Abstract
In this paper, we consider the harmonic maps from a Riemannian manifold with non-positive pinching curvature to any Finsler manifold, and we can prove that there is no non-degenerate harmonic maps from a classical bounded symmetric domain to any Finsler manifold with moderate divergent energy. In particular, we obtain that any harmonic map from a classical bounded symmetric domain to any Riemannian manifold with finite energy has to be constant, which improves the Xin’s result in Acta Math Sinica 15:277–292, 1999.
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1 Introduction
Bounded symmetric domains were introduced by Cartan [1, 2] and were systematically studied by Hua, Look, Siegel and Xin [4–11]. 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
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
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:
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:
-
(i)
Regularity: \(F(x,Y)\) is smooth in \(TM \backslash 0\);
-
(ii)
Positive homogeneity: \(F(x,\lambda Y)=\lambda F(x, Y)\) for \(\lambda >0\);
-
(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:
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
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
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
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
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
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
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
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]
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
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
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
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
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
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 \(x\in M, \widetilde{x}\in \widetilde{M}, r(x)=\widetilde{r}(\widetilde{x})\ne 0\)
where \(Ric\hbox { and }\widetilde{Ric}\) denote the Ricci curvatures of \(M\) and \(\widetilde{M}\), respectively. Then at any differentiable point \(\widetilde{x}\) of \(\widetilde{r}\)
In particular, when \(Ric \le -b^2\), where \(b\) is a positive constant, then
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
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
Substituting (3.2) into (3.1), we obtain that
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
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 \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\)
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 \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\)
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
Substituting (3.7) into Lemma 2.2 yields that
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
Integrating (3.9) implies that
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
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 \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\)
i.e.,
Proof
For any vector field \(\hbox {d}\phi X\in \Gamma (\phi ^{-1}T\overline{M})\), we have that
where \(\{e_i\}\) is a local orthonormal frame field on \(M\).
When \(\phi \) is a harmonic map. Integrating (3.12) implies that
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
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
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
such that
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.,
Then there is a \(R_0>1\), for \(R>R_0\) we have that
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
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
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
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)
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
Substituting (3.21) into (3.19), we have that
By \(Ric\le -b^2\) and Theorem 2.6, we get that
Set \(C=a\) in (3.22). Since \(g(x)=x coth (x)\) is non-decreasing function and \(b\ge a\), we obtain that
When \(-a^2\le K\le 0\), the Hessian comparison theorem implies that
Then
where \(\delta >0\) is a constant.
By (3.22), (3.23) and (3.25), we have that
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.
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Li, J. Harmonic maps from bounded symmetric domains to Finsler manifolds. Annali di Matematica 194, 569–579 (2015). https://doi.org/10.1007/s10231-013-0389-8
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DOI: https://doi.org/10.1007/s10231-013-0389-8