On VT-harmonic maps

Abstract

VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat VT-harmonic maps and their parabolic analogues with PDE tools. We establish a Jäger–Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for VT-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.

1 Introduction

Let $(M^m,g)$ be a compact Riemannian manifolds with nonempty boundary $\partial M$ and $(N^n, {\tilde{g}})$ a complete Riemannian manifold without boundary. Let $d: N \times N \rightarrow {\mathbb {R}}$ be the distance function on N and $B_{(1+\sigma )R}(p):= \{ q \in N: d(p, q) \le (1+\sigma )R \}$ a regular ball in N, that is, disjoint from the cut locus of its center p and of radius $(1+\sigma )R < \frac{\pi }{2\sqrt{\kappa }}$, where $\kappa = \max \{ 0, \sup _{B_{(1+\sigma )R}(p)}K_N \}$ and $\sup _{B_{(1+\sigma )R}(p)}K_N$ is an upper bound of the sectional curvature K of N on $B_{(1+\sigma )R}(p)$, and $\sigma >0$ is any given constant.

Let $V \in \Gamma (TM)$, $T \in \Gamma (\otimes ^{1, 2}TN)$. We call a map $u: M \rightarrow N$ a VT-harmonic map if u satisfies

$$\begin{aligned} \tau (u)+du(V)+ \mathrm{Tr}_g T(du, du) = 0, \end{aligned}$$

(1.1)

where $\tau (u)=\text{ tr }D du$ is the tension field of the map u. This is a generalization the notion of a V-harmonic map that has been studied in recent years as a common framework including Hermitian, affine and Weyl harmonic maps into Riemannian manifolds, that is, the domain possessed a connection different from the Levi-Civita connection, but the target was a Riemannian manifold with its Levi-Civita connection. This generalized the standard harmonic map system $\tau (u)=0$ to a system of the form $\tau (u)+du(V)=0$ with a vector field V on the domain. Here, we want to consider targets that are of the same type as the domain. That leads to the system (1.1) with an additional term arising from a 2-tensor T on the target. As this new term $\mathrm{Tr}_g T(du, du)$, in contrast to the term du(V), is analytically of the same weight as the elliptic operator $\tau (u)$ (which includes the Laplace–Beltrami operator of the domain), this makes the analysis more difficult and subtle. This is the problem that we are addressing in this paper.

In local coordinates $\{ x^\alpha \}$ on M and $\{ y^i \}$ on N, respectively, we can write (1.1) as

$$\begin{aligned} \Delta _M u^i + \Gamma ^{i}_{jk}(u)\frac{\partial u^j}{\partial x^\alpha }\frac{\partial u^k}{\partial x^\beta }g^{\alpha \beta } + V^\alpha \frac{\partial u^i}{\partial x^\alpha } + T^{i}_{jk}(u)\frac{\partial u^j}{\partial x^\alpha }\frac{\partial u^k}{\partial x^\beta }g^{\alpha \beta }=0, \end{aligned}$$

(1.2)

where $\Delta _M$ is the Laplacian on (M, g), $\Gamma ^{i}_{jk}$ stands for the Christoffel symbols of $(N, {\tilde{g}})$, $V:=V^\alpha \frac{\partial }{\partial x^\alpha }$ and $T:=T^{i}_{jk}\frac{\partial }{\partial y^i}\otimes dy^j\otimes dy^k$. This is a second-order semilinear elliptic system on the manifold (M, g).

As is already the case for V-harmonic maps, in general, (1.1) is neither in divergence form, nor has a variational structure. Chen et al. [5] established a Jäger–Kaul type maximum principle for V-harmonic maps by using the method of [8], and combining this with the continuity method, the existence of V-harmonic maps into a regular ball could be proved. Therefore, it is natural to ask whether a maximum principle holds for VT-harmonic maps. However, the case of VT-harmonic maps is harder to deal with than V-harmonic maps since we now have an additional quadratic term arising from the tensor T. Due to this additional structure, the construction of the elliptic operator in [5] is no longer valid in our case. To overcome this difficulty, we use another construction as in [7] to compensate this term and obtain the following maximum principle for VT-harmonic maps:

Theorem 1

Let $u_1, u_2 \in C^0(M, N)$ be two VT-harmonic maps into a geodesic ball $B_{R}(p)$. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla T| + \max |T| \le C_0, \end{aligned}$$

(1.3)

then the function $\Theta : M \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \Theta := \frac{q_{\frac{\kappa }{4}}(\rho )}{(q_\kappa ((1+\sigma )R)-q_\kappa (\rho _1))^{\frac{1}{2}}\cdot (q_\kappa ((1+\sigma )R)-q_\kappa (\rho _2))^{\frac{1}{2}}} \end{aligned}$$

(1.4)

satisfies the maximum principle, namely

$$\begin{aligned} \max _M \Theta \le \max _{\partial M}\Theta . \end{aligned}$$

Here the expression of $q_\kappa$ is given in Sect. 2, and $\rho := d(u_1, u_2), \rho _i:= d(p, u_i), i=1,2.$

In particular, if $u_1=u_2$ on the boundary $\partial M$, then $u_1\equiv u_2$ on M.

Remark

The explicit expression of the constant $C_0$ in the above and in the subsequent results can be seen in (3.5). Importantly, $C_0\rightarrow \infty$ for $R\rightarrow 0$. Thus, we can also satisfy the condition on T by making the target ball sufficiently small.

For the heat flow of VT-harmonic maps, an analogous result holds. For $T>0$, we set

$$\begin{aligned} M_T:= M\times [0, T] \end{aligned}$$

and denote the parabolic boundary of $M_T$ by

$$\begin{aligned} \partial _p M_T:=(M\times \{0\})\cup (\partial M\times [0, T]). \end{aligned}$$

We consider the heat flow of VT-harmonic maps

$$\begin{aligned} \partial _t u= \tau (u)+du(V) + \mathrm{Tr}_g T(du, du) \end{aligned}$$

(1.5)

and have

Theorem 2

Let $u_1, u_2 \in C^0(M, N)$ be two solutions of heat flow Eq. (1.5) for VT-harmonic maps into a geodesic ball $B_{R}(p)$. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla T| + \max |T| \le C_0, \end{aligned}$$

then the function $\Theta : M_T \rightarrow {\mathbb {R}}$ defined by (1.4) with M replaced by $M_T$ satisfies the maximum principle:

$$\begin{aligned} \max _{M_T} \Theta \le \max _{\partial _p M_T}\Theta . \end{aligned}$$

In particular, if $u_1=u_2$ on the boundary $\partial _p M_T$, then $u_1\equiv u_2$ on $M_T$.

As an application of the above maximum principle, we obtain the existence of VT-harmonic maps into a geodesic ball.

Theorem 3

Let $M, N, V, T, B_R(p)$ be as in Theorem 2. Suppose $u_0 \in H^{2, q}(M, N)(q>m)$ with $u_0(M) \subset B_R(p)$. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla T| + \max |T| \le C_0, \end{aligned}$$

then the initial boundary value problem

$$\begin{aligned} \left\{ \begin{array}{rll} \displaystyle &{}\partial _tu= \tau (u) + du(V) + \mathrm{Tr}_g T(du, du), \\ \displaystyle &{} u-u_0 \in H^{2, q}_0(M, N),\quad u(0)=u_0, \quad u(M\times [0,\infty ))\subset B_R(p), \end{array} \right. \end{aligned}$$

(1.6)

admits a unique global solution u which subconverges to a unique solution $u \in H^{2, q}(M, N)$ of the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{rll} \displaystyle &{} \tau (u) + du(V) + \mathrm{Tr}_g T(du, du) = 0, \\ \displaystyle &{} u-u_0 \in H^{2, q}_0(M, N), \end{array} \right. \end{aligned}$$

(1.7)

such that $u(M)\subset B_R(p)$.

Furthermore, based on Theorem 3, we shall also establish the existence of VT-harmonic maps from complete noncompact Riemannian manifolds by using a gradient estimate and the compact exhaustion method.

Theorem 4

Let $(M^m, g)$ be a complete noncompact Riemannian manifold and $(N^n, {\tilde{g}})$ be a complete Riemannian manifold with sectional curvature bounded above by a positive constant $\kappa$. Let $B_R(p)$ be a geodesic ball with radius $R< \frac{\pi }{2(1+\sigma )\sqrt{\kappa }}$ and $u_0: M\rightarrow N$ a smooth map with $u_0(M) \subset B_R(p)$. Suppose $\Vert V\Vert _{L^\infty }< +\infty .$

For appropriate $\sigma$ and R, there exists a constant $C'_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla T| + \max |T| \le C'_0, \end{aligned}$$

then there exists a VT-harmonic map $u \in C^{\infty }(M, N)$ homotopic to $u_0$ such that $u(M)\subset B_R(p)$.

2 Preliminaries

Let us first give some notations:

$$\begin{aligned} s_\kappa (t):= & {} \left\{ \begin{array}{ll} \displaystyle t &{} \quad \kappa =0 \\ \displaystyle \frac{1}{\sqrt{\kappa }}\sin \sqrt{\kappa }t&{} \quad \kappa>0, \end{array} \right. \qquad q_\kappa (t):=\left\{ \begin{array}{ll} \displaystyle \frac{t^2}{2} &{} \quad \kappa =0 \\ \displaystyle \frac{1}{\kappa }(1-\cos \sqrt{\kappa }t)&{} \quad \kappa>0. \end{array} \right. \\ a_\kappa (t):= & {} \left\{ \begin{array}{ll} \displaystyle 0 &{} \quad \mathrm{if} \ \ t=0\\ \displaystyle \frac{1-s'_\kappa (t)}{s_\kappa (t)} &{} \quad \mathrm{if} \,\, t>0, \end{array} \right. \quad \quad b_\kappa (t):=\left\{ \begin{array}{ll} \displaystyle 0 &{}\quad \mathrm{if} \ \ t=0\\ \displaystyle \frac{1-s'_\kappa (t)}{2s_\kappa (t)}\left( 1+ \frac{t}{s_\kappa (t)} \right) \ &{} \quad \mathrm{if} \ \ t>0. \end{array}\right. \end{aligned}$$

In local coordinates $\{ x^\alpha \}$ on M and $\{ y^i \}$ on N, respectively, the energy density of u is

$$\begin{aligned} e(u):= g^{\alpha \beta }{\tilde{g}}_{ij}(u(x))\frac{\partial u^i}{\partial x^\alpha }\frac{\partial u^j}{\partial x^\beta }. \end{aligned}$$

Assume the metric of N satisfies:

$$\begin{aligned} 0<{\widetilde{\lambda }}(y)(\delta _{ij})\le ({\widetilde{g}}_{ij}(y))\le {\widetilde{\Lambda }}(y)(\delta _{ij}), \quad \forall y\in N. \end{aligned}$$

Denote $\lambda :=\min \nolimits _N{{\widetilde{\lambda }}}$ and $\Lambda :=\max \nolimits _N{{\widetilde{\Lambda }}}$

$\forall y_1, y_2 \in B_R(p)$, there exists a unit speed geodesic $\gamma : [0, \rho ] \rightarrow B_R(p)\subset N$ with $\gamma (0)= y_1, \gamma (\rho )=y_2$, where $\rho =\mathrm{dist}(y_1, y_2)$. For any $v_j \in T_{y_j}N, j=1, 2$, let X be the unique Jacobi field along $\gamma$ with $X(0)=v_1, X(\rho )= v_2$. Then, we define a pseudo-distance

$$\begin{aligned} \delta (v_1, v_2):=\left\{ \begin{array}{ll} \displaystyle \left( \rho \int ^{\rho }_{0}|{\dot{X}}|^2 \right) ^{\frac{1}{2}}&{}\quad \mathrm{if} \ \ \rho >0,\\ \displaystyle |v_1 - v_2| &{}\quad \mathrm{if} \ \ \rho =0. \end{array} \right. \end{aligned}$$

Another pseudo-distance is given by

$$\begin{aligned} \delta _0(v_1, v_2) := \left| v_1 - \bar{{\bar{v}}}_2 \right| , \end{aligned}$$

where $\bar{{\bar{v}}}_2 \in T_{y_1}N$ stands for the vector obtained by parallel displacement of $v_2 \in T_{y_2}N$ along $\gamma$. Let $L(T_x M, T_y N)$ be the space of all linear maps from $T_x M$ to $T_y N$. The pseudo-distance $\delta$ on the tangent bundle can be extended to a pseudo-distance on the fibers, that is, for $q_1, q_2 \in \cup _{y\in B_R(p)}L(T_xM, T_yN)$ (disjoint union), we define their pseudo-distance as

$$\begin{aligned} \delta (q_1, q_2) := \left( \sum _{\alpha =1}^{m}\delta ^2(q_1(e_\alpha ), q_2(e_\alpha ))\right) ^{\frac{1}{2}}, \end{aligned}$$

where $\{ e_1, \ldots , e_m \}$ is an orthonormal base for $T_{x}M$.

We have the following relationship between these two pseudo-distances:

Lemma 1

([4]) There is a positive constant C depending only on $B_R(p)$ and the geometry of N such that for any $y_j \in B_R(p)$ and $v_j \in T_{y_j}N, j=1, 2$, we have

$$\begin{aligned} \delta _{0}^2(v_1, v_2) - C(|v_1|^2+|v_2|^2)\rho ^2 \le \delta ^2(v_1, v_2)\le \delta _{0}^2(v_1, v_2) + C(|v_1|^2+|v_2|^2)\rho ^2. \end{aligned}$$

Remark 1

In fact, by the proof in [4] and using a well-known expression of the curvature operator (see, e.g., Lemma 4.3.3 in [12]), it is not hard to see that if the sectional curvature K on $B_R(p)$ satisfies $\theta \le \left. K\right| _{B_R(p)} \le \kappa$ for a constant $\theta < 0$, then the constant C can be expressed as $14(\kappa - \theta )$.

The following estimates will also be important for us:

Lemma 2

([7]) Let (M, g) be a compact Riemannian manifolds with nonempty boundary $\partial M$ and $(N, {\tilde{g}})$ a complete manifold without boundary and $B_R(p)$ a regular ball in N. Let

$$\begin{aligned} \begin{aligned} g_1:=&q_\kappa \circ d(p, \cdot ): B_R(p) \rightarrow {\mathbb {R}},\\ h:=&q_{\frac{\kappa }{4}}\circ d: B_R(p) \times B_R(p) \rightarrow {\mathbb {R}}. \end{aligned} \end{aligned}$$

Then,

$$\begin{aligned} \nabla ^2g_1(u,u) \ge s'_\kappa (\tau )|u|^2 \end{aligned}$$

(2.1)

hold for $u \in T_x N, x \in B_R(p)$ and $\tau := d(p, x)$.

$$\begin{aligned} \nabla ^2h(v, v) \ge -s_{\frac{\kappa }{4}}(\rho )a_\kappa (\rho )\sum _{i=1}^2|v_i|^2 \end{aligned}$$

(2.2)

and

$$\begin{aligned} \nabla ^2h(v,v)\ge s'_{\frac{\kappa }{4}}(\rho )\delta ^2(v_1, v_2) - s_{\frac{\kappa }{4}}(\rho )b_\kappa (\rho )\sum _{i=1}^2|v_i|^2 \end{aligned}$$

(2.3)

holds for $v=v_1\oplus v_2, v_j \in T_{y_j}N, y_j \in B_R(p), j=1, 2, \rho =\mathrm{dist}(y_1, y_2)$.

3 The maximum principle

Proof of Theorem 1

Let

$$\begin{aligned} \begin{aligned} \psi (x):=&q_{\frac{\kappa }{4}}\circ d(u_1(x), u_2(x)),\\ \psi _i(x):=&q_\kappa \circ d(p, u_i(x)), \ \ \ i=1, 2, \\ \varPhi (x):=&\frac{1}{2}\sum _{i=1}^2 \omega \circ \psi _i(x), \ \ \ \mathrm{where} \ \ \ \omega (t):=-\log (q_\kappa ((1+\sigma )R)-t). \end{aligned} \end{aligned}$$

We consider the operator

$$\begin{aligned} {\mathcal {L}}_V(\cdot ):= e^\varPhi \cdot \mathrm{div}(e^{-2\varPhi }\nabla \cdot )+e^{-\varPhi }\cdot V(\cdot ). \end{aligned}$$

By direct computation, we obtain

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_V (\Theta ) =&{\mathcal {L}}_V (e^\varPhi \cdot \psi )=\Delta \psi + \psi (\Delta \varPhi - |\nabla \varPhi |^2) + \psi V(\varPhi ) + V(\psi )\\ =&\Delta _V \psi + \psi (\Delta _V \varPhi - |\nabla \varPhi |^2). \end{aligned} \end{aligned}$$

Define $U, U_1, U_2: M \rightarrow N\times N$ by

$$\begin{aligned} U(x):= (u_1(x), u_2(x)), \ \ \ U_i(x):=(p, u_i(x)), \ \ \ i=1, 2. \end{aligned}$$

Let $\nu :=\frac{\kappa }{4}, h:= q_\nu \circ d, \phi :=q_\kappa \circ d,$ then $\psi =h\circ U, \psi _i=\phi \circ U_i$.

For any $x \in M$, we let ${\tilde{\gamma }}$ be the unique geodesic connecting $u_1(x)$ and $u_2(x)$. Choosing a parallel orthonormal frame $\{ E_i(t) \}$ along ${\tilde{\gamma }}$ with $E_1 = {\tilde{\gamma }}'$, and a local orthonormal frame $\left. \{ e_\alpha \}\right| _{\alpha =1}^{m}$ around x, assuming that $\frac{\partial }{\partial y^i} := a^{j}_iE_j$, we have

$$\begin{aligned} \begin{aligned}&\delta _0(T(du_1(e_\alpha ), du_1(e_\alpha )), T(du_2(e_\alpha ), du_2(e_\alpha )))\\&\quad =\delta _0\left( (u_1)^{i}_{\alpha }(u_1)^{j}_{\alpha }T^{r}_{ij}(u_1)\frac{\partial }{\partial y^r}(u_1), (u_2)^{i}_{\alpha }(u_2)^{j}_{\alpha }T^{r}_{ij}(u_2)\frac{\partial }{\partial y^r}(u_2)\right) \\&\quad = \delta _0\left( (u_1)^{i}_{\alpha }(u_1)^{j}_{\alpha }T^{r}_{ij}(u_1)a^{\mu }_{r}(u_1)E_\mu (u_1), (u_2)^{i}_{\alpha }(u_2)^{j}_{\alpha }T^{r}_{ij}(u_2)a^{\mu }_{r}(u_2)E_\mu (u_2)\right) \\&\quad \le \sum _{\mu } \left| T^{r}_{ij}(u_1)a^{\mu }_r(u_1)(u_1)^{i}_{\alpha }(u_1)^{j}_\alpha - T^{r}_{ij}(u_2)a^{\mu }_r(u_2)(u_2)^{i}_{\alpha }(u_2)^{j}_\alpha \right| \\&\quad \le \sum _{\mu } \left| (T^{r}_{ij}(u_1)-T^{r}_{ij}(u_2))a^{\mu }_r (u_1)(u_1)^{i}_{\alpha }(u_1)^{j}_\alpha \right| \\&\qquad + \sum _{\mu } \left| T^{r}_{ij}(u_2)( a^{\mu }_r (u_1) - a^{\mu }_r (u_2) )(u_1)^{i}_{\alpha }(u_1)^{j}_\alpha \right| \\&\qquad + \sum _{\mu }\left| T^{r}_{ij}(u_2)a^{\mu }_r(u_2) (u_1)^{j}_\alpha \left( (u_1)^{i}_\alpha - (u_2)^{i}_\alpha \right) \right| \\&\qquad + \sum _{\mu }\left| T^{r}_{ij}(u_2)a^{\mu }_r(u_2) (u_2)^{i}_{\alpha } \left( (u_1)^{j}_\alpha - (u_2)^{j}_\alpha \right) \right| . \end{aligned} \end{aligned}$$

Denote $A= (a^{k}_i)$, then $AA^T=G:=({\tilde{g}}_{ik})$. Since

$$\begin{aligned} \sum _{i,k}|a^{k}_i|^2 = \Vert A\Vert ^2 = \mathrm{tr}(AA^T) =\mathrm{tr}(G)\le n\Lambda , \quad \quad \left| (u_1)^{i}_{\alpha }(u_1)^{j}_\alpha \right| \le \frac{1}{\lambda }|du_1|^2, \end{aligned}$$

we then obtain

$$\begin{aligned} \begin{aligned}&\delta _0(T(du_1(e_\alpha ), du_1(e_\alpha )), T(du_2(e_\alpha ), du_2(e_\alpha )))\\&\quad \le \sum _{\mu =1}^n\left( \frac{\sqrt{n\Lambda }}{\lambda } \rho \max | \nabla T| |du_1|^2 + \frac{{\widetilde{C}}_1}{\lambda }\rho \max |T||du_1|^2 \right. \\&\left. \qquad + \frac{\sqrt{n\Lambda }}{\lambda }\max |T|(|du_1|+|du_2|) |du_1-du_2|\right) \\&\quad = \frac{n\sqrt{n\Lambda }}{\lambda } \rho \max | \nabla T| |du_1|^2 + \frac{n{\widetilde{C}}_1}{\lambda }\rho \max |T||du_1|^2 \\&\qquad + \frac{n\sqrt{n\Lambda }}{\lambda }\max |T|(|du_1|+|du_2|) |du_1-du_2|, \end{aligned} \end{aligned}$$

where ${\widetilde{C}}_1>0$ is a constant depending only on the bound of $(da_{r}^\mu )$ on $B_{\frac{\pi }{2\sqrt{\kappa }}}(p)$. By Lemma 1, we get

$$\begin{aligned} \begin{aligned}&\delta (T(du_1(e_\alpha ), du_1(e_\alpha )), T(du_2(e_\alpha ), du_2(e_\alpha )))\\&\quad \le \delta _0(T(du_1(e_\alpha ), du_1(e_\alpha )), T(du_2(e_\alpha ), du_2(e_\alpha ))) + \sqrt{C}\rho (\left| T(du_1(e_\alpha ), du_1(e_\alpha )) \right| \\&\qquad + \left| T(du_2(e_\alpha ), du_2(e_\alpha )) \right| ) \\&\quad \le \frac{n\sqrt{n\Lambda }}{\lambda } \rho \max | \nabla T| \sum _{i=1}^2 |du_i|^2 + \left( \frac{n{\widetilde{C}}_1}{\lambda }+\sqrt{C}\right) \rho \max |T|\sum _{i=1}^2 |du_i|^2\\&\qquad + \frac{n\sqrt{n\Lambda }}{\lambda }\max |T|(|du_1|+|du_2|) |du_1-du_2|, \end{aligned} \end{aligned}$$

where $C=14(\kappa -\theta )$, and the constant $\theta$ is a lower bound of the sectional curvature of N on $B_{\frac{\pi }{2\sqrt{\kappa }}}(p)$. The Cauchy inequality implies that

$$\begin{aligned}&\frac{n\sqrt{n\Lambda }}{\lambda }\max |T||du_i| |du_1-du_2|s_\nu (\rho )\le \frac{\varepsilon _1n^2}{2}|du_1-du_2|^2 \\&\quad + \frac{n\Lambda }{2\varepsilon _1 \lambda ^2}s_\nu ^2(\rho )\max |T|^2 |du_i|^2. \end{aligned}$$

By using the formula (2.13) in [7], it follows that

$$\begin{aligned} \begin{aligned}&\langle (\nabla h)\circ U, -T(dU(e_\alpha ), dU(e_\alpha ))\rangle \\&\quad = -s_\nu (\rho ) \langle (\nabla d)\circ U, T(dU(e_\alpha ), dU(e_\alpha )) \rangle \\&\quad = -s_\nu (\rho ) \langle {\tilde{e}}_1(U)\oplus {\tilde{e}}_2(U), T(du_1(e_\alpha ), du_1(e_\alpha )) \oplus T(du_2(e_\alpha ), du_2(e_\alpha )) \rangle \\&\quad \ge -s_\nu (\rho ) \delta (T(du_1(e_\alpha ), du_1(e_\alpha )), T(du_2(e_\alpha ), du_2(e_\alpha )))\\&\quad \ge -s_\nu (\rho ) \left\{ \frac{n\sqrt{n\Lambda }}{\lambda } \rho \max | \nabla T| \sum _{i=1}^2 |du_i|^2 + \left( \frac{n{\widetilde{C}}_1}{\lambda }+\sqrt{C}\right) \rho \max |T|\sum _{i=1}^2 |du_i|^2 \right\} \\&\qquad -\varepsilon _1 n^2|du_1-du_2|^2 - \frac{n\Lambda }{2\varepsilon _1\lambda ^2}s_\nu ^2(\rho )\max |T|^2 \sum _{i=1}^2|du_i|^2, \end{aligned} \end{aligned}$$

where ${\tilde{e}}_1 = -{\tilde{\gamma }}'(0), {\tilde{e}}_2 = {\tilde{\gamma }}'(\rho )$. Since

$$\begin{aligned} \begin{aligned}&\delta ^{2}_{0}(du_1(e_\alpha ), du_2(e_\alpha )) = \delta ^{2}_0 \left( (u_1)^{i}_\alpha \frac{\partial }{\partial y^i}(u_1), (u_2)^{i}_\alpha \frac{\partial }{\partial y^i}(u_2)\right) \\&\quad = \delta ^{2}_0\left( (u_1)^{i}_\alpha a^{j}_i(u_1)E_j(u_1), (u_2)^{i}_\alpha a^{j}_i(u_2)E_j(u_2) \right) = \sum _j \left[ (u_1)^{i}_\alpha a^{j}_i(u_1) - (u_2)^{i}_\alpha a^{j}_i(u_2) \right] ^2 \\&\quad = \sum _j \left[ (u_1)^{i}_\alpha (a^{j}_i(u_1)-a^{j}_i(u_2)) + a^{j}_i(u_2)((u_1)^{i}_\alpha - (u_2)^{i}_\alpha ) \right] ^2 \\&\quad \ge \frac{1}{2}\sum _j \left[ a^{j}_i(u_2)((u_1)^{i}_\alpha - (u_2)^{i}_\alpha ) \right] ^2 - \sum _j \left[ (u_1)^{i}_\alpha (a^{j}_i(u_1)-a^{j}_i(u_2)) \right] ^2 \\&\quad = \frac{1}{2}\sum _j\sum _{ik}{\tilde{g}}_{ik}(u_2)((u_1)^{i}_\alpha - (u_2)^{i}_\alpha )((u_1)^{k}_\alpha - (u_2)^{k}_\alpha ) - \sum _j \left[ (u_1)^{i}_\alpha (a^{j}_i(u_1)-a^{j}_i(u_2)) \right] ^2 \\&\quad \ge \frac{n\lambda }{2} | (du_1-du_2)(e_\alpha ) |^2 - n{\widetilde{C}}_1 \rho ^2 |du_1(e_\alpha )|^2, \end{aligned} \end{aligned}$$

then by Lemma 1, we have

$$\begin{aligned} \begin{aligned}&\delta ^{2}(du_1(e_\alpha ), du_2(e_\alpha )) \ge \delta ^{2}_{0}(du_1(e_\alpha ), du_2(e_\alpha )) -C\left( |du_1(e_\alpha )|^2+|du_2(e_\alpha )|^2 \right) \rho ^2 \\&\quad \ge \frac{n\lambda }{2} | (du_1-du_2)(e_\alpha ) |^2 - \left( n{\widetilde{C}}_1 +C\right) \rho ^2\sum _{i=1}^2 |du_i(e_\alpha )|^2. \end{aligned} \end{aligned}$$

Namely,

$$\begin{aligned} \delta ^2(du_1, du_2) \ge \frac{n\lambda }{2} | du_1-du_2 |^2 - \left( n{\widetilde{C}}_1 +C\right) \rho ^2\sum _{i=1}^2 |du_i|^2. \end{aligned}$$

(3.1)

Therefore,

$$\begin{aligned} \begin{aligned}&\langle (\nabla h)\circ U, -T(dU(e_\alpha ), dU(e_\alpha ))\rangle \\&\quad \ge -s_\nu (\rho ) \left\{ \frac{n\sqrt{n\Lambda }}{\lambda } \rho \max | \nabla T| \sum _{i=1}^2 |du_i|^2 + \left( \frac{n{\widetilde{C}}_1}{\lambda }+\sqrt{C}\right) \rho \max |T|\sum _{i=1}^2 |du_i|^2 \right\} \\&\qquad -\frac{2n\varepsilon _1}{\lambda }\left[ \delta ^2(du_1, du_2)+ \left( n{\widetilde{C}}_1 +C\right) \rho ^2\sum _{i=1}^2 |du_i|^2\right] - \frac{n\Lambda }{2\varepsilon _1\lambda ^2}s_\nu (\rho )\rho \max |T|^2 |\sum _{i=1}^2du_i|^2 \\&\quad = - s_\nu (\rho )\rho \left\{ \frac{n\sqrt{n\Lambda }}{\lambda } \max | \nabla T| + \left( \frac{n{\widetilde{C}}_1}{\lambda }+\sqrt{C}\right) \max |T| + \frac{n\Lambda }{2\varepsilon _1\lambda ^2}\max |T|^2 \right\} \sum _{i=1}^2 |du_i|^2\\&\qquad -\frac{2n\varepsilon _1}{\lambda }\left[ \delta ^2(du_1, du_2)+ \left( n{\widetilde{C}}_1 +C\right) \rho ^2\sum _{i=1}^2 |du_i|^2\right] . \end{aligned} \end{aligned}$$

The above inequality and (2.3) imply that

$$\begin{aligned} \begin{aligned} \Delta _V \psi =&\Delta _V(h\circ U) = \sum _\alpha \nabla ^2 h(dU(e_\alpha ), dU(e_\alpha )) + \langle (\nabla h)\circ U, \tau _V(U) \rangle .\\ =&\sum _\alpha \nabla ^2 h(dU(e_\alpha ), dU(e_\alpha )) + \sum _\alpha \langle (\nabla h)\circ U, -T(dU(e_\alpha ), dU(e_\alpha ))\rangle \\ \ge&\left( s'_\nu (\rho )-\frac{2n\varepsilon _1}{\lambda }\right) \delta ^2(du_1, du_2) - s_\nu (\rho ) b_\kappa (\rho ) \sum _{i=1}^2 |du_i|^2 \\&- s_\nu (\rho )\rho \left\{ \frac{n\sqrt{n\Lambda }}{\lambda } \max | \nabla T| + \left( \frac{n{\widetilde{C}}_1}{\lambda }+\sqrt{C}\right) \max |T|\right. \\&\left. + \frac{n\Lambda }{2\varepsilon _1\lambda ^2}\max |T|^2 \right\} \sum _{i=1}^2 |du_i|^2\\&-\frac{2n\varepsilon _1}{\lambda }\left( n{\widetilde{C}}_1 +C\right) \rho ^2\sum _{i=1}^2 |du_i|^2. \end{aligned} \end{aligned}$$

Choosing $\varepsilon _1= \frac{\lambda }{2n}\cos (\sqrt{\kappa }R)>0$, then we obtain

$$\begin{aligned} \begin{aligned} \Delta _V \psi \ge&- s_\nu (\rho ) b_\kappa (\rho ) \sum _{i=1}^2 |du_i|^2 - \cos (\sqrt{\kappa }R) \left( n{\widetilde{C}}_1 +C\right) \rho ^2\sum _{i=1}^2 |du_i|^2 \\&- s_\nu (\rho )\rho \left\{ \frac{n\sqrt{n\Lambda }}{\lambda } \max | \nabla T| + \left( \frac{n{\widetilde{C}}_1}{\lambda }+\sqrt{C}\right) \max |T| \right. \\&\left. + \frac{n^2\Lambda }{\lambda ^3\cos (\sqrt{\kappa }R)}\max |T|^2 \right\} \sum _{i=1}^2 |du_i|^2. \end{aligned} \end{aligned}$$

(3.2)

It follows from (2.1) that

$$\begin{aligned} \begin{aligned} \Delta _V \psi _i&= \Delta _V(\phi \circ U_i) = \sum _\alpha \nabla ^2 \phi (dU_i(e_\alpha ), dU_i(e_\alpha )) + \langle (\nabla \phi )\circ U_i, \tau _V(U_i) \rangle \\&\ge s_\kappa '(\rho _i)|du_i|^2 +s_\kappa (\rho _i) \langle (\nabla d)\circ U_i, 0\oplus (-T(du_i(e_\alpha ), du_i(e_\alpha ))) \rangle \\&\ge s_\kappa '(\rho _i)|du_i|^2 -s_\kappa (\rho _i) \left| T(du_i(e_\alpha ), du_i(e_\alpha )) \right| \\&\ge s_\kappa '(\rho _i)|du_i|^2 -s_\kappa (\rho _i) \max |T| |du_i|^2. \end{aligned} \end{aligned}$$

(3.3)

It is easy to check that

$$\begin{aligned} \omega ''=\omega '^2, \quad \quad \omega '\circ \psi _i=\frac{1}{q_\kappa ((1+\sigma )R)-q_\kappa (\rho _i)}. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned}&{\mathcal {L}}_V (\Theta ) = \Delta _V\psi +\psi (\Delta _V\varPhi - |\nabla \varPhi |^2)\\&\quad = \Delta _V\psi + \frac{1}{2}\psi \sum _{i=1}^2 (\omega '\circ \psi _i)\Delta _V \psi _i + \sum _{i=1}^2 \left[ \frac{1}{2}\psi (\omega ''\circ \psi _i) - \frac{1}{4}\psi (\omega '\circ \psi _i)^2\right] |\nabla \psi _i|^2\\&\quad = \Delta _V\psi + \frac{1}{2}\psi \sum _{i=1}^2 (\omega '\circ \psi _i)\Delta _V \psi _i + \frac{\psi }{4}\sum _{i=1}^2(\omega '\circ \psi _i)^2|\nabla \psi _i|^2\\&\quad \ge -\, s_\nu (\rho ) b_\kappa (\rho ) \sum _{i=1}^2 |du_i|^2 - \cos (\sqrt{\kappa }R) \left( n{\widetilde{C}}_1 +C\right) \rho ^2\sum _{i=1}^2 |du_i|^2 \\&\qquad -\, s_\nu (\rho )\rho \left\{ \frac{n\sqrt{n\Lambda }}{\lambda } \max | \nabla T| + \left( \frac{n{\widetilde{C}}_1}{\lambda }+\sqrt{C}\right) \max |T|\right. \\&\left. \qquad +\, \frac{n^2\Lambda }{\lambda ^3\cos (\sqrt{\kappa }R)}\max |T|^2 \right\} \sum _{i=1}^2 |du_i|^2 \\&\qquad +\, \frac{1}{2}\psi \sum _{i=1}^2 \frac{1}{q_\kappa ((1+\sigma )R)-q_\kappa (\rho _i)}\left[ s_\kappa '(\rho _i)|du_i|^2 -s_\kappa (\rho _i) \max |T| |du_i|^2 \right] \\&\quad = \psi \sum _{i=1}^2 \left\{ \frac{s_{\kappa }'(\rho _i)}{2(q_\kappa ((1+\sigma )R)-q_\kappa (\rho _i))} - \frac{b_\kappa (\rho )s_\nu (\rho )}{q_\nu (\rho )} \right. \\&\left. \qquad -\, \frac{s_\nu (\rho )\rho }{q_\nu (\rho )}\left[ \frac{n\sqrt{n\Lambda }}{\lambda } \max | \nabla T| +\left( \frac{n{\widetilde{C}}_1}{\lambda }+\sqrt{C}\right) \max |T| \right. \right. \\&\qquad \left. +\, \frac{n^2\Lambda }{\lambda ^3\cos (\sqrt{\kappa }R)}\max |T|^2 \right] - \frac{s_\kappa (\rho _i)}{2(q_\kappa ((1+\sigma )R)-q_\kappa (\rho _i))}\max |T|\\&\left. \qquad - \frac{\rho ^2}{q_\nu (\rho )}\cos (\sqrt{\kappa }R) \left( n{\widetilde{C}}_1 +C\right) \right\} |du_i|^2 \\&\quad \ge \psi \sum _{i=1}^2 \left\{ \frac{s_{\kappa }'(\rho _i)}{2(q_\kappa ((1+\sigma )R)-q_\kappa (\rho _i))} - \frac{b_\kappa (\rho )s_\nu (\rho )}{q_\nu (\rho )}\right. \\&\left. \qquad - \left[ \frac{2n\sqrt{n\Lambda }}{\lambda } \max | \nabla T| +\left( \frac{2n{\widetilde{C}}_1}{\lambda }+2\sqrt{C}\right) \max |T| \right. \right. \\&\qquad \left. \left. + \frac{2n^2\Lambda }{\lambda ^3\cos (\sqrt{\kappa }R)}\max |T|^2 \right] - \frac{\kappa R^2}{1-\cos (\sqrt{\kappa }R)}\cos (\sqrt{\kappa }R) \left( n{\widetilde{C}}_1 +C\right) \right. \\&\qquad \left. - \frac{\sqrt{\kappa }\sin (\sqrt{\kappa }R)}{2\left[ \cos (\sqrt{\kappa }R)-\cos ((1+\sigma )\sqrt{\kappa }R)\right] }\max |T| \right\} |du_i|^2, \end{aligned} \end{aligned}$$

(3.4)

where we have used the fact that $\frac{s_\nu (\rho )\rho }{q_\nu (\rho )}$ is nonincreasing in (0, 2R] and $\frac{\rho ^2}{q_\nu (\rho )}$ is increasing in (0, 2R]. Direct computation gives us

$$\begin{aligned} \frac{b_\kappa (\rho )s_\nu (\rho )}{q_\nu (\rho )}= \frac{\kappa }{4}(1+\frac{1}{s'_{\frac{\kappa }{4}}(\rho )})(1+\frac{\rho }{s_\kappa (\rho )})=:\alpha (\rho ), \end{aligned}$$

and $\alpha (t)$ is increasing in [0, 2R] and $\beta _R(t)=\frac{s'_\kappa (t) }{2(q_\kappa ((1+\sigma )R)-q_\kappa (t))}$ is increasing in [0, R]. Hence, we obtain

$$\begin{aligned} \begin{aligned}&\frac{s_{\kappa }'(\rho _i)}{2(q_\kappa ((1+\sigma )R)-q_\kappa (\rho _i))} - \frac{b_\kappa (\rho )s_\nu (\rho )}{q_\nu (\rho )} \\&\quad \ge \frac{1}{2q_\kappa ((1+\sigma )R) }- \frac{\kappa }{4}\left( 1+\frac{1}{s'_{\frac{\kappa }{4}}(2R)}\right) \left( 1+\frac{2R}{s_\kappa (2R)}\right) \\&\quad = \frac{\kappa }{2[1-\cos ((1+\sigma )\sqrt{\kappa }R)]}- \frac{\kappa [1+\cos (\sqrt{\kappa }R)][2\sqrt{\kappa }R+\sin (2\sqrt{\kappa }R)]}{4\cos (\sqrt{\kappa }R)\sin (2\sqrt{\kappa }R)}. \end{aligned} \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_V(\Theta )\ge&\psi \sum _{i=1}^2 \left\{ \frac{\kappa }{2[1-\cos ((1+\sigma )\sqrt{\kappa }R)]}- \frac{\kappa [1+\cos (\sqrt{\kappa }R)][2\sqrt{\kappa }R+\sin (2\sqrt{\kappa }R)]}{4\cos (\sqrt{\kappa }R)\sin (2\sqrt{\kappa }R)} \right. \\&\quad - \frac{2n\sqrt{n\Lambda }}{\lambda } \max | \nabla T|\\&\quad - \left( \frac{2n{\widetilde{C}}_1}{\lambda }+2\sqrt{C}+ \frac{\sqrt{\kappa }\sin (\sqrt{\kappa }R)}{2\left[ \cos (\sqrt{\kappa }R)-\cos ((1+\sigma )\sqrt{\kappa }R)\right] }\right) \max |T| \\&\quad \left. - \frac{2n^2\Lambda }{\lambda ^3\cos (\sqrt{\kappa }R)}\max |T|^2 - \frac{\left( n{\widetilde{C}}_1 +C\right) \kappa R^2\cos (\sqrt{\kappa }R)}{1-\cos (\sqrt{\kappa }R)} \right\} |du_i|^2.\\ \end{aligned} \end{aligned}$$

Clearly, by choosing appropriate $\sigma$ and R, we obtain

$$\begin{aligned}&\frac{\kappa }{2[1-\cos ((1+\sigma )\sqrt{\kappa }R)]}- \frac{\kappa [1+\cos (\sqrt{\kappa }R)][2\sqrt{\kappa }R+\sin (2\sqrt{\kappa }R)]}{4\cos (\sqrt{\kappa }R)\sin (2\sqrt{\kappa }R)}\\&\quad - \frac{\left( n{\widetilde{C}}_1 +C\right) \kappa R^2\cos (\sqrt{\kappa }R)}{1-\cos (\sqrt{\kappa }R)} > 0. \end{aligned}$$

Hence, if

$$\begin{aligned} \begin{aligned}&\frac{2n\sqrt{n\Lambda }}{\lambda } \max | \nabla T| +\left( \frac{2n{\widetilde{C}}_1}{\lambda }+2\sqrt{C}+ \frac{\sqrt{\kappa }\sin (\sqrt{\kappa }R)}{2\left[ \cos (\sqrt{\kappa }R)-\cos ((1+\sigma )\sqrt{\kappa }R)\right] }\right) \max |T|\\&\qquad + \frac{2n^2\Lambda }{\lambda ^3\cos (\sqrt{\kappa }R)}\max |T|^2 \\&\quad \le \frac{\kappa }{2[1-\cos ((1+\sigma )\sqrt{\kappa }R)]}- \frac{\kappa [1+\cos (\sqrt{\kappa }R)][2\sqrt{\kappa }R+\sin (2\sqrt{\kappa }R)]}{4\cos (\sqrt{\kappa }R)\sin (2\sqrt{\kappa }R)}\\&\qquad - \frac{\left( n{\widetilde{C}}_1 +C\right) \kappa R^2\cos (\sqrt{\kappa }R)}{1-\cos (\sqrt{\kappa }R)}, \end{aligned} \end{aligned}$$

(3.5)

then we have

$$\begin{aligned} {\mathcal {L}}_V(\Theta ) \ge 0. \end{aligned}$$

(For $\sqrt{\kappa }R \rightarrow 0$, we use the Taylor expansions of $\sin$ and $\cos$ to obtain positive values on the right-hand side of (3.5).) It is easy to see that there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, so that if

$$\begin{aligned} \max |\nabla T| + \max |T| \le C_0, \end{aligned}$$

(3.6)

then (3.5) holds true; consequently, ${\mathcal {L}}_V(\Theta ) \ge 0$. Applying the ordinary maximum principle, we have

$$\begin{aligned} \max _M \Theta \le \max _{\partial M}\Theta . \end{aligned}$$

$\square$

Proof of Theorem 2

We consider a parabolic operator of the form

$$\begin{aligned} \widetilde{{\mathcal {L}}}_V := {\mathcal {L}}_V - e^{-\varPhi }\partial _t. \end{aligned}$$

By using

$$\begin{aligned} \left( \Delta _V -\partial _t\right) \psi= & {} \left( \Delta _V-\partial _t\right) (h\circ U) = \sum _\alpha \nabla ^2 h(dU(e_\alpha ), dU(e_\alpha ))\\&\quad + \langle (\nabla h)\circ U, \tau _V(U) -\partial _t U \rangle \end{aligned}$$

and

$$\begin{aligned} \left( \Delta _V-\partial _t\right) \psi _i= & {} \left( \Delta _V-\partial _t\right) (\phi \circ U_i) = \sum _\alpha \nabla ^2 \phi (dU_i(e_\alpha ), dU_i(e_\alpha )) \\&\quad + \langle (\nabla \phi )\circ U_i, \tau _V(U_i) - \partial _t U_i \rangle , \end{aligned}$$

as in the proof of Theorem 1, we can conclude that $\widetilde{{\mathcal {L}}}_V(\Theta )\ge 0$ on $M_T$. From the parabolic maximum principle, we have

$$\begin{aligned} \max _{M_T} \Theta \le \max _{\partial _p M_T}\Theta . \end{aligned}$$

4 Existence results

Using the maximum principle obtained in the last section, we shall prove the existence of solutions of the Dirichlet problem for $VT-$harmonic maps.

Proof of Theorem 3

Let us choose normal coordinates $\{ y^i \}_{i=1,2,\ldots ,n}$ centered at p, then any $VT-$harmonic map $u: M\rightarrow B_R(p)\subset N$ can be written as

$$\begin{aligned} u=(u^1, \ldots , u^n) \in (H^{2, q}(M))^n \end{aligned}$$

which satisfies the elliptic system

$$\begin{aligned} \Delta _M u^i + \Gamma ^{i}_{jk}(u)\frac{\partial u^j}{\partial x^\alpha }\frac{\partial u^k}{\partial x^\beta }g^{\alpha \beta } + V^\alpha \frac{\partial u^i}{\partial x^\alpha } + T^{i}_{jk}(u)\frac{\partial u^j}{\partial x^\alpha } \frac{\partial u^k}{\partial x^\beta }g^{\alpha \beta } = 0, \ \ \ i=1, 2, \ldots , n. \end{aligned}$$

For simplicity of notation, we write it in a concise form

$$\begin{aligned} \Delta u+ \Gamma (du, du) +du(V) + \mathrm{Tr}_g T(du, du) = 0. \end{aligned}$$

Now we consider the initial boundary value problem for the heat flow of $VT-$harmonic maps

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _t u = \Delta u + \Gamma (du, du) + du(V) + \mathrm{Tr}_g T(du, du), \\ \displaystyle u-u_0 \in H^{2, q}_0(M, N), \quad u(0)=u_0, \\ \displaystyle u(M\times [0, +\infty )) \subset B_R(p). \end{array} \right. \end{aligned}$$

(4.1)

As in the proof of Theorem 3 in [5], by a continuity method that rests on the maximum principle Theorem 2, we can conclude the global existence of a solution u(x, t) of the above flow (4.1). This solution satisfies

$$\begin{aligned} \Vert u(\cdot , t) \Vert _{1+\alpha } \le C(q, M, V, T, N, u_0, R), \quad \forall t\in (0, +\infty ) \end{aligned}$$

for some $\alpha >0$. Consequently, by the parabolic regularity theory, we have the uniform estimate

$$\begin{aligned} \Vert u\Vert _{C^{1+\alpha , 2+\alpha }(M)} \le C. \end{aligned}$$

(4.2)

For $u_1(x, t) = u(x, t), u_2(x, t) = u(x, t+\sigma _1), \sigma _1>0, \forall (x, t)\in M\times (0, +\infty )$, as in the proof of Theorem 2, the function $\Theta$ satisfies

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle (\Delta -\partial _t)(\frac{\Theta }{\sigma _{1}^2}) + \langle V-2\nabla \varPhi , \nabla (\frac{\Theta }{\sigma _{1}^2}) \rangle \ge 0, \\ \displaystyle \left. \Theta \right| _{\partial M} = 0. \end{array} \right. \end{aligned}$$

By the ordinary maximum principle for functions, it follows that (see pp.178–179 in [17])

$$\begin{aligned} \left( \frac{\Theta }{\sigma _{1}^2}\right) \le C(t-t_0)^{-k}, \quad \forall t\ge t_0 \end{aligned}$$

for any positive integer k and some $t_0>0$. Letting $\sigma _1 \rightarrow 0$, then we obtain $|u_t| \rightarrow 0$ as $t\rightarrow +\infty$, from which together with (4.2), we have u subconverges to a $VT-$harmonic map $u_\infty$ satisfying (1.7) and $u_\infty (M)\subset B_R(p)$. $\square$

With the Schauder and higher regularity estimates, we can improve Theorem 3 to the following

Theorem 5

Let $M, N, V, T, B_R(p)$ be as in Theorem 1. Suppose $u_0 \in C^0(M, N)$ with $u_0(M) \subset B_R(p)$. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla T| + \max |T| \le C_0, \end{aligned}$$

(4.3)

then the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \tau (u) + du(V) + \mathrm{Tr}_g T(du, du) = 0, \\ \displaystyle \left. u\right| _{\partial M}= \left. u_0\right| _{\partial M} \end{array} \right. \end{aligned}$$

admits a unique solution $u \in C^\infty (M, N)\cap C^0({\overline{M}}, N)$ such that $u(M)\subset B_R(p)$.

5 Applications

5.1 Weyl harmonic maps (c.f. [14])

Let $(M, [g], {}^W\nabla )$ be a Weyl manifold. According to the definition, there exists a 1-form $\Theta$ such that ${}^W g= \Theta \otimes g$ for any $g \in [g]$. Equivalently, ${}^W\nabla$ is defined by

$$\begin{aligned} {}^W\nabla _{X}Y = \nabla _X Y - \frac{1}{2}\Theta (X)Y - \frac{1}{2}\Theta (Y)X + \frac{1}{2}g(X, Y)\Theta ^{\sharp }, \ \ \ \forall X, Y \in \Gamma (TM), \end{aligned}$$

where $\nabla$ is the Levi-Civita connection and $\Theta ^\sharp$ the vector field dual to $\Theta$ w.r.t. g. Let $\Gamma ^\gamma _{\alpha \beta }, {}^{W}\Gamma ^{\gamma }_{\alpha \beta }$ be the Christoffel symbols corresponding to $\nabla$ and ${}^W\nabla$, respectively.

Let $(N, [{\tilde{g}}], {}^W{\tilde{\nabla }})$ be also a Weyl manifold, and correspondingly, we denote by ${\widetilde{\Theta }}$ the 1-form, and $\Gamma ^{k}_{ij}, {}^{W}{\tilde{\Gamma }}^{k}_{ij}$ are the Christoffel symbols for the Levi-Civita connection ${\tilde{\nabla }}$ and Weyl connection ${}^W{\tilde{\nabla }}$, respectively. Let $u: (M, [g], {}^W\nabla ) \rightarrow (N, [{\tilde{g}}], {}^W{\tilde{\nabla }})$ be the usual smooth map.

Let

$$\begin{aligned}\begin{aligned} V:=&(\Gamma ^\gamma _{\alpha \beta } - {}^W\Gamma ^\gamma _{\alpha \beta } )g^{\alpha \beta }\frac{\partial }{\partial x^\gamma },\\ T({\tilde{X}}, {\tilde{Y}}) :=&- \frac{1}{2}{\widetilde{\Theta }}({\tilde{X}}){\tilde{Y}} - \frac{1}{2}{\widetilde{\Theta }}({\tilde{Y}}){\tilde{X}} + \frac{1}{2}g({\tilde{X}}, {\tilde{Y}}){\widetilde{\Theta }}^{\sharp }, \ \ \ \forall {\tilde{X}}, {\tilde{Y}} \in \Gamma (TN). \end{aligned} \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned} \tau (g, {}^W\nabla , {}^W{\tilde{\nabla }}) =&g^{\alpha \beta }( u^{k}_{\alpha \beta }+ {}^{W}{\tilde{\Gamma }}^{k}_{ij}u^{i}_\alpha u^{j}_\beta - {}^W\Gamma ^\sigma _{\alpha \beta }u^{k}_\sigma )\partial _{y^k} \\ =&[g^{\alpha \beta }( u^{k}_{\alpha \beta }+ \Gamma ^{k}_{ij}u^{i}_\alpha u^{j}_\beta +T^{k}_{ij}u^{i}_\alpha u^{j}_\beta - \Gamma ^{\sigma }_{\alpha \beta }u^{k}_\sigma )+ V^\sigma u^{k}_\sigma ]\partial _{y^k} \\ =&\tau (g, \nabla , {\tilde{\nabla }}) +du(V) + \mathrm{Tr}_g (du, du). \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \tau (g, {}^W\nabla , {}^W{\tilde{\nabla }}) =0 \quad \mathrm{iff} \quad \tau (u) + du(V) + \mathrm{Tr}_g T(du, du) = 0. \end{aligned}$$

Corollary 1

Let $(M, [g], {}^W\nabla )$ be a compact Wey manifold with nonempty boundary $\partial M$ and $(N, [{\tilde{g}}], {}^W{\tilde{\nabla }})$ a complete Weyl manifold with sectional curvature bounded from above by $\kappa \ge 0$. Let $u_0: M \rightarrow N$ be a continuous map with $u_0(M) \subset B_R(p)$, a geodesic ball with radius $R< \frac{\pi }{2(1+\sigma )\sqrt{\kappa }}$. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla {\widetilde{\Theta }}| + \max |{\widetilde{\Theta }}| \le C_0, \end{aligned}$$

then there exists a unique Weyl harmonic map $u: M\rightarrow B_R(p) \subset N$ with $u=u_0$ on $\partial M$.

5.2 Affine harmonic maps (c.f. [9, 10])

Let $(M, g, {\tilde{\nabla }}), (N, h, {\tilde{\nabla }}')$ both be affine manifolds, where ${\tilde{\nabla }}$ is a global flat and torsion-free connection on M and ${\tilde{\nabla }}'$ is a torsion-free connection on N. Then, we have

$$\begin{aligned} \tau (g, {\tilde{\nabla }}, {\tilde{\nabla }}') = g^{\alpha \beta }(u^{k}_{\alpha \beta }+{\tilde{\Gamma }}'^{k}_{ij}u^{i}_{\alpha }u^{j}_\beta )\partial _{y^k}, \end{aligned}$$

where ${\tilde{\Gamma }}'^{k}_{ij}$ are the Christoffel symbols of ${\tilde{\nabla }}'$.

Regarding (M, g) and (N, h) as Riemannian manifolds, let $\Gamma ^{\gamma }_{\alpha \beta }$ and $\Gamma ^{i}_{jk}$ be the Christoffel symbols of the Levi-Civita connections $\nabla$ and $\nabla '$ of (M, g) and (N, h), respectively. We then have the usual tension field

$$\begin{aligned} \tau (g, \nabla , \nabla ' ) = g^{\alpha \beta } (u^{k}_{\alpha \beta } - \Gamma ^{\gamma }_{\alpha \beta }u^{k}_\gamma + \Gamma ^{k}_{ij}u^{i}_\alpha u^{j}_\beta )\partial _{y^k}. \end{aligned}$$

Let

$$\begin{aligned} V:= & {} g^{\alpha \beta }\Gamma ^{\gamma }_{\alpha \beta }\partial _{x^\gamma }, \\ T^{k}_{ij}:= & {} {\tilde{\Gamma }}'^{k}_{ij} - \Gamma ^{k}_{ij}. \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned} \tau (g, {\tilde{\nabla }}, {\tilde{\nabla }}') =\tau (g, \nabla , \nabla ' ) +du(V) + \mathrm{Tr}_g (du, du). \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \tau (g, {\tilde{\nabla }}, {\tilde{\nabla }}') =0 \quad \mathrm{iff} \quad \tau (u) + du(V) + \mathrm{Tr}_g T(du, du) = 0. \end{aligned}$$

Corollary 2

Let $(M, g, {\tilde{\nabla }})$ be a compact affine manifold with nonempty boundary $\partial M$ and $(N, h, {\tilde{\nabla }}')$ a complete affine manifold with sectional curvature bounded from above by $\kappa \ge 0$, where ${\tilde{\nabla }}$ is a global flat and torsion-free connection on M and ${\tilde{\nabla }}'$ is a torsion-free connection on N. Let $u_0: M \rightarrow N$ be a continuous map with $u_0(M) \subset B_R(p)$, a geodesic ball with radius $R< \frac{\pi }{2(1+\sigma )\sqrt{\kappa }}$. Denote $T^{k}_{ij}:= {\tilde{\Gamma }}'^{k}_{ij} - \Gamma ^{k}_{ij}$, where ${\tilde{\Gamma }}'^{k}_{ij}$ and $\Gamma ^{i}_{jk}$ stand for the Christoffel symbols of ${\tilde{\nabla }}'$ and $\nabla '$, respectively. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla T| + \max |T|\le C_0, \end{aligned}$$

then there exists a unique affine harmonic map $u: M\rightarrow B_R(p) \subset N$ with $u=u_0$ on $\partial M$.

5.3 Hermitian harmonic maps (c.f. [11, 15])

Let $(M^m, g, {\tilde{\nabla }}), (N^n, h, {\tilde{\nabla }}')$ are both Hermitian manifolds, where ${\tilde{\nabla }}$ and ${\tilde{\nabla }}'$ are holomorphic torsion-free connections on M and N, respectively. Direct calculation gives us

$$\begin{aligned} \tau (g, {\tilde{\nabla }}, {\tilde{\nabla }}') = g^{\alpha {\bar{\beta }}}\left( \frac{\partial ^2 u^i}{\partial z^\alpha \partial {\bar{z}}^\beta } + \Gamma '^{i}_{jk}\frac{\partial u^{j}}{\partial z^\alpha }\frac{\partial u^k}{\partial {\bar{z}}^\beta } \right) \frac{\partial }{\partial w^i} + \overline{g^{\alpha {\bar{\beta }}}\left( \frac{\partial ^2 u^i}{\partial z^\alpha \partial {\bar{z}}^\beta } + \Gamma '^{i}_{jk}\frac{\partial u^{j}}{\partial z^\alpha }\frac{\partial u^k}{\partial {\bar{z}}^\beta } \right) }\frac{\partial }{\partial {\bar{w}}^i}, \end{aligned}$$

where $\Gamma '^{i}_{jk}$ are the Christoffel symbols of ${\tilde{\nabla }}'$.

Let J be the almost complex structure, and $\{e_A\}=\{e_1, \ldots , e_m, Je_1, \ldots , Je_m\}$ a local basis of M. Let $\nabla , \nabla '$ be the Levi-Civita connections on M and N, respectively, and $\Gamma ^{i}_{jk}$the Christoffel symbols of $\nabla '$. Set

$$\begin{aligned} V:= {\tilde{\nabla }}_{e_A}e_A - \nabla _{e_A}e_A \quad \mathrm{and} \quad T^{i}_{jk} := \Gamma '^{i}_{jk} - \Gamma ^{i}_{jk}, \end{aligned}$$

then we have

$$\begin{aligned} \tau (g, {\tilde{\nabla }}, {\tilde{\nabla }}') = \tau (u) + du(V) + \mathrm{Tr}_g T(du, du). \end{aligned}$$

Namely,

$$\begin{aligned} \tau (g, {\tilde{\nabla }}, {\tilde{\nabla }}') =0 \quad \mathrm{iff} \quad \tau (u) + du(V) + \mathrm{Tr}_g T(du, du) = 0. \end{aligned}$$

Corollary 3

Let $(M^m, g, {\tilde{\nabla }})$ be a compact Hermitian manifold with nonempty boundary $\partial M$ and $(N^n, h, {\tilde{\nabla }}')$ a complete Hermitian manifold with sectional curvature bounded from above by $\kappa \ge 0$, where ${\tilde{\nabla }}$ and ${\tilde{\nabla }}'$ are holomorphic torsion-free connections on M and N, respectively. Let $u_0: M \rightarrow N$ be a continuous map with $u_0(M) \subset B_R(p)$, a geodesic ball with radius $R< \frac{\pi }{2(1+\sigma )\sqrt{\kappa }}$. Denote $T^{k}_{ij}:= \Gamma '^{k}_{ij} - \Gamma ^{k}_{ij}$, where $\Gamma '^{k}_{ij}$ and $\Gamma ^{i}_{jk}$ stand for the Christoffel symbols of ${\tilde{\nabla }}'$ and $\nabla '$, respectively. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla T| + \max |T| \le C_0, \end{aligned}$$

then there exists a unique Hermitian harmonic map $u: M\rightarrow B_R(p) \subset N$ with $u=u_0$ on $\partial M$.

5.4 Magnetic harmonic maps

We now consider a case that, in contrast to the previous ones, does not arise from a structure different from the Riemannian, but from on additional structure on a Riemannian manifold. Let $(\Sigma ^m, g)$ be an m-dimensional compact oriented Riemannian manifold with nonempty boundary, $(N, {\widetilde{g}})$ a Riemannian manifold of dimension n. Let $u: (\Sigma ^m, g)\rightarrow (N, {\widetilde{g}})$ be a map and $Z\in \Gamma (\mathrm{Hom}(\Lambda ^mTN, TN))\cong \Gamma (\Lambda ^mT^*N\otimes TN)$.

Consider the following system:

$$\begin{aligned} \tau (u) + Z(du(e_1) \wedge \cdots \wedge du(e_m)) = 0, \end{aligned}$$

(5.1)

where $\{ e_1, \ldots , e_m \}$ is a positively oriented local orthonormal frame of $\Sigma ^m$. In string theory, it can be interpreted as the motion equation of an $(m-1)$-brane under an extrinsic magnetic force (c.f. [13]). In [13], the author obtained the global existence of the heat flow in one- dimensional case.

Using a similar method as above, in the two-dimensional case, we can obtain the following

Theorem 6

Let $u_1, u_2 \in C^0(\Sigma ^2, N)$ be two magnetic harmonic maps into a geodesic ball $B_{R}(p)$. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla Z| + \max |Z| \le C_0, \end{aligned}$$

then the function $\Theta : \Sigma ^2 \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \Theta := \frac{q_{\frac{\kappa }{4}}(\rho )}{(q_\kappa ((1+\sigma )R)-q_\kappa (\rho _1))^{\frac{1}{2}}\cdot (q_\kappa ((1+\sigma )R)-q_\kappa (\rho _2))^{\frac{1}{2}}} \end{aligned}$$

(5.2)

satisfies the maximum principle, namely

$$\begin{aligned} \max _{\Sigma ^2} \Theta \le \max _{\partial \Sigma ^2}\Theta . \end{aligned}$$

Here $\rho := d(u_1, u_2), \rho _i:= d(p, u_i), i=1,2.$

In particular, if $u_1=u_2$ on the boundary $\partial \Sigma ^2$, then $u_1\equiv u_2$ on $\Sigma ^2$.

For the heat flow of magnetic harmonic maps, an analogous result holds. For $T>0$, we set

$$\begin{aligned} \Sigma ^2_T:= \Sigma ^2\times [0, T] \end{aligned}$$

and denote the parabolic boundary of $\Sigma ^2_T$ by

$$\begin{aligned} \partial _p \Sigma ^2_T:=(\Sigma ^2\times \{0\})\cup (\partial \Sigma ^2\times [0, T]). \end{aligned}$$

For the heat flow of magnetic harmonic maps

$$\begin{aligned} \partial _t u= \tau (u)+Z(du(e_1)\wedge du(e_2)), \end{aligned}$$

(5.3)

we have

Theorem 7

Let $u_1, u_2 \in C^0(\Sigma ^2, N)$ be two solutions of heat flow Eq. (5.3) for magnetic harmonic maps into a geodesic ball $B_{R}(p)$. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla Z| + \max |Z| \le C_0, \end{aligned}$$

then the function $\Theta : \Sigma ^2_T \rightarrow {\mathbb {R}}$ defined by (5.2) with $\Sigma ^2$ replaced by $\Sigma ^2_T$ satisfies the maximum principle:

$$\begin{aligned} \max _{\Sigma ^2_T} \Theta \le \max _{\partial _p \Sigma ^2_T}\Theta . \end{aligned}$$

In particular, if $u_1=u_2$ on the boundary $\partial _p \Sigma ^2_T$, then $u_1\equiv u_2$ on $\Sigma ^2_T$.

As an application of the above maximum principle, we obtain the existence of magnetic harmonic maps into a geodesic ball.

Theorem 8

Let $\Sigma ^2, N, Z, B_R(p)$ be as in Theorem 7. Suppose $u_0 \in H^{2, q}(\Sigma ^2, N)(q>2)$ with $u_0(\Sigma ^2) \subset B_R(p)$. For appropriate $\sigma$ and R, there exists a constant $C_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla Z| + \max |Z| \le C_0, \end{aligned}$$

then the initial boundary value problem

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _tu= \tau (u) +Z(du(e_1)\wedge du(e_2)), \\ \displaystyle u-u_0 \in H^{2, q}_0(\Sigma ^2, N),\quad u(0)=u_0, \quad u(\Sigma ^2\times [0,\infty )) \subset B_R(p), \end{array} \right. \end{aligned}$$

(5.4)

admits a unique global solution u which subconverges to a unique solution $u \in H^{2, q}(\Sigma ^2, N)$ of the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \tau (u) + Z(du(e_1)\wedge du(e_2)) = 0, \\ \displaystyle u-u_0 \in H^{2, q}_0(\Sigma ^2, N), \end{array} \right. \end{aligned}$$

(5.5)

such that $u(\Sigma ^2)\subset B_R(p)$.

Remark 2

In local coordinates $\{ x^\alpha \}$ on M and $\{ y^j \}$ on N, respectively, the term $\mathrm{Tr}_g T(du, du)$ can be written as $T^i_{jk}(u) \frac{\partial u^j}{\partial x^\alpha }\frac{\partial u^k}{\partial x^\beta }g^{\alpha \beta }$, where $T:=T^i_{jk}\frac{\partial }{\partial y^i}\otimes dy^j \otimes dy^k$. Correspondingly, the term $Z(du(e_1)\wedge du(e_2))$ in (5.3) could be written as $(Z^i_{jk}-Z^i_{kj})\frac{\partial u^j}{\partial x^\alpha }\frac{\partial u^k}{\partial x^\beta }g^{\alpha \beta }$. Using $Z^i_{jk}-Z^i_{kj}$ in place of $T^i_{jk}$ in the proof of the results for VT-harmonic maps, we can conclude the above theorems for magnetic harmonic maps.

6 VT-harmonic maps from complete manifolds into geodesic balls

In this section, we shall establish the existence of VT-harmonic maps from complete noncompact manifolds into geodesic balls in complete Riemannian manifolds with sectional curvature bounded above by a positive constant.

Before proving the existence theorem, we first give the following Bochner formula:

Lemma 3

Let $(M^m, g)$ and $(N^n, {\tilde{g}})$ be Riemannian manifolds. Let $\mathrm{Ric}_V:= \mathrm{Ric} - \frac{1}{2}L_V g$, where $\mathrm{Ric}$ is the Ricci curvature of M and $L_V$ is the Lie derivative. Suppose u is a VT-harmonic map from M to N, then

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Delta _V e(u) =&|\nabla du|^2 + \sum _{\alpha =1}^m\langle du(\mathrm{Ric}_V (e_\alpha )), du(e_\alpha ) \rangle \\&- \sum _{\alpha , \beta =1}^m R^N(du(e_\alpha ), du(e_\beta ), du(e_\alpha ), du(e_\beta ))\\&- \sum _{\alpha , \beta =1}^m \langle (\nabla _{e_\alpha }T)(du(e_\beta ), du(e_\beta )), du(e_\alpha )\rangle \\&- \sum _{\alpha , \beta =1}^m \langle 2T((\nabla _{e_\alpha }du)(e_\beta ), du(e_\beta )), du(e_\alpha ) \rangle , \end{aligned} \end{aligned}$$

(6.1)

where $\{e_\alpha \}$ is a local orthonormal frame of M.

Proof

By the proof of Proposition 1.3.5 in [18] (c.f. also in [1]), we have

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Delta e(u) =&|\nabla du|^2 + \sum _{\alpha =1}^m \langle \nabla _{e_\alpha }\tau (u), du(e_\alpha ) \rangle + \sum _{\alpha =1}^m\langle du(\mathrm{Ric} (e_\alpha )), du(e_\alpha ) \rangle \\&- \sum _{\alpha , \beta =1}^m R^N(du(e_\alpha ), du(e_\beta ), du(e_\alpha ), du(e_\beta )). \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&\sum _{\alpha =1}^m\langle \nabla _{e_\alpha }du(V), du(e_\alpha ) \rangle = \sum _{\alpha =1}^m\langle (\nabla _{e_\alpha }du)(V)+ du(\nabla _{e_\alpha }V), du(e_\alpha ) \rangle \\&\quad = \sum _{\alpha =1}^m\langle (\nabla _{V}du)(e_\alpha )+ du(\nabla _{e_\alpha }V), du(e_\alpha ) \rangle = \sum _{\alpha =1}^m \langle \nabla _V du(e_\alpha ) + du(\nabla _{e_\alpha }V), du(e_\alpha ) \rangle \\&\quad = \frac{1}{2}V|du|^2 + \sum _{\alpha , \beta =1}^m \langle \nabla _{e_\alpha }V, e_\beta \rangle \langle du(e_\beta ), du(e_\alpha ) \rangle \\&\quad = \frac{1}{2}V|du|^2 + \frac{1}{2}\sum _{\alpha , \beta =1}^m (L_V g)(e_\alpha , e_\beta )\langle du(e_\alpha , du(e_\beta )) \rangle \end{aligned}, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\nabla _{e_\alpha }(\mathrm{Tr}_g T(du,du)) = \nabla _{e_\alpha } (T(du(e_\beta ), du(e_\beta ))) \\&\quad = (\nabla _{e_\alpha }T)(du(e_\beta ), du(e_\beta ))+ 2T((\nabla _{e_\alpha }du)(e_\beta ), du(e_\beta )). \end{aligned} \end{aligned}$$

Therefore, we get

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Delta e(u) =&|\nabla du|^2 - \frac{1}{2}V|du|^2 - \frac{1}{2}\sum _{\alpha , \beta =1}^m (L_V g)(e_\alpha , e_\beta )\langle du(e_\alpha , du(e_\beta )) \rangle \\&- \sum _{\alpha ,\beta =1}^m \langle (\nabla _{e_\alpha }T)(du(e_\beta ), du(e_\beta ))+ 2T((\nabla _{e_\alpha }du)(e_\beta ), du(e_\beta )), du(e_\alpha ) \rangle \\&+ \sum _{\alpha =1}^m\langle du(\mathrm{Ric} (e_\alpha )), du(e_\alpha ) \rangle - \sum _{\alpha , \beta =1}^m R^N(du(e_\alpha ), du(e_\beta ), du(e_\alpha ), du(e_\beta )), \end{aligned} \end{aligned}$$

which implies that (6.1) holds. $\square$

Using the above Bochner formula and the estimate of $\Delta _V r$ in [6] (here r denotes the distance function on M), we establish the gradient estimate for VT-harmonic maps.

Theorem 9

Let $(M^m, g)$ be a complete noncompact Riemannian manifold with

$$\begin{aligned} \mathrm{Ric}_V:= \mathrm{Ric} - \frac{1}{2}L_Vg \ge -A, \end{aligned}$$

where $A \ge 0$ is a constant, $\mathrm{Ric}$ is the Ricci curvature of M and $L_V$ is the Lie derivative. Let $(N^n, {\widetilde{g}})$ be a complete Riemannian manifold with sectional curvature bounded from above by a positive constant $\kappa$. Let $u: M\rightarrow N$ be a VT-harmonic map such that $u(M) \subset B_{{\widetilde{R}}}(p)$, where $B_{{\widetilde{R}}}(p)$ is a regular ball in N, i.e., disjoint from the cut locus of p and ${\widetilde{R}} < \frac{\pi }{2\sqrt{\kappa }}$. Suppose $\Vert V\Vert _{L^\infty }< +\infty , \Vert T \Vert _{L^\infty }<+\infty , \Vert \nabla T \Vert _{L^\infty }<+\infty$ and

$$\begin{aligned} \left( 1+(m+1)^2 - \frac{1}{(m+1)^2} \right) \Vert T\Vert ^2_{L^\infty } + \frac{\sqrt{\kappa }}{\cos (\sqrt{\kappa }\widetilde{R)}}\Vert T\Vert _{L^\infty } < \frac{\kappa }{\min \{m, n\}}. \end{aligned}$$

(6.2)

Then, we have

$$\begin{aligned} |\nabla u|\le C_6(\sqrt{A} + 1 ), \end{aligned}$$

where $C_6>0$ is a constant depending only on $m, n, \kappa , {\tilde{R}}, V, T$.

Proof

Let $r, \rho$ be the respective distance functions on M and N from some fixed points ${\tilde{p}}\in M, p \in N$. Let $B_a({\tilde{p}})$ be a geodesic ball of radius a around ${\tilde{p}}$. Define $\varphi :=\cos (\sqrt{\kappa }\rho )$. Then, the Hessian comparison theorem implies

$$\begin{aligned} \mathrm{Hess}^N(\varphi ) \le -\kappa (\cos (\sqrt{\kappa }\rho )){\widetilde{g}}. \end{aligned}$$

(6.3)

Define $f: B_a({\tilde{p}}) \rightarrow {\mathbb {R}}$ by

$$\begin{aligned} f:= (a^2-r^2)\frac{|\nabla u|}{\varphi \circ u}. \end{aligned}$$

Denote $\psi := \frac{|\nabla u|}{\varphi \circ u}$. Clearly, f achieves its maximum at some interior point of $B_a({\tilde{p}})$, say q. WLOG, we assume that $\nabla u(q) \ne 0$. Then, from

$$\begin{aligned} \begin{aligned} \nabla f(q) =&0, \\ \Delta _V f(q) \le&0, \end{aligned} \end{aligned}$$

we obtain at q:

$$\begin{aligned}&\frac{\nabla r^2}{a^2-r^2}=\frac{\nabla \psi }{\psi }, \end{aligned}$$

(6.4)

$$\begin{aligned}&\frac{\Delta _V\psi }{\psi } - \frac{\Delta _V r^2}{a^2-r^2} - \frac{2\langle \nabla r^2, \nabla \psi \rangle }{(a^2-r^2)\psi }\le 0. \end{aligned}$$

(6.5)

It follows from the above two inequalities that

$$\begin{aligned} \frac{\Delta _V\psi }{\psi } - \frac{\Delta _V r^2}{a^2-r^2} - \frac{2|\nabla r^2|^2}{(a^2-r^2)^2}\le 0. \end{aligned}$$

(6.6)

By formula (2.4) in [6] (see also [16]), we have

$$\begin{aligned} \Delta _Vr^2 = 2r\Delta _V r + 2|\nabla r|^2 \le 2r(A(r-r_0)+{\widetilde{C}}_0)+2, \end{aligned}$$

(6.7)

where $r_0>0$ is a sufficiently small constant and ${\widetilde{C}}_0 := \max _{\partial B_{r_0}({\tilde{p}})}\Delta _V r$.

Let $\{e_\alpha \}$ be a local orthonormal frame field of M and s the rank of u at the point. We shall compute in normal coordinates at the considered point of N. By Newton’s inequality, we get

$$\begin{aligned} \begin{aligned}&\sum _{\alpha , \beta }R^N(du(e_\alpha ), du(e_\beta ), du(e_\alpha ), du(e_\beta )) = \sum _{\alpha \ne \beta }R^N(du(e_\alpha ), du(e_\beta ), du(e_\alpha ), du(e_\beta ))\\&\quad \le 2\kappa \sum _{1\le \alpha < \beta \le s}\left( \sum _i(u^i_{\alpha })^2\right) \left( \sum _j(u^j_{\beta })^2\right) \le 2 \kappa \cdot \frac{s(s-1)}{2} \cdot \frac{1}{s^2} \left( \sum _{\alpha , i} (u^i_{\alpha })^2 \right) ^2 \\&\quad = \frac{s-1}{s}\kappa |du|^4 \le \frac{s_0-1}{s_0}\kappa |du|^4, \end{aligned} \end{aligned}$$

where we have used the fact that $s_0:=\min \{m, n\}\ge s$ in the third $"\le "$.

The Cauchy–Schwarz inequality gives us

$$\begin{aligned}&\left| \langle (\nabla _{e_\alpha }T)(du(e_\beta ), du(e_\beta )), du(e_\alpha ) \rangle \right| \le \Vert \nabla T\Vert _{L^\infty } |du|^2 \cdot |du| \\&\quad \le \varepsilon _2 e(u) + \frac{1}{4\varepsilon _2}\Vert \nabla T\Vert ^2_{L^\infty } e(u)^2, \\&\left| \langle 2T((\nabla _{e_\alpha }du)(e_\beta ), du(e_\beta )), du(e_\alpha ) \rangle \right| \le 2\Vert T\Vert _{L^\infty } |\nabla du||du| \cdot |du| \le \varepsilon _3 |\nabla du|^2 \\&\quad + \frac{1}{\varepsilon _3}\Vert T\Vert ^2_{L^\infty } e(u)^2. \end{aligned}$$

The formula (3.12) in [2] (see also [3]) implies that for any $\epsilon >0$

$$\begin{aligned} |\nabla du|^2 \ge \frac{1-\epsilon }{m-1}|\tau (u)|^2 + \left( \frac{m}{m-1}- \frac{1}{(m-1)\epsilon } \right) |\nabla \sqrt{e(u)}|^2. \end{aligned}$$

Choosing $\epsilon = m$, then we have

$$\begin{aligned} |\nabla du|^2 \ge -|\tau (u)|^2 + \left( 1+ \frac{1}{m} \right) |\nabla \sqrt{e(u)}|^2. \end{aligned}$$

By the VT-harmonic map equation (1.1), it is easy to see that

$$\begin{aligned} |\tau (u)|^2 \le e(u)\Vert V\Vert ^2_{L^\infty } + e(u)^2\Vert T\Vert ^2_{L^\infty }. \end{aligned}$$

Hence, from the Bochner formula (6.1), we obtain

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Delta _V e(u) \ge&\left( -(1-\varepsilon _3) \Vert V\Vert ^2_{L^\infty } -A - \varepsilon _2\right) e(u) +(1-\varepsilon _3) \left( 1+ \frac{1}{m} \right) |\nabla \sqrt{e(u)}|^2 \\&+\left( -(1-\varepsilon _3) \Vert T\Vert ^2_{L^\infty } - \kappa \left( 1- \frac{1}{s_0}\right) - \frac{1}{4\varepsilon _2}\Vert \nabla T\Vert ^2_{L^\infty } -\frac{1}{\varepsilon _3}\Vert T\Vert ^2_{L^\infty }\right) e(u)^2. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} \frac{1}{2}\Delta _V e(u) = \frac{1}{2}\Delta _V |\nabla u|^2 = |\nabla |\nabla u||^2 + |\nabla u|\Delta _V |\nabla u|, \end{aligned}$$

therefore

$$\begin{aligned} \begin{aligned} \Delta _V |\nabla u| \ge&\left[ (1-\varepsilon _3) \left( 1+ \frac{1}{m} \right) -1\right] \frac{ |\nabla |\nabla u||^2 }{|\nabla u|} + \left( -(1-\varepsilon _3) \Vert V\Vert ^2_{L^\infty } -A - \varepsilon _2\right) |\nabla u|\\&+\left[ -(1-\varepsilon _3) \Vert T\Vert ^2_{L^\infty } - \kappa \left( 1- \frac{1}{s_0}\right) - \frac{1}{4\varepsilon _2}\Vert \nabla T\Vert ^2_{L^\infty } -\frac{1}{\varepsilon _3}\Vert T\Vert ^2_{L^\infty }\right] |\nabla u|^3. \end{aligned} \end{aligned}$$

Choose $\varepsilon _3=\frac{1}{(m+1)^2}$, then

$$\begin{aligned} \begin{aligned} \Delta _V |\nabla u| \ge&\frac{1}{m+1} \frac{ |\nabla |\nabla u||^2 }{|\nabla u|} + \left[ -\left( 1-\frac{1}{(m+1)^2}\right) \Vert V\Vert ^2_{L^\infty } -A - \varepsilon _2\right] |\nabla u|\\&+\left[ -\left( 1-\frac{1}{(m+1)^2}\right) \Vert T\Vert ^2_{L^\infty } - \kappa \left( 1- \frac{1}{s_0}\right) \right. \\&\left. - \frac{1}{4\varepsilon _2}\Vert \nabla T\Vert ^2_{L^\infty } -(m+1)^2\Vert T\Vert ^2_{L^\infty }\right] |\nabla u|^3. \end{aligned} \end{aligned}$$

(6.8)

For simplicity in the following computation, we denote

$$\begin{aligned} \begin{aligned} C_1:=&\frac{1}{m+1}, \ \ C_2:= -\left( 1-\frac{1}{(m+1)^2}\right) \Vert V\Vert ^2_{L^\infty } -A - \varepsilon _2, \\ C_3:=&-\left( 1-\frac{1}{(m+1)^2}\right) \Vert T\Vert ^2_{L^\infty } - \kappa \left( 1- \frac{1}{s_0}\right) - \frac{1}{4\varepsilon _2}\Vert \nabla T\Vert ^2_{L^\infty } -(m+1)^2\Vert T\Vert ^2_{L^\infty }. \end{aligned} \end{aligned}$$

By direct calculation, we have

$$\begin{aligned} \begin{aligned} \nabla \psi =&\frac{\nabla |\nabla u|}{\varphi \circ u} - \frac{|\nabla u|\nabla (\varphi \circ u)}{(\varphi \circ u)^2}, \\ \Delta _V \psi =&\frac{\Delta _V |\nabla u|}{\varphi \circ u} - \frac{|\nabla u|\Delta _V(\varphi \circ u)}{(\varphi \circ u)^2} - \frac{2}{\varphi \circ u}\langle \nabla \psi , \nabla (\varphi \circ u) \rangle \\ \ge&C_1 \frac{ |\nabla |\nabla u||^2 }{(\varphi \circ u)|\nabla u|} + C_2 \frac{|\nabla u|}{\varphi \circ u}+ C_3 \frac{|\nabla u|^3}{\varphi \circ u} - \frac{\psi \Delta _V (\varphi \circ u)}{\varphi \circ u} - \frac{2\langle \nabla \psi , \nabla (\varphi \circ u) \rangle }{\varphi \circ u}. \end{aligned} \end{aligned}$$

The Cauchy–Schwarz implies that

$$\begin{aligned} \begin{aligned}&- \frac{2\langle \nabla \psi , \nabla (\varphi \circ u) \rangle }{\varphi \circ u}\\&\quad = -(2-2C_1)\frac{\langle \nabla \psi , \nabla (\varphi \circ u)\rangle }{\varphi \circ u} - 2C_1 \frac{\langle \nabla \psi , \nabla (\varphi \circ u) \rangle }{\varphi \circ u} \\&\quad = -(2-2C_1)\frac{\langle \nabla \psi , \nabla (\varphi \circ u)\rangle }{\varphi \circ u} - 2C_1 \frac{\langle \nabla |\nabla u|, \nabla (\varphi \circ u) \rangle }{(\varphi \circ u)^2} + 2C_1 \frac{|\nabla (\varphi \circ u)|^2|\nabla u|}{(\varphi \circ u)^3} \\&\quad \ge -(2-2C_1)\frac{\langle \nabla \psi , \nabla (\varphi \circ u)\rangle }{\varphi \circ u} - C_1 \frac{ |\nabla |\nabla u||^2 }{(\varphi \circ u)|\nabla u|} + C_1 \frac{|\nabla (\varphi \circ u)|^2|\nabla u|}{(\varphi \circ u)^3}. \end{aligned} \end{aligned}$$

From the above two inequalities and (6.4), we get

$$\begin{aligned} \begin{aligned} \frac{\Delta _V \psi }{\psi } \ge&C_2 + C_3 |\nabla u|^2 - \frac{\Delta _V (\varphi \circ u)}{\varphi \circ u} +C_1 \frac{|\nabla (\varphi \circ u)|^2}{(\varphi \circ u)^2} - (2-2C_1)\frac{\langle \nabla r^2, \nabla (\varphi \circ u)\rangle }{(a^2-r^2)(\varphi \circ u)}. \end{aligned}\nonumber \\ \end{aligned}$$

(6.9)

Since

$$\begin{aligned} \left| \frac{\langle \nabla r^2, \nabla (\varphi \circ u)\rangle }{(a^2-r^2)(\varphi \circ u)} \right| \le \frac{2r|\nabla (\varphi \circ u)|\cdot |\nabla r|}{(a^2-r^2)(\varphi \circ u)} \le \frac{2r|\nabla (\varphi \circ u)|}{(a^2-r^2)(\varphi \circ u)} \le \frac{2\sqrt{\kappa }r|\nabla u|}{(a^2-r^2)(\varphi \circ u)}\nonumber \\ \end{aligned}$$

(6.10)

and from (6.3),

$$\begin{aligned} \begin{aligned} \Delta _V(\varphi \circ u) =&g^{\alpha \beta }\nabla ^2 \varphi (\partial _\alpha u, \partial _\beta u) + \langle (\nabla \varphi )\circ u, \tau (u) + du(V) \rangle \\ =&g^{\alpha \beta }u^j_{x^\alpha }u^k_{x^{\beta }}\mathrm{Hess}^N(\varphi )\left( \frac{\partial }{\partial y^j}, \frac{\partial }{\partial y^k} \right) + \langle (\nabla \varphi )\circ u, -\mathrm{Tr}_gT(du, du) \rangle \qquad \\ \le&\left( -\kappa \cos (\sqrt{\kappa }\rho ) + \sqrt{\kappa }\Vert T\Vert _{L^\infty } \right) |\nabla u|^2. \end{aligned} \end{aligned}$$

(6.11)

Therefore, from (6.6), (6.7), (6.9)–(6.11), we obtain

$$\begin{aligned} \begin{aligned}&C_2+\left( C_3+\kappa -\frac{\sqrt{\kappa }\Vert T\Vert _{L^\infty }}{\cos (\sqrt{\kappa }\rho \circ u)} \right) |\nabla u|^2 - \frac{4(1-C_1)\sqrt{\kappa }r}{(a^2-r^2)(\varphi \circ u)}|\nabla u|\\&\quad - \frac{2Ar^2-2Ar_0r+2{\widetilde{C}}_0 r+2}{a^2-r^2} - \frac{8r^2}{(a^2-r^2)^2} \le 0. \end{aligned} \end{aligned}$$

(6.12)

Since the condition (6.2) tells us

$$\begin{aligned} \left( 1+(m+1)^2 - \frac{1}{(m+1)^2} \right) \Vert T\Vert ^2_{L^\infty } + \frac{\sqrt{\kappa }}{\cos (\sqrt{\kappa }\widetilde{R)}}\Vert T\Vert _{L^\infty } < \frac{\kappa }{s_0}, \end{aligned}$$

there exists a constant $\varepsilon _0>0$, such that

$$\begin{aligned} \left( 1+(m+1)^2 - \frac{1}{(m+1)^2} \right) \Vert T\Vert ^2_{L^\infty } + \frac{\sqrt{\kappa }}{\cos (\sqrt{\kappa }\widetilde{R)}}\Vert T\Vert _{L^\infty }+\varepsilon _0 < \frac{\kappa }{s_0}. \end{aligned}$$

Choosing $\varepsilon _2= \frac{\Vert \nabla T\Vert ^2_{L^\infty }+1}{4\varepsilon _0}$, then we have

$$\begin{aligned} \begin{aligned}&C_3+\kappa -\frac{\sqrt{\kappa }\Vert T\Vert _{L^\infty }}{\cos (\sqrt{\kappa }\rho \circ u)}= -\left( 1-\frac{1}{(m+1)^2}\right) \Vert T\Vert ^2_{L^\infty } - \kappa \left( 1- \frac{1}{s_0}\right) \\&\qquad - \frac{1}{4\varepsilon _2}\Vert \nabla T\Vert ^2_{L^\infty } -(m+1)^2\Vert T\Vert ^2_{L^\infty } +\kappa -\frac{\sqrt{\kappa }\Vert T\Vert _{L^\infty }}{\cos (\sqrt{\kappa }\rho \circ u)}\\&\quad \ge \frac{\kappa }{s_0} -\left( 1-\frac{1}{(m+1)^2}\right) \Vert T\Vert ^2_{L^\infty } - \frac{1}{4\varepsilon _2}\Vert \nabla T\Vert ^2_{L^\infty } -(m+1)^2\Vert T\Vert ^2_{L^\infty } -\frac{\sqrt{\kappa }\Vert T\Vert _{L^\infty }}{\cos (\sqrt{\kappa }{\widetilde{R}})}\\&\quad = \frac{\kappa }{s_0} -\left[ \left( 1+(m+1)^2- \frac{1}{(m+1)^2} \right) \Vert T\Vert ^2_{L^\infty } + \frac{1}{4\varepsilon _2}\Vert \nabla T\Vert ^2_{L^\infty } + \frac{\sqrt{\kappa }\Vert T\Vert _{L^\infty }}{\cos (\sqrt{\kappa }{\widetilde{R}})} \right] \\&\quad> \frac{\kappa }{s_0} -\left[ \left( 1+(m+1)^2- \frac{1}{(m+1)^2} \right) \Vert T\Vert ^2_{L^\infty } + \varepsilon _0 + \frac{\sqrt{\kappa }\Vert T\Vert _{L^\infty }}{\cos (\sqrt{\kappa }{\widetilde{R}})} \right] =: C_4 >0. \end{aligned} \end{aligned}$$

Therefore, it follows from (6.12) that

$$\begin{aligned} \begin{aligned}&C_4|\nabla u|^2 - \frac{4(1-C_1)\sqrt{\kappa }r}{(a^2-r^2)(\varphi \circ u)}|\nabla u| -\left( \frac{2Ar^2-2Ar_0r+2{\widetilde{C}}_0 r+2}{a^2-r^2} + \frac{8r^2}{(a^2-r^2)^2}-C_2\right) \\&\quad \le 0. \end{aligned} \end{aligned}$$

Note the elementary fact that if $ax^2-bx -c \le 0$ with a, b, c all positive, then

$$\begin{aligned} x \le \frac{b}{a} + \sqrt{\frac{c}{a}}. \end{aligned}$$

Hence, at the point q,

$$\begin{aligned} \begin{aligned} |\nabla u| \le&\frac{4(1-C_1)\sqrt{\kappa }r}{C_4(a^2-r^2)\cos (\sqrt{\kappa } {\widetilde{R}})} +\sqrt{\frac{1}{C_4}\left( \frac{2Ar^2-2Ar_0r+2{\widetilde{C}}_0 r+2}{a^2-r^2} + \frac{8r^2}{(a^2-r^2)^2} -C_2 \right) }. \end{aligned} \end{aligned}$$

From this, we can derive the upper bound of f, and it is easy to conclude that at every point of $B_{\frac{a}{2}}({\tilde{p}})$, we have

$$\begin{aligned} \begin{aligned} |\nabla u| \le&C_5 \left( \sqrt{A+\frac{\Vert \nabla T\Vert ^2_{L^\infty }+1}{4\varepsilon _0}+ \left( 1- \frac{1}{(m+1)^2}\right) \Vert V\Vert _{L^\infty }} + \frac{1}{a} +\frac{1}{\sqrt{a}} \right) \\ \le&C_6 \left( \sqrt{A} + 1 + \frac{1}{a}+ \frac{1}{\sqrt{a}} \right) , \end{aligned} \end{aligned}$$

(6.13)

where $C_6>0$ is a constant depending only on $m, n, \kappa , {\widetilde{R}}, V, T$.

For any fixed $x\in M$, letting $a\rightarrow \infty$ in (6.13), we obtain $|\nabla u| \le C_6(\sqrt{A} + 1 )$. $\square$

Proof of Theorem 4

We first choose a constant $C'_0$ depending only on $\kappa , \sigma , R$ and the geometry of N, such that if

$$\begin{aligned} \max |\nabla T| + \max |T| \le C'_0, \end{aligned}$$

then both the condition (4.3) in Theorem 5 and the condition (6.2) in Theorem 9 hold. Let $\{\Omega _i\}$ be a compact exhaustion of M. By Theorem 5, we have a sequence of maps $\{u_i\}$ which solve the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{rll} \displaystyle &{} \tau (u_i) + du_i(V) + \mathrm{Tr}_g T(du_i, du_i) = 0, \\ \displaystyle &{} \left. u_i\right| _{\partial \Omega _i}= \left. u_0\right| _{\partial \Omega _i}, \\ &{} u_i \ \ \mathrm{homotopic} \ \ \mathrm{to} \ \ u_0 \ \ \mathrm{rel.} \ \ \partial \Omega _i, \end{array} \right. \end{aligned}$$

where $u_i \in C^\infty (\Omega _i, N)\cap C^0(\overline{\Omega _i}, N)$ such that $u_i(\Omega _i)\subset B_R(p)$.

For any compact set $K \subset M$, there exists an integer $i_0 >0$, such that $K\subset \Omega _i$ for $i>i_0$. Then, by (6.8),

$$\begin{aligned} \begin{aligned} \Delta _V |\nabla u_i| \ge&\frac{1}{m+1} \frac{ |\nabla |\nabla u_i||^2 }{|\nabla u_i|} + \left[ -\left( 1-\frac{1}{(m+1)^2}\right) \Vert V\Vert ^2_{L^\infty } -{\widetilde{A}} - \frac{\Vert \nabla T\Vert ^2_{L^\infty }+1}{4\varepsilon _0}\right] |\nabla u_i|\\&+\left[ -\left( 1+(m+1)^2-\frac{1}{(m+1)^2}\right) \Vert T\Vert ^2_{L^\infty } - \kappa \left( 1- \frac{1}{s_0}\right) - \varepsilon _0 \right] |\nabla u_i|^3, \end{aligned}\nonumber \\ \end{aligned}$$

(6.14)

where ${\widetilde{A}}$ is a positive constant depending only on the bounds for Ricci curvature of K and $\Vert V\Vert _{C^1(K)}$.

Since K is compact, there exist finitely many such geodesic balls $\{B_{a_j}({\tilde{p}}_j)\}_{j=1}^{k_0}\subset M$, such that $\cup _{j=1}^{k_0}B_{a_j}({\tilde{p}}_j)\supset K$. Hence, for any $q\in K$, there is a geodesic ball, say $B_{a_{j_0}}({\tilde{p}}_{j_0}) \ \ (1\le j_0 \le k_0)$, containing q. Then, by Theorem 9, we can conclude that

$$\begin{aligned} |\nabla u_i|(q) \le C_{7j_0}(\sqrt{{\widetilde{A}}}+1). \end{aligned}$$

Hence,

$$\begin{aligned} \sup _{K}|\nabla u_i| \le \max _{1\le j_0 \le k_0} \left\{ C_{7j_0}(\sqrt{{\widetilde{A}}}+1) \right\} =: C_8, \end{aligned}$$

where $C_8$ is a positive constant independent of i. Then, by the standard elliptic theory, $u_i$ subconverges to a VT-harmonic map $u \in C^\infty (M, N)$ with $u(M)\subset B_R(p)$ and u is homotopic to $u_0$. $\square$