1 Introduction

Bounded symmetric domains were introduced by Cartan [1, 2] and were systematically studied by Hua, Look, Siegel and Xin [411]. Those are important Cartan–Hadamard manifolds whose further geometrical and analytical properties should be explored and discovered. By the work done by Wong [8], a bounded symmetric domain can be viewed as a pseudo-Grassmannian manifold of all the spacelike subspaces of dimension \(m\) in pseudo-Euclidean space \(R^{m+n}_n\) of index \(n\). Let \(C^{n+m}_m\) be an \((n + m)\)-dimensional complex vector space. For any \(u=(x,y)=(x_1,\ldots ,x_n,x_{n+1},\ldots ,x_{n+m})\in C^{n+m}_m\), define a pseudo-Euclidean inner product

$$\begin{aligned} \langle u,u\rangle =|x_1|^2+\cdots +|x_n|^2-|x_{n+1}|^2-\cdots -|x_{n+m}|^2. \end{aligned}$$

Let \(A\) be an \(n\)-subspace. If the induced inner product on \(A\) is positive definite, \(A\) is called a spacelike \(n\)-subspace. The set of all the \(n\)-dimensional spacelike subspaces forms a pseudo-Grassmannian manifold \(G^m_{n,m}(C)\). \(\mathfrak {R}_I(n,m)\), the first type of bounded symmetric domains, can be identified with \(G^m_{n,m}(C)\). The Bergman metric corresponds to the canonical metric in \(G^m_{n,m}(C)\). The second and third types of bounded symmetric domains \(\mathfrak {R}_{II}(n),\mathfrak {R}_{III}(n)\) are totally geodesic submanifolds of \(G^m_{n,m}(C)\). The fourth type of bounded symmetric domains \(\mathfrak {R}_{IV}(n)\) is identified with \(G^2_{n,2}\) (see [8]).

Harmonic maps between Riemannian manifolds are defined as the critical points of energy functionals. They are important in both classical and modern differential geometry. As is well known, any harmonic map from a space form of non-positive sectional curvature to any Riemannian manifold with finite energy has to be constant [6]. Xin considered the case of Cartan–Hadamard manifolds with some pinching condition and obtained that

Theorem A

([9]). Let \(M\) be an \(n (n\ge 3)\)-dimensional Cartan–Hadamard manifold with sectional curvature \(K : -a^2\le K\le 0\) and Ricci curvature bounded from above by \(-b^2\). Let \(\phi \) be a harmonic map from \(M\) to any Riemannian manifold with moderate divergent energy. If \(b\ge 2a\), then \(\phi \) is constant.

For the bounded symmetric domains, this implies that

Theorem B

([9]). Let

$$\begin{aligned} M=\left\{ \begin{array}{rl} \mathfrak {R}_I(n,m),~~~~~&{} when ~~~~n+m\ge 8,\\ \mathfrak {R}_{II}(n),~~~~~&{} when ~~~~n\ge 7,\\ \mathfrak {R}_{III}(n),~~~~~&{} when ~~~~n\ge 5,\\ \mathfrak {R}_{IV}(n),~~~~~&{} when ~~~~n\ge 8, \end{array}\right. \end{aligned}$$

Then any harmonic map from \(M\) to any Riemannian manifold with moderate divergent energy has to be constant.

In [10], by studying the harmonic maps from certain k\(\ddot{a}\)hler manidolds and the geometry of classical bounded symmetric domains, Xin proved that

Theorem C

([10]). A harmonic map of finite energy from a classical bounded symmetric domain (except \(\mathfrak {R}_{IV}(2)=\mathbb {H}^2\times \mathbb {H}^2\)) to any Riemannian manifold has to be constant.

Xin [10] raised the question as follows:

Question. Is there any harmonic map of finite energy from \(\mathfrak {R}_{IV}(2)\) to any Riemannian manifold?

The main purpose of this paper is to find a way to answer this question. In this paper, we consider the harmonic maps from a Riemannian manifold with non-positive pinching curvature to any Finsler manifold. We can prove the following

Main Theorem. There is no non-degenerate harmonic map from a classical bounded symmetric domain to any Finsler manifold with moderate divergent energy.

Corollary 1

Any harmonic map from a classical bounded symmetric domain to any Riemannian manifold with finite energy has to be constant.

2 Preliminaries

We shall use the following convention of index ranges unless otherwise stated:

$$\begin{aligned} 1\le i,j,\ldots \le n;~~1\le \alpha ,\beta ,\ldots \le m; ~~1\le \lambda , \mu ,\ldots \le n-1. \end{aligned}$$

Let \(M\) be an \(n\)-dimensional smooth manifold and \(\pi :TM\rightarrow M\) be the natural projection from the tangent bundle. Let \((x, Y)\) be a point of \(TM\) with \(x\in M, Y\in T_{x}M\) and let \((x^{i}, Y^{i})\) be the local coordinates on \(TM\) with \(Y=Y^{i}\frac{\partial }{\partial x^{i}}\). A Finsler metric on \(M\) is a function \(F:TM \rightarrow [0, +\infty )\) satisfying the following properties:

  1. (i)

    Regularity: \(F(x,Y)\) is smooth in \(TM \backslash 0\);

  2. (ii)

    Positive homogeneity: \(F(x,\lambda Y)=\lambda F(x, Y)\) for \(\lambda >0\);

  3. (iii)

    Strong convexity: The fundamental quadratic form \(g=g_{ij}\hbox {d}x^{i}\otimes \hbox {d}x^{j}\) is positive definite, where \(g_{ij}=\frac{\partial ^2(F^2)}{2\partial Y^i\partial Y^j}\).

Then \((M, F)\) is called an \(n\)-dimensional Finsler manifold. \(F\) determines the Hilbert form and Cartan tensor as follows:

$$\begin{aligned} \omega =\frac{\partial F}{\partial Y^{i}}\hbox {d}x^{i},\quad A=A_{ijk}\hbox {d}x^i\otimes \hbox {d}x^{j}\otimes \hbox {d}x^{k}, \quad A_{ijk}=F\frac{\partial g_{ij}}{2\partial Y^k}. \end{aligned}$$

The natural dual of the \(Hilbert\) form \(\omega \) is \(\ell =\frac{Y^i}{F}\frac{\partial }{\partial x^i}\) which is called the distinguished section.

Let \(\pi \) be the canonical projection \(TM \backslash 0 \rightarrow M\). It is well known that there is a unique connection the Chern connection \(\nabla \) on \(\pi ^{*}TM\) with \(\nabla \frac{\partial }{\partial x^i}=\omega ^j_i\frac{\partial }{\partial x^j}\hbox { and }\omega _i^j=\Gamma ^j_{ik}hx^k\) satisfying that

$$\begin{aligned} dg_{ij}-g_{ik}\omega ^k_j-g_{jk}\omega ^k_i=2A_{ijk}\frac{\delta Y^k}{F}, \end{aligned}$$
(2.1)

where \(\delta Y^i=dY^i+N^i_jdx^j,\,N^i_j=\gamma ^i_{jk}Y^k-\frac{1}{F}A^i_{jk}\gamma ^k_{st}Y^sY^t\hbox { and }\gamma ^i_{jk}\) are the formal Christoffel symbols of the second kind for \(g_{ij}\).

The curvature 2-forms of the Chern connection \(\nabla \) are

$$\begin{aligned} d\omega ^i_j-\omega ^k_j\wedge \omega ^i_k=\Omega ^i_j =\frac{1}{2}R^i_{jkl}\hbox {d}x^k\wedge \hbox {d}x^l+\frac{1}{F}P^i_{jkl}\hbox {d}x^k\wedge \delta Y^l, \end{aligned}$$
(2.2)

where \(R^i_{jkl}\hbox { and }P^i_{jkl}\) are the components of the \(hh\)-curvature tensor and \(hv\)-curvature tensor of the Chern connection, respectively.

Take a \(g\)-orthonormal frame \(\{e_i=u_i^j\frac{\partial }{\partial x^j}\}\) with \(e_n=\ell =\frac{Y^i}{F}\frac{\partial }{\partial x^i}\) for each fiber of \(\pi ^{*}TM\hbox { and }\{\omega ^i\}\) with \(\omega ^n=\omega \) is its dual coframe. The collection \(\{\omega ^i, \omega _n^i\}\) forms an orthonormal basis for \(T^*(TM\backslash \{0\})\) with respect to the Sasaki type metric \(g_{ij}\hbox {d}x^i\otimes \hbox {d}x^j+g_{ij}\delta Y^i\otimes \delta Y^j\). The pullback of the Sasaki metric from \(TM\backslash \{0\}\) to the sphere bundle \(SM\) is a Riemannian metric

$$\begin{aligned} \widehat{g}=g_{ij}\hbox {d}x^i\otimes \hbox {d}x^j+\delta _{\lambda \mu }\omega _n^\lambda \otimes \omega _n^\mu . \end{aligned}$$
(2.3)

Let \(\psi =\psi _i\omega ^i\in \Gamma (\pi ^*T^*M)\). With respect to the Chern connection \(\nabla \), the covariant derivatives of \(\psi _i\) are defined by

$$\begin{aligned} d\psi _i-\psi _j\omega ^j_i=\psi _{i|j}\omega ^j+\psi _{i;\lambda }\omega ^\lambda _n. \end{aligned}$$

where “|” and “;” denote the horizontal and the vertical covariant differentials with respect to the Chern connection, respectively.

Then we have

Lemma 2.1

([3]). For \(\psi =\psi _i\omega ^i\in \Gamma (\pi ^*T^*M)\), we have

$$\begin{aligned} div_{\widehat{g}}\psi =\sum _i \psi _{i|i}+\sum _{\lambda ,\mu }\psi _\lambda P_{\mu \mu \lambda } =(\nabla _{e_i^H}\psi )e_i+\sum _{\lambda ,\mu }\psi _\lambda P_{\mu \mu \lambda }, \end{aligned}$$

where \(e_i^H=u^j_i\frac{\delta }{\delta x^j} =u^j_i(\frac{\partial }{\partial x^j}-N_j^k\frac{\partial }{\partial Y^k})\) denotes the horizontal part of \(e_i\) and \(P_{\mu \mu \lambda }=P_{\mu \mu \lambda }^n\).

Lemma 2.2

([3]). Let \((M, F)\) be a Finsler manifold. Then any function \(f\) on the sphere bundle \(SM\) satisfies

$$\begin{aligned} \int \limits _{S_xM}g^{ij}(F^2f)_{Y^iY^j}\Omega \hbox {d}\tau =2n\int \limits _{S_xM}f\Omega \hbox {d}\tau , \end{aligned}$$

where \((F^2f)_{Y^iY^j}=\frac{\partial }{\partial Y^j}\frac{\partial }{\partial Y^i}(F^2f), \hbox {d}\tau =\sum _i(-1)^{i-1}Y^idY^1\wedge \cdots \wedge \widehat{dY^i}\wedge \cdots \wedge dY^n\hbox { and }\Omega =det(\frac{g_{ij}}{F})\).

For any fixed \(x\in M,\,S_xM=\{Y\in T_xM|F(Y)=1\}\) has a natural Riemannian metric

$$\begin{aligned} \widehat{r}_x=\sum _{\lambda } \theta ^{\lambda }_n\otimes \theta ^{\lambda }_n,~~~~~~where~~\theta ^{\lambda }_n=\omega ^{\lambda }_n|_{S_xM}. \end{aligned}$$

On the Riemannian manifold \((S_xM,\widehat{r}_x)\), considering an 1-form \(\Psi =\nu \Psi _idY^i\), where \(\nu =\sqrt{det(g_{ij})}\), we have [3]

$$\begin{aligned} div_{\widehat{r}_x}\Psi =r^{ij}[\nu \Psi _i]_{Y^j}-(n-1)\nu \Psi _iY^i-Fg^{ij}\nu \Psi _i\eta _j, \end{aligned}$$
(2.4)

where \(r^{ij}=F^2g^{ij}-Y^iY^j\hbox { and }\eta _i=Fg^{jk}C_{jki}\).

Using the fact \(\frac{\partial \nu }{\partial Y^i}=\frac{\nu \eta _i}{F}\), (2.4) implies the following

Lemma 2.3

Let \((M,F)\) be a Finsler manifold and \(\Psi =\nu \Psi _idY^i\) be a global section on \(T^*(S_xM)\), where \(\nu =\sqrt{det(g_{ij})}\). Then we have

$$\begin{aligned} div_{\widehat{r}_x}\Psi =F^2\nu g^{ij}(\Psi _i)_{Y^j}-(n-2)\nu \Psi _iY^i. \end{aligned}$$

Let \(\phi : (M^n, F)\rightarrow (\overline{M}^m, \overline{F})\) be a non-degenerate smooth map between Finsler manifolds, i.e., \(ker(\hbox {d}\phi )=\emptyset \), and \(\widetilde{\nabla }\) be the pullback Chern connection on \(\pi ^*(\phi ^{-1}T\overline{M})\). We have

Lemma 2.4

$$\begin{aligned} \begin{array}{rl} X\langle \hbox {d}\phi U, \hbox {d}\phi V\rangle =&{} \langle \widetilde{\nabla }_X( \hbox {d}\phi U), \hbox {d}\phi V\rangle +\langle \hbox {d}\phi U, \widetilde{\nabla }_X( \hbox {d}\phi V)\rangle \\ &{}+\,2\overline{C}(\hbox {d}\phi U, \hbox {d}\phi V, (\widetilde{\nabla }_X (\hbox {d}\phi F\ell )),\end{array} \end{aligned}$$

where \(\overline{C}_{ijk}=\frac{1}{\overline{F}}\overline{A}_{ijk}\hbox { and }X, U, V \in \Gamma (\pi ^*TM)\).

The energy density of \(\phi \) is the function \(e(\phi ):SM\rightarrow R\) defined by

$$\begin{aligned} \displaystyle e(\phi )(x, Y)=\frac{1}{2}|\hbox {d}\phi |^2=\frac{1}{2}g^{ij}(x,Y)\phi ^\alpha _i\phi ^\beta _j\overline{g}_{\alpha \beta }(\overline{x}, \overline{Y}), \end{aligned}$$
(2.5)

where \(\hbox {d}\phi (\frac{\partial }{\partial x^i})=\phi _i^\alpha \frac{\partial }{\partial \overline{x}^\alpha }\hbox { and }\overline{Y}=\overline{Y}^\alpha \frac{\partial }{\partial \overline{ x}^\alpha }=Y^i\phi _i^\alpha \frac{\partial }{\partial \overline{ x}^\alpha }.\)

We define the energy functional \(E(\phi )\) by

$$\begin{aligned} E(\phi )=\frac{1}{C_{n-1}}\int \limits _{SM}e(\phi )\hbox {d}V_{SM}, \end{aligned}$$
(2.6)

where \(\hbox {d}V_{SM}=\Omega \hbox {d}\tau \wedge \hbox {d}x, \hbox {d}x=dx^1\wedge \cdots \wedge dx^n\hbox { and }C_{n-1}\) denotes the volume of the unit Euclidean sphere \(S^{n-1}\).

We call \(\phi \) a harmonic map if it is a critical point of the energy functional.

Theorem 2.5

([3]). \(\phi \) is harmonic map if and only if

$$\begin{aligned} \displaystyle \int \limits _{SM}\langle V, (\widetilde{\nabla }_{\ell ^H}\hbox {d}\phi )\ell \rangle \hbox {d}V_{SM}=0, \end{aligned}$$

for any vector \(V\in \Gamma (\phi ^{-1}T\overline{M})\).

We need the following

Theorem 2.6

([9]). Let \(M\hbox { and }\widetilde{M}\) be \(n\)-dimensional complete Riemannian manifolds, where \(M\) is simply connected without focal points. Let \(r\hbox { and }\widetilde{r}\) be the distance functions from \(x_0\in M\hbox { and }\widetilde{x}_0\in \widetilde{M}\), respectively. Suppose for any \(x\in M, \widetilde{x}\in \widetilde{M}, r(x)=\widetilde{r}(\widetilde{x})\ne 0\)

$$\begin{aligned} Ric_{(x)}\left( \frac{\partial }{\partial r}, \frac{\partial }{\partial r}\right) \le \frac{1}{n-1}\widetilde{Ric}_{(\widetilde{x})}\left( \frac{\partial }{\partial \widetilde{r}},\frac{\partial }{\partial r}\right) , \end{aligned}$$

where \(Ric\hbox { and }\widetilde{Ric}\) denote the Ricci curvatures of \(M\) and \(\widetilde{M}\), respectively. Then at any differentiable point \(\widetilde{x}\) of \(\widetilde{r}\)

$$\begin{aligned} \triangle r\ge \frac{1}{n-1}\widetilde{\triangle }\widetilde{r}. \end{aligned}$$

In particular, when \(Ric \le -b^2\), where \(b\) is a positive constant, then

$$\begin{aligned} \triangle r\ge b coth (br). \end{aligned}$$
(2.7)

3 The proof of main theorem

Let \(\phi : M^n\rightarrow \overline{M}\) be a smooth map form a Riemannian manifold to a Finsler manifold. For any vector field \( \hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\), by Lemmas 2.1, 2.4, \(\nabla _{X^H}\ell =0\hbox { and }\overline{C}(\hbox {d}\phi \ell , \bullet ,\bullet )=0\) we have that

$$\begin{aligned}&\displaystyle div_{\widehat{g}}\left( \frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle X\right) \nonumber \\&\quad =\displaystyle \nabla _{X^H}\left( \frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle \right) +\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle \langle \nabla _{e_i^H}X,e_i\rangle \nonumber \\&\quad =\displaystyle \langle (\widetilde{\nabla }_{X^H}\hbox {d}\phi )\ell , \hbox {d}\phi \ell \rangle +\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle \langle \nabla _{e_i^H}X,e_i\rangle \nonumber \\&\quad =\displaystyle \langle \widetilde{\nabla }_{\ell ^H}\hbox {d}\phi (X), \hbox {d}\phi \ell \rangle -\langle \hbox {d}\phi (\nabla _{\ell }X), \hbox {d}\phi \ell \rangle +\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle \langle \nabla _{e_i}X,e_i\rangle . \end{aligned}$$
(3.1)

Let \(\psi =\langle \hbox {d}\phi X, \hbox {d}\phi \ell \rangle \omega ^n\) which is a global section on \(\pi ^*T^*M\). By Lemma 2.1 and \(P_{aan}=0\), we know that

$$\begin{aligned} div_{\widehat{g}}\psi&= \widetilde{\nabla }_{\ell ^H}\langle \hbox {d}\phi X, \hbox {d}\phi \ell \rangle \nonumber \\&= \langle \widetilde{\nabla }_{\ell ^H}\hbox {d}\phi (X), \hbox {d}\phi \ell \rangle +\langle \hbox {d}\phi (X), (\widetilde{\nabla }_{\ell ^H}\hbox {d}\phi ) \ell \rangle . \end{aligned}$$
(3.2)

Substituting (3.2) into (3.1), we obtain that

$$\begin{aligned}&div_{\widehat{g}}(\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell X)-div_{\widehat{g}}(\langle \hbox {d}\phi X, \hbox {d}\phi \ell \rangle \omega ^n)\nonumber \\&\quad =\langle \hbox {d}\phi (X),(\widetilde{\nabla }_{\ell ^H}\hbox {d}\phi ) \ell \rangle -\langle \hbox {d}\phi (\nabla _{\ell }X), \hbox {d}\phi \ell \rangle +\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle \langle \nabla _{e_i}X,e_i\rangle . \end{aligned}$$
(3.3)

Since \(M\) is a Riemannian manifold, we have that \(d \phi X\) only depends on the local coordinates \((x^i)\) for \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\). When \(\phi \) is a harmonic map. Integrating (3.3) implies that

$$\begin{aligned}&\displaystyle \int \limits _{\partial (SM)}\Big \{\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle \langle X, \mathbf{n}\rangle -\langle \hbox {d}\phi X, \hbox {d}\phi \ell \rangle \langle \ell , \mathbf{n}\rangle \Big \}\hbox {d}V_{\partial SM}\nonumber \\&\quad =\displaystyle \int \limits _{SM}\Big \{-\langle \hbox {d}\phi (\nabla _{\ell }X), \hbox {d}\phi \ell \rangle +\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle \langle \nabla _{e_i}X,e_i\rangle \Big \}\hbox {d}V_{SM}, \end{aligned}$$
(3.4)

where \(\mathbf{n}\) is the unit normal vector of the boundary \(\partial (SM)\) in \(SM\hbox { and }\ell \) is the natural dual of \(\omega ^n\).

From (3.4), we obtain the following immediately

Proposition 3.1

Let \(\phi \) be a harmonic map from a Riemannian manifold \(M\) to any Finsler manifold. If \(e(\phi )|_{\partial (SM)}=0\), then for any vector field \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\)

$$\begin{aligned} \int \limits _{SM}\Big \{-\langle \hbox {d}\phi (\nabla _{\ell }X),\hbox {d}\phi \ell \rangle +\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle tr\nabla X\Big \}\hbox {d}V_{SM}=0, \end{aligned}$$
(3.5)

where \(tr\nabla X=\langle \nabla _{e_i}X,e_i\rangle _g\).

Theorem 3.2

Let \(\phi \) be a harmonic map from a Riemannian manifold \(M\) to any Finsler manifold. If \(e(\phi )|_{\partial (SM)}=0\), then for any vector field \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\)

$$\begin{aligned} \int \limits _{SM}\Big \{e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \Big \}\hbox {d}V_{SM}=0, \end{aligned}$$
(3.6)

where \(tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle =\langle \hbox {d}\phi (\nabla _{e_i}X), \phi _*(e_i)\rangle \).

Proof

Since \(M\) is a Riemannian manifold, we have that \(tr\nabla X=g^{ij}\langle \nabla _{\frac{\partial }{\partial x^i}}X, \frac{\partial }{\partial x^j}\rangle _g\) only depends on the local coordinates \((x^i)\) for \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\). Denote \(f=\frac{1}{2}\langle \hbox {d}\phi \ell , \hbox {d}\phi \ell \rangle tr\nabla X\in C^\infty (SM)\). Using the fact \(\overline{C}(\hbox {d}\phi \ell , \bullet ,\bullet )=0\) and Lemma 2.4, we obtain that

$$\begin{aligned} \begin{array}{rl} \displaystyle g^{ij}(F^2f)_{Y^iY^j} =&{}\displaystyle g^{ij}\left[ \frac{1}{2}Y^kY^l\left\langle \hbox {d}\phi \frac{\partial }{\partial x^k}, \hbox {d}\phi \frac{\partial }{\partial x^k}\right\rangle tr\nabla X\right] _{Y^iY^j}\\ =&{}\displaystyle \sum _i\langle \hbox {d}\phi e_i, \hbox {d}\phi e_i\rangle tr\nabla X. \end{array} \end{aligned}$$
(3.7)

Substituting (3.7) into Lemma 2.2 yields that

$$\begin{aligned} \int \limits _{S_xM}\frac{n}{2}|\hbox {d}\phi \ell |^2tr\nabla X\Omega \hbox {d}\tau =\int \limits _{S_xM}\frac{1}{2}|\hbox {d}\phi |^2tr\nabla X\Omega \hbox {d}\tau . \end{aligned}$$
(3.8)

Considering \(\Psi =\frac{1}{F}\langle \hbox {d}\phi (\nabla _{\frac{\partial F\ell }{\partial Y^i}}X), \hbox {d}\phi \ell \rangle dY^i=\frac{1}{F}\langle \hbox {d}\phi (\nabla _{\frac{\partial }{\partial x^i}}X), \hbox {d}\phi \ell \rangle dY^i\). Since \(\hbox {d}\phi (\nabla _{\frac{\partial }{\partial x^i}}X)\) only depends on \((x^i)\) for \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\), by Lemma 2.3 we have that

$$\begin{aligned} div_{\widehat{r}_x}\psi&= F^2\nu g^{ij}\left[ \frac{1}{F^2}\left\langle \hbox {d}\phi (\nabla _{\frac{\partial }{\partial x^i}}X), \hbox {d}\phi F\ell \right\rangle \right] _{Y^j}-(n-2)\nu \frac{1}{F}\left\langle \hbox {d}\phi (\nabla _{\frac{\partial }{\partial x^i}}X), \hbox {d}\phi \ell \right\rangle Y^i\nonumber \\&= F^2\nu g^{ij}\left[ -\frac{2Y^kg_{jk}}{F^4}\langle \hbox {d}\phi (\nabla _{\frac{\partial }{\partial x^i}}X),\hbox {d}\phi F\ell \rangle +\frac{1}{F^2}\langle \hbox {d}\phi (\nabla _{\frac{\partial }{\partial x^i}}X), \hbox {d}\phi \frac{\partial }{\partial x^j}\rangle \right] \nonumber \\&-(n-2)\nu \frac{1}{F}\langle \hbox {d}\phi (\nabla _{\frac{\partial }{\partial x^i}}X), \hbox {d}\phi \ell \rangle Y^i\nonumber \\&= \nu \left[ \langle \hbox {d}\phi (\nabla _{e_i}X), \hbox {d}\phi e_i\rangle -n\langle \hbox {d}\phi (\nabla _{\ell }X), \hbox {d}\phi \ell \rangle \right] . \end{aligned}$$
(3.9)

Integrating (3.9) implies that

$$\begin{aligned} n\int \limits _{S_xM}\langle \hbox {d}\phi (\nabla _{\ell }X), \hbox {d}\phi \ell \rangle \Omega \hbox {d}\tau =\int \limits _{S_xM}tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \Omega \hbox {d}\tau , \end{aligned}$$
(3.10)

where \(tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle =\langle \hbox {d}\phi (\nabla _{e_i}X), \hbox {d}\phi e_i\rangle \).

It can be seen from (3.8), (3.10) and Proposition 3.1 that

$$\begin{aligned} \int \limits _{SM}\Big \{e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \Big \} \hbox {d}V_{SM}=0. \end{aligned}$$

This completes the proof of Theorem 3.2. \(\square \)

For a harmonic map \(\phi :M^n\rightarrow \overline{M}\) between Riemannian manifolds, We also have that

Theorem \(\mathbf{3.2}^{\prime }\). Let \(\phi \) be a harmonic map from a Riemannian manifold \(M\) into any Riemannian manifold. If \(e(\phi )|_{\partial M}=0\), then for any vector field \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\)

$$\begin{aligned} \int \limits _{SM}\Big \{e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \Big \}\hbox {d}V_{SM}=0,\quad (3.6^{\prime }) \end{aligned}$$

i.e.,

$$\begin{aligned} \int \limits _M\Big \{e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \Big \}\hbox {d}V_M=0. \end{aligned}$$
(3.11)

Proof

For any vector field \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\), we have that

$$\begin{aligned} \hbox {div}(e(\phi )X)&= \nabla _Xe(\phi )+e(\phi )\langle \nabla _{e_i}X,e_i\rangle \nonumber \\&= \hbox {div}(\langle \phi _*X, \phi _*(e_i)\rangle e_i)-\langle \phi _*X,\tau (\phi )\rangle \nonumber \\&-\langle \hbox {d}\phi (\nabla _{e_i}X), \phi _*(e_i)\rangle +e(\phi )\langle \nabla _{e_i}X,e_i\rangle , \end{aligned}$$
(3.12)

where \(\{e_i\}\) is a local orthonormal frame field on \(M\).

When \(\phi \) is a harmonic map. Integrating (3.12) implies that

$$\begin{aligned}&\displaystyle \int \limits _{\partial M}\Big \{e(\phi )\langle X, \mathbf{n}\rangle -\langle \phi _*X, \phi _*(\mathbf{n})\rangle \Big \}\hbox {d}V_M\nonumber \\&\quad \displaystyle =\int \limits _M\Big \{e(\phi )\langle \nabla _{e_i}X,e_i\rangle -\langle \hbox {d}\phi (\nabla _{e_i}X),\phi _*(e_i)\rangle \Big \}\hbox {d}V_M, \end{aligned}$$
(3.13)

where \(\mathbf{n}\) is the unit normal vector of \(\partial M\) in \(M\),

By (3.13) and our assumption condition \(e(\phi )|_{\partial M}=0\), we obtain that

$$\begin{aligned} \int \limits _M\Big \{e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \Big \}\hbox {d}V_M=0. \end{aligned}$$
(3.14)

For a harmonic map \(\phi \) between Riemannian manifolds, we have that \(e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \) only depends on the local coordinates \((x^i)\) for \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\). Then

$$\begin{aligned}&\displaystyle \frac{1}{C_{n-1}}\int \limits _{SM}\Big \{e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \Big \}\hbox {d}V_{SM}\nonumber \\&\quad =\displaystyle \frac{1}{C_{n-1}}\int \limits _{M}\Big \{\int \limits _{S_xM}[e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X),\hbox {d}\phi \rangle ]\Omega \hbox {d}\tau \Big \}\hbox {d}x\nonumber \\&\quad =\displaystyle \int \limits _M\Big \{e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X),\hbox {d}\phi \rangle \Big \}\hbox {d}V_M. \end{aligned}$$
(3.15)

We complete the proof of Theorem \(3.2^{\prime }\). \(\square \)

Definition 1

The energy of a map \(\phi \) from a Riemannian manifold \(M\) to any Finsler manifold is called moderate divergent energy if there exists a positive function \(\psi (r)\) satisfying

$$\begin{aligned} \int \limits ^\infty _{R_0}\frac{\mathrm{d}r}{r\psi (r)}=\infty , \end{aligned}$$

such that

$$\begin{aligned} \lim _{R\rightarrow \infty }\int \limits _{SB_R(x_0)}\frac{e(\phi )}{\psi (r)}*1<\infty . \end{aligned}$$

where \(SB_R(x_0)\subseteq SM\) is a geodesic ball of radius \(R\) and centered at \(x_0\) in \(SM\) whose boundary is the geodesic sphere \(\partial (SB_R(x_0))\).

Remark

For a map \(\phi \) between Riemannian manifolds, the \(\lim _{R\rightarrow \infty }\int _{SB_R(x_0)}\frac{e(\phi )}{\psi (r)}*1<\infty \) is equivalent to \(\lim _{R\rightarrow \infty }\int _{B_R(x_0)}\frac{e(\phi )}{\psi (r)}*1<\infty \), where \(B_R(x_0)\subseteq M\) is a geodesic ball of radius \(R\) and centered at \(x_0\) in \(M\).

Theorem 3.3

Let \(\phi \) be a harmonic map from a Riemannian manifold \(M\) to any Finsler manifold. If the energy of \(\phi \) is moderate divergent energy, then \(e(\phi )|_{\partial (SM)}=0\).

Proof

We assume that \(\int _{\partial (SM)}e(\phi )*1\ne 0\). Let \(\int _{\partial (SM)}e(\phi )*1=\delta >0\), i.e.,

$$\begin{aligned} \lim _{R\rightarrow \infty }\int \limits _{\partial (SB_R(x_0))}e(\phi )*1=\delta >0. \end{aligned}$$
(3.16)

Then there is a \(R_0>1\), for \(R>R_0\) we have that

$$\begin{aligned} \int \limits _{\partial (SB_R(x_0))}e(\phi ) *1\ge \varepsilon , \end{aligned}$$
(3.17)

where \(\varepsilon \) is a sufficiently small constant.

For any positive function \(\psi (r)\) satisfying \(\int ^\infty _{R_0}\frac{\mathrm{d}r}{r\psi (r)}=\infty \), we have that

$$\begin{aligned} \begin{array}{rl} \displaystyle \lim _{R\rightarrow \infty }\int \limits _{SB_R(x_0)}\frac{e(\phi )}{\psi (r)}*1=&{} \displaystyle \int \limits _0^\infty \left[ \frac{1}{\psi (R)}\int \limits _{\partial (SB_R(x_0))}e(\phi )*1\right] \hbox {d}R\\ \ge &{}\displaystyle \varepsilon \int \limits _{R_0}^\infty \frac{1}{\psi (R)}\hbox {d}R\\ \ge &{}\displaystyle \varepsilon \int \limits _{R_0}^\infty \frac{1}{R\psi (R)} \hbox {d}R=\infty , \end{array} \end{aligned}$$

which is a contradiction to the assumption that the energy of \(\phi \) is moderate divergent energy. This completes the proof of Theorem 3.3. \(\square \)

Corollary 3.4

Let \(\phi \) be a harmonic map from a Riemannian manifold \(M\) to any Riemannian manifold. If the energy of \(\phi \) is moderate divergent energy, then \(e(\phi )|_{\partial M}=0\).

Theorem 3.5

Let \(M\) be an \(n (n\ge 3)\)-dimensional Cartan–Hadamard Riemannian manifold with sectional curvature \(K : -a^2\le K\le 0\) and Ricci curvature bounded from above by \(-b^2\). If \(b\ge a\), then there is no non-degenerate harmonic maps from \(M\) to any Finsler manifold with moderate divergent energy.

Remark

This theorem improves the Xin’s result in [10].

Proof

For the harmonic map \(\phi \). It can be seen from Theorems 3.2 and 3.3 that

$$\begin{aligned} \lim _{R\rightarrow \infty }\int \limits _{B_R(x_0)}\left[ \int \limits _{S_xB_R(x_0)}\Big \{e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \Big \}\Omega \hbox {d}\tau \right] \hbox {d}x=0, \end{aligned}$$
(3.18)

where \(B_R(x_0)\subseteq M\) be a geodesic ball of radius \(R\) and centered at \(x_0\) whose boundary is the geodesic sphere \(S_R(x_0)\).

The square of the distance function \(r^2\) from \(x_0\) in Riemannian manifold \(M\) is a smooth function. Set \(X=sh (Cr)\frac{\partial }{\partial r}=\frac{e^{Cr}-e^{-Cr}}{2}\frac{\partial }{\partial r}\), where \(C\) is constant and \(\frac{\partial }{\partial r}\) denotes the unit radial vector. Obviously, the unit normal vector to \(S_R(x_0)\) is \(\frac{\partial }{\partial r}\). Let \(\{e_\lambda , \frac{\partial }{\partial r}\}\) is an orthonormal frame field on \(B_R(x_0)\), where \(\lambda =1,\ldots , n-1\). Then we have that

$$\begin{aligned} \begin{array}{rl} \nabla _{\frac{\partial }{\partial r}}X=&{}C ch(Cr)\frac{\partial }{\partial r},\\ \nabla _{e_\lambda }X=&{}sh(Cr)\sum _{\mu }h_{\lambda \mu }e_\mu =sh(Cr)\sum _{\mu }Hess(r)(e_\lambda , e_\mu )e_\mu , \end{array} \end{aligned}$$
(3.19)

where \(Hess(r)\) is the Hessian of the distance function \(r\hbox { and }-h_{\lambda \mu }\) is the components of the second fundamental form of \(S_R(x_0)\) in \(B_R(x_0)\).

Then we obtain that by (3.19)

$$\begin{aligned}&\displaystyle e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \nonumber \\&\quad =\displaystyle e(\phi )\Big \{Cch(Cr)+sh(Cr)\sum _\lambda h_{\lambda \lambda }\Big \}\nonumber \\&\qquad \displaystyle -Cch(Cr)|\hbox {d}\phi \left( \frac{\partial }{\partial r}\right) |^2-sh(Cr)h_{\lambda \mu }\left\langle \hbox {d}\phi (e_\lambda ), \hbox {d}\phi (e_\mu )\right\rangle . \end{aligned}$$
(3.20)

We can choose an orthonormal frame field \(\{e_\lambda \}\) on \(S_R(x_0)\) such that \(h_{\lambda \mu }=\delta _{\lambda \mu }h_{\lambda \lambda }\). It follows from (3.20) that

$$\begin{aligned}&\displaystyle e(\phi )tr\nabla X-tr\langle \hbox {d}\phi (\nabla X), \hbox {d}\phi \rangle \nonumber \\&\quad =\displaystyle \frac{1}{2}\Big \{sh(Cr)\sum _\lambda h_{\lambda \lambda }-Cch(Cr)\Big \}|\hbox {d}\phi \left( \frac{\partial }{\partial r}\right) |^2\nonumber \\&\qquad \displaystyle +\frac{1}{2}\sum _\lambda \Big \{Cch(Cr)+\sum _\mu sh(Cr) h_{\mu \mu }-2sh(Cr)h_{\lambda \lambda }\Big \}|\hbox {d}\phi (e_\lambda )|^2. \end{aligned}$$
(3.21)

Substituting (3.21) into (3.19), we have that

$$\begin{aligned}&\displaystyle \lim _{R\rightarrow \infty }\int \limits _{B_R(x_0)}\left[ \int \limits _{S_xB_R(x_0)}\Big \{[sh(Cr)\sum _\lambda h_{\lambda \lambda }-Cch(Cr)]|\hbox {d}\phi (\frac{\partial }{\partial r})|^2\right. \nonumber \\&\quad \left. \displaystyle +\sum _\lambda [Cch(Cr)+\sum _\mu sh(Cr) h_{\mu \mu }-2sh(Cr)h_{\lambda \lambda }]|\hbox {d}\phi (e_\lambda )|^2\Big \}\Omega \hbox {d}\tau \right] \hbox {d}x=0.\qquad \quad \end{aligned}$$
(3.22)

By \(Ric\le -b^2\) and Theorem 2.6, we get that

$$\begin{aligned} \sum _\mu h_{\mu \mu }\ge b coth(br). \end{aligned}$$

Set \(C=a\) in (3.22). Since \(g(x)=x coth (x)\) is non-decreasing function and \(b\ge a\), we obtain that

$$\begin{aligned} sh(Cr)\sum _\lambda h_{\lambda \lambda }-Cch(Cr)\ge sh(ar)[b coth(br)-a coth(ar)]\ge 0. \end{aligned}$$
(3.23)

When \(-a^2\le K\le 0\), the Hessian comparison theorem implies that

$$\begin{aligned} \frac{1}{r}\le h_{\lambda \lambda }\le a coth (ar),~~~\forall \lambda . \end{aligned}$$
(3.24)

Then

$$\begin{aligned}&\displaystyle C ch(Cr)+\sum _\mu sh(Cr) h_{\mu \mu }-2sh(Cr)h_{\lambda \lambda }\nonumber \\&\quad =\displaystyle a ch(ar)+\sum _{\mu \ne \lambda } sh(ar) h_{\mu \mu }-sh(ar)h_{\lambda \lambda }\nonumber \\&\quad \ge \displaystyle \sum _{\mu \ne \lambda } sh(ar) h_{\mu \mu }\ge \delta , \end{aligned}$$
(3.25)

where \(\delta >0\) is a constant.

By (3.22), (3.23) and (3.25), we have that

$$\begin{aligned} \hbox {d}\phi (e_\lambda )=0, \forall \lambda . \end{aligned}$$

Put \(C=0\) in (3.22). Then (3.22), together with \(\hbox {d}\phi (e_\lambda )=0, \forall \lambda \), yields that \(\hbox {d}\phi (\frac{\partial }{\partial r})=0\). We get that \(e(\phi )=0\) which is a contradiction to the assumption that \(\phi \) is a non-degenerate harmonic maps. \(\square \)

Corollary 3.6

Let \(M\) be an \(n (n\ge 3)\)-dimensional Cartan–Hadamard Riemannian manifold with sectional curvature \(K:-a^2\le K\le 0\) and Ricci curvature bounded from above by \(-b^2\). If \(b\ge a\), then there is no non-degenerate harmonic maps from \(M\) to any Riemannian manifold with moderate divergent energy.

From [7] we know the Ricci curvatures of bounded symmetric domains and estimate the bounds of the sectional curvatures of bounded symmetric domains as follows

 

Dim.

Sec. curvature

Ric. curvature

\(\mathfrak {R}_I(n,m) (min(n,m)\ge 2)\)

\(2nm\)

\(-4\le k\le 0\)

\(-2(n+m)\)

\(\mathfrak {R}_{II}(n) (n\ge 2)\)

\(n(n+1)\)

\( -4\le k\le \) 0

\(-2(n+1)\)

\(\mathfrak {R}_{III}(n) (n\ge 4) \)

\(n(n-1)\)

\( -2\le k\le 0\)

\(-2(n-1)\)

\(\mathfrak {R}_{IV}(n) (n\ge 2)\)

\(2n \)

\( -2\le k\le 0 \)

\(-n\)

This table together with Theorem 3.5 yields immediately.

Main Theorem. There is no non-degenerate harmonic map from a classical bounded symmetric domain to any Finsler manifold with moderate divergent energy.

Corollary 3.7

Any harmonic map from a classical bounded symmetric domain to any Riemannian manifold with moderate divergent energy has to be constant.

Using the fact that any harmonic map of finite energy must be moderate divergent energy, we have immediately

Corollary 3.8

Any harmonic map from a classical bounded symmetric domain to any Riemannian manifold with finite energy has to be constant.