{2+\alpha}$ map $u_{0}\in
C^{2+\alpha}(S^{1},N),$ there exists a positive number $T_{max}>0$ and a map
$u\in C^{2+\alpha,1+\alpha/2}(S^{1}\times[0,T_{max}],N)\cap
C^{\infty}(S^{1}\times(0,T_{max}),N)$ such that

$\left\\{\begin{array}[]{l}\tau_{V}\left(u\right)(x)=-\operatorname{Tr}_{g}T(du,du)+\frac{\partial
u}{\partial t}(x),(x,t)\in S^{1}\times(0,T_{max}),\\\
u(x,0)=u_{0}(x)\end{array}\right.$

holds. Here, $T_{max}$ is a constant depending on $N,M,T,f$ and $\alpha$
alone.

###### Proof.

It follows from the inverse functon theorem using the equation (4.1) after
embedding $N$ into $\mathbb{R}^{q}$. One can refer to the proof of Theorem 16
in Koh [10] and adjust it. ∎

We can improve Lemma 2.1 since the domain manifold is $S^{1}$.

######  Lemma 4.2.

Let $u:S^{1}\times[0,T_{max})\rightarrow N$ be a smooth solution of (1.17).
The sectional cuervature of N is bounded above by a positive constant
$\kappa$. The tensor $T$ satisfy

$4C_{0}|T(X^{\sharp},Y)|^{2}+\langle(\nabla T)(X^{\sharp},Y,Z),X\rangle\leq
0;$ (4.2)

where $X,Y,Z\in\Gamma(TN)$, constant $C_{0}>1$, $X^{\sharp}$ denotes the dual
1-form of $X$. Then for all $t\in[0,T_{Max})$, we have

$\displaystyle(\frac{\partial}{\partial
t}|du|^{2}-\Delta_{V}|du|^{2})\leq-2(1+\frac{1}{4C_{0}})|\nabla
du|^{2}+C|du|^{2};$ (4.3)

and

$\displaystyle\frac{1}{2}(\frac{\partial}{\partial
t}|\frac{du}{dt}|^{2}-\Delta_{V}|\frac{du}{dt}|^{2})\leq-2(1+\frac{1}{4C_{0}})|\nabla\frac{du}{dt}|^{2}+\kappa
k(u)e(u).$ (4.4)

Here $C$ is an constant depending only on the bounds if $L_{V}g$ .

######  Remark 4.1.

One can compare the conditions in Lemma 2.1 with that in Lemma 4.2.

###### Proof.

Let $e_{1}$ be a local orthonormal frame of $S^{1}$ at the considered point.
By the formula in Lemma 2.1,

$\displaystyle\frac{1}{2}\triangle|du|^{2}=$ $\displaystyle|\nabla
du|^{2}+\langle\nabla_{e_{1}}\tau(u),du(e_{1})\rangle+\langle
du(Ric^{M}(e_{1})),du(e_{1})\rangle$ $\displaystyle=$ $\displaystyle|\nabla
du|^{2}+\bigg{\langle}\nabla_{e_{1}}\bigg{(}\frac{\partial u}{\partial
t}-T(du(e_{1}),du(e_{1})-du(V)\bigg{)},du(e_{1}\bigg{\rangle}$
$\displaystyle+\langle du(Ric^{M}(e_{1})),du(e_{1})\rangle$ $\displaystyle=$
$\displaystyle|\nabla
du|^{2}-\bigg{\langle}\nabla_{e_{1}}\bigg{(}T(du(e_{1}),du(e_{1}))+du(V)\bigg{)},du(e_{1})\bigg{\rangle}+\frac{1}{2}\frac{\partial}{\partial
t}|du|^{2}$

Therefore

$\displaystyle\frac{1}{2}\triangle_{V}|du|^{2}-\frac{1}{2}\frac{\partial}{\partial
t}|du|^{2}=$ $\displaystyle|\nabla
du|^{2}+\langle\nabla_{e_{1}}\big{(}T(du(e_{1})),du(e_{1}))\big{)},du(e_{1})\rangle$
$\displaystyle+\frac{1}{2}\langle du(L_{V}g(e_{1})),du(e_{1})\rangle$

We choose the normal coordinates at the considered point of $N$, by (2.7), we
get

$\displaystyle-\langle\nabla_{e_{1}}\big{(}T(du(e_{1}),du(e_{1}))\big{)},du(e_{1})\rangle$
$\displaystyle=\frac{1}{2\epsilon}|T(du(e_{1})^{\sharp},du(e_{1}))|^{2}+\langle(\nabla
T)(du(e_{1})^{\sharp},du(e_{1}),du(e_{1})),du(e_{1})\rangle+\epsilon|\nabla
du|^{2}.$

Therefore, we get

$\displaystyle\frac{1}{2}(\triangle_{V}|du|^{2}-\frac{\partial}{\partial
t}|du|^{2})$ $\displaystyle=(1-\epsilon)|\nabla
du|^{2}+\bigg{(}\frac{1}{2\epsilon}|T(du(e_{1})^{\sharp},du(e_{1}))|^{2}+\frac{1}{2}\langle(\nabla
T)(du(e_{1})^{\sharp},du(e_{1}),du(e_{1})),du(e_{1})\rangle\bigg{)}$
$\displaystyle-\frac{1}{2}\sum_{1}\langle du(L_{V}g(e_{1})),du(e_{1})\rangle.$

In the last equality, we used the fact that the domain manifold is $S^{1}$.
Therefore, we choose $\epsilon=\frac{1}{4C_{0}}$,

$\frac{1}{2}(\triangle_{V}|du|^{2}-\frac{\partial}{\partial
t}|du|^{2})\geq-(1-\frac{1}{4C_{0}})|\nabla du|^{2}+C_{1}|du|^{2}.$ (4.5)

where the constant $C_{1}$ depends only on $L_{V}g.$ By (2.9), we get

$\displaystyle\frac{1}{2}\triangle_{V}|\frac{du}{dt}|^{2}-\frac{1}{2}\frac{\partial}{\partial
t}|\frac{du}{dt}|^{2}$ $\displaystyle=$
$\displaystyle|\nabla_{\partial_{t}}du|^{2}+\sum_{1=1}^{1}\langle\nabla_{\partial_{t}}\big{(}T(du(e_{1}),du(e_{1}))+du(V)\big{)},du(\partial_{t})\rangle$
$\displaystyle-\sum_{1,1}\langle
R^{N}(du(\partial_{t}),du(e_{1}))du(\partial_{t}),du(e_{1})\rangle.$

Therefore choosing $\epsilon=\frac{1}{4C_{0}}$,

$\displaystyle(\Delta_{V}-\frac{\partial}{\partial t})|\frac{du}{dt}|^{2}$
$\displaystyle\geq(1-\epsilon)|\nabla_{\partial_{t}}du|^{2}$
$\displaystyle-\bigg{(}\sum_{\gamma}\big{(}\langle
R^{N}(du(\partial_{t}),du(e_{\gamma}))du(\partial_{t}),du(e_{\gamma})\rangle+\frac{1}{2\epsilon}|T(du(\partial_{t})^{\sharp},du(e_{\gamma}))|^{2}$
$\displaystyle+\frac{1}{2}\langle(\nabla
T)(du(\partial_{t})^{\sharp},du(e_{1}),du(e_{1})),du(\partial_{t})\rangle\bigg{)}$
$\displaystyle\geq(1-\frac{1}{4C_{0}})|\nabla\frac{du}{dt}|^{2}-\kappa
k(u)e(u).$

∎

######  Theorem 4.2 (Energy estimates).

Let $u\in C^{2,1}\left(S^{1}\times[0,T),N\right)\cap
C^{\infty}\left(S^{1}\times(0,T),N\right)$ be a solution to (1.17) .The
sectional cuervature of N is bounded above by a positive constant $\kappa$.
The tensor $T$ satisfies the condition (4.2). Then the following statements
hold:

(1) For all $(x,t)\in S^{1}\times[0,T_{max})$,

$e\left(u\right)(x,t)\leq e^{CT_{max}}\sup_{x\in S^{1}}e(u_{0})(x).$

Here, $C$ are the constants defined in Lemma (2.2).

(2) For all $(x,t)\in S^{1}\times[0,T_{max})$,

$\left|\frac{\partial u}{\partial t}(x,t)\right|\leq
e^{CT_{max}}\sup_{x\in\Sigma}\left|\frac{\partial u}{\partial t}(x,0)\right|.$

here $C=2\kappa e^{\lambda T_{max}}\max_{\Sigma}e(u_{0}).$

###### Proof.

By the inequality,

$\displaystyle(\frac{\partial}{\partial
t}|du|^{2}-\triangle_{V}|du|^{2})\leq-2(1+\frac{1}{4C_{0}})|\nabla
du|^{2}+C|du|^{2};$ (4.6)

$\displaystyle(\frac{\partial}{\partial
t}|\frac{du}{dt}|^{2}-\triangle_{V}|\frac{du}{dt}|^{2})\leq-2(1+\frac{1}{4C_{0}})|\nabla\frac{du}{dt}|^{2}+2\kappa
k(u)e(u).$ (4.7)

Then applying the maximum principle to the function $e^{-Ct}e(u)$ and
$e^{-Ct}\kappa(u)$ respectively, the lemma follows easily as in Theorem 2.2 .
∎

######  Theorem 4.3.

Let $u\in C^{2,1}\left(S^{1}\times[0,T_{max}),N\right)\cap
C^{\infty}\left(S^{1}\times(0,T_{max}),N\right)$ be a solution to (1.17) . The
tensor $T$ satisfies the condition (4.2). Then

$\begin{split}|u(\cdot,t)|_{C^{2+\alpha}(S^{1},N)}+|\frac{\partial u}{\partial
t}(\cdot,t)|_{C^{\alpha}(S^{1},N)}\leq C,\end{split}$

for $t\in[0,T_{max})$, C is a constant depending only on
$S^{1},N,T,T_{max},\alpha.$

###### Proof.

Firstly, applying elliptic Schauder estimates to the equation

$\begin{split}\tau_{V}\left(u\right)(x)=-\operatorname{Tr}_{g}T(du,du)+\frac{\partial
u_{t}}{\partial t}(x).\end{split}$

Secondly, applying parabolic Schauder estimates to the equation

$\begin{split}\frac{\partial u_{t}}{\partial
t}(x)-\tau_{V}\left(u\right)(x)=\operatorname{Tr}_{g}T(du,du).\end{split}$

Then the theorem follows from Theorem 4.2 . ∎

Inspired by Branding [2], similar to Theorem 2.3, we also have

######  Theorem 4.4 (Long time existence and convergence ).

Let $(N,h)$ be Riemannian manifolds. The tensor $T$ satisfies the condition
(4.2). For a given $C^{2+\alpha}$ map $u_{0}\in C^{2+\alpha}(S^{1},M)$, there
exists a global map $u\in
C^{2+\alpha,1+\alpha/2}(S^{1}\times[0,+\infty),N)\cap
C^{\infty}(S^{1}\times(0,+\infty),N)$ , solves (1.17). Furthermore, if we have
uniform bounds

$\int_{S^{1}}e(u)\mathrm{d}s\leq
C,\int_{0}^{\infty}\int_{S^{1}}k(u)\mathrm{d}s\mathrm{d}t\leq C.$

for all $t\in[0,\infty)$, then the solution $u(x,t)$ of the evolution equation
(1.17) subconverges in $C^{2}\left(S^{1},N\right)$ to a $VT$ geodesic
$u_{\infty}$.

###### Proof.

The upper bound in Theorem 4.2 is uniformly bounded for $t\in(0,T).$ The rest
of proof is same as that in Theorem 1 in Koh [10], since the heat flow of
magenetic geodesic is similar to that of VT geodesic, the first order term
$du(V)$ and the quadratic term $\operatorname{Tr}_{g}T(du,du)$ are uniformly
bounded. One can refer to the proof of Theorem 2.3. ∎

######  Theorem 4.5 (uniqueness).

Let $u,v:S^{1}\times\left[0,T_{\max}\right)\rightarrow\mathbb{R}^{q}$ be
smooth solutions of (4.1). The tensor $T$ satisfies the condition (4.2), then
the following inequality holds for all $t\in\left[0,T_{\max}\right),$

$\left|u-v\right|^{2}\leq\left|u_{0}-v_{0}\right|^{2}e^{Ct}$

for some positive constant $C$. In particular, if $u_{0}=v_{0}$ then
$u(x,t)=v(x,t)$ for all $t\in[0,T_{\max})$.

###### Proof.

The proof is almost the same as that in Lemma 4.8 in [1], once we have the
bounds in Theroem 2.2. So, we also have

$\begin{split}\frac{\partial h}{\partial t}-\Delta_{V}h\leq Ch\end{split}$

The result then follows by application of the maximum principle ∎

In order to apply Theorem 4.4, we consider we consider the energy functional

$\begin{split}E(u)=\int_{M}\left(\frac{1}{2}f|du|^{2}+f\left(\operatorname{Tr}_{g}u^{*}\Phi\right)\right)dv_{g}\end{split}$

where $\left\langle T(Y,Z),X\right\rangle=\left(L_{X}\Phi\right)(Y,Z),f\in
C^{1}(M),f>0.$ Let $V=\nabla\log f$, its critical point of $E(u)$ is
$\tau_{f}(u)+\operatorname{Tr}_{g}T(du,du)=0,$ where
$\tau_{f}(u)=\tau(u)+du(\nabla\log f)$.

Similar to Lemma 2.2, we also have

######  Lemma 4.3 (Monotonicity formula).

Let $N$ be a Riemannian manifold, $u$ is the solution of (1.17) . Then the
energy functional $E(u)$ satisfies that

$\frac{d}{dt}E(u)=-\int_{S^{1}}|\frac{\partial u}{\partial
t}|^{2}\mathrm{d}s\leq 0.$ (4.8)

In particular,

$\int_{S^{1}}\frac{1}{2}f|du|^{2}+f\operatorname{Tr}_{g}\left(u^{*}\Phi\right)\mathrm{d}s+\int_{0}^{t}\int_{S^{1}}f|\frac{\partial
u}{\partial t}|^{2}\ \leq E(u_{0})\mathrm{d}s\mathrm{d}t<\infty.$ (4.9)

###### Proof.

$\begin{split}\frac{d}{dt}E(u)=&-\int_{S^{1}}f\langle\frac{\partial
u}{\partial
t},\tau_{f}(u)\rangle+f(u^{*}(L_{du(\partial_{t})}\Phi)(e_{1},e_{1})\mathrm{d}s\\\
=&-\int_{S^{1}}f\langle\tau_{f}(u)+T(du(e_{1}),du(e_{1})),\frac{\partial
u}{\partial
t}\rangle\mathrm{d}s=-\int_{S^{1}}f|\frac{du}{dt}|^{2}\mathrm{d}s.\\\
\end{split}$

where $\tau_{f}(u)=\tau(u)+du(\nabla\log f)$. ∎

By Theorem 4.4, we get

######  Theorem 4.6.

If the tensor $T$ is befined by (1.4), $\|\Phi\|_{L^{\infty}}<\frac{1}{2}$,
the tensor $T$ satisfies the condition (4.2), then the solution $u$ of (1.17)
exists globally and subconverges to a VT geodesic $u_{\infty}$.

###### Proof.

By (4.9), it is easy to see that we have uniform bounds

$\int_{S^{1}}e(u)\mathrm{d}s\leq
C,\int_{0}^{\infty}\int_{S^{1}}k(u)\mathrm{d}s\mathrm{d}t\leq C.$

for all $t\in[0,\infty)$. The theorem follows from Theorem 4.4. ∎

So, it is immediate that we have

######  Corollary 4.1.

If the tensor $T$ is befined by (1.4), $\|\Phi\|_{L^{\infty}}<\frac{1}{2}$,
the tensor $T$ satisfies the condition (4.2), then each homotopy class of
curves in N contains a $VT$ geodesic.

######  Theorem 4.7.

If the tensor $T$ is befined by (1.4), $\|\Phi\|_{L^{\infty}}<\frac{1}{2}$,
the tensor $T$ satisfies the condition (4.2), assume $E\left(u_{0}\right)\leq
0$ and, in case $E\left(u_{0}\right)=0$, furthermore assume that $u_{0}$ is
nontrivial. Then $u_{\infty}$ is a nontrivial closed $VT$ geodesic.

###### Proof.

By (4.9), the proof is the same as Propositon 3.22 in [2]. ∎

## References

  * [1] Branding, V. The heat flow for the full bosonic string.  Annals of Global Analysis and Geometry 50, 4 (2016), 347–365.
  * [2] Branding, V., and Hanisch, F. Magnetic geodesics via the heat flow.  Asian Journal of Mathematics 21, 6 (2017), 995–1014.
  * [3] Cao, X. Existence of generalized harmonic map andLiouville type theorems on Riemannian manifold.  PhD thesis, Wuhan university, 2018.  MR number.
  * [4] Cao, X., and Chen, Q. Existence of harmonic maps with two-form and scalar potentials.  Vietnam J. Math. 49, 2 (2021), 349–361.
  * [5] Cao, X., and Chen, Q. Existence for VT-harmonic maps from compact manifolds with boundary.  Science China Mathematics (2022), 1–8.
  * [6] Chen, Q., Jost, J., and Qiu, H. On $VT$-harmonic maps.  Ann. Global Anal. Geom. 57, 1 (2020), 71–94.
  * [7] Eells, J., and Sampson, J. H. Harmonic mappings of Riemannian manifolds.  American Journal of Mathematics 86, 1 (1964), 109–160.
  * [8] Hamilton, R. S. Harmonic Maps of Manifolds with Boundary.  Springer Berlin Heidelberg, 1975.
  * [9] Jost, J., and Yau, S. T. A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry.  Acta Mathematica 170, 2 (1993), 221–254.
  * [10] Koh, D. On the evolution equation for magnetic geodesics.  Calculus of Variations and Partial Differential Equations 36, 3 (2009), 453–480.
  * [11] Lin, F., and Wang, C. The analysis of harmonic maps and their heat flows. Hackensack, NJ: World Scientific, 2008.
  * [12] Ni, L. Hermitian harmonic maps from complete hermitian manifolds to complete riemannian manifolds.  Mathematische Zeitschrift 232, 2 (1999), 331–355.
  * [13] Ottarsson, S. Closed geodesics on riemannian manifolds via the heat flow.  Journal of Geometry and Physics 2, 1 (1985), 49–72.