The existence of harmonic maps from Finsler manifolds of Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler manifold and study the characterisation of harmonic maps,in the spirit of Ishihara.Using heat quation method we show that any map from a compact Finsler manifold M to a compact Riemannian manifold with non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class.     

Second variation of harmonic maps between Finsler manifolds

The first and second variation formulas of the energy functional for a nonde-generate map between Finsler manifolds is derived. As an application, some nonexistence theorems of nonconstant stable harmonic maps from a Finsler manifold to a Riemannian manifold are given.     

Existence and Liouville theorems for V -harmonic maps from complete manifolds

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.     

稳定调和映照的不存在性

讨论了从Finsler流形到Riemannian流形的稳定调和映照的不存在性,得到了一个拼挤定理.     

On weakly harmonic maps from Finsler to Riemannian manifolds

We prove global C0,α$C^{0,\alpha }$-estimates for harmonic maps from Finsler manifolds into regular balls of Riemannian target manifolds generalizing results of Giaquinta, Hildebrandt, and Hildebrandt, Jost and Widman from Riemannian to Finsler domains. As consequences we obtain a Liouville theorem for entire harmonic maps on simple Finsler manifolds, and an existence theorem for harmonic maps from Finsler manifolds into regular balls of a Riemannian target.     

Strictly convex functions on complete Finsler manifolds

The purpose of the present paper is to investigate the influence of strictly convex functions on the metric structures of complete Finsler manifolds. More precisely we discuss the properties of the group of isometries and the exponential maps on a complete Finsler manifold admitting strictly convex functions.     

The heat flow and harmonic maps between complete manifolds

We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As an application, we solve the Dirichlet problem at infinity for proper harmonic maps between two hyperbolic manifolds for a class of boundary maps. The boundary map under consideration has finite many points at which either it is not differentiable or has vanishing energy density.     

Projective vector fields on Finsler manifolds

In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.     

Locally minimizing harmonic maps from noncompact manifolds

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.     

Some results on P-harmonic maps and exponentially harmonic maps between Finsler manifolds

This paper studies the stability of P-harmonic maps and exponentially harmonic maps from Finsler manifolds to Riemannian manifolds by an extrinsic average variational method in the calculus of variations. It generalizes Li＇s results in [2] and [3].     

Positivities and vanishing theorems on complex Finsler manifolds

In this paper, we study the relations between curvatures and geometries of a compact complex Finsler manifold. We first introduce various definitions of curvatures and then vanishing theorems are established under some positivity assumptions of curvatures. In particular, we get some criteria of a compact complex Finsler manifold to be of negative Kodaira dimension.     

Harmonic maps from closed Riemannian manifolds with positive scalar curvature

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal.     

Gromov hyperbolicity of negatively curved Finsler manifolds

Let (M, F) be a closed C ^∞ Finsler manifold. The lift of the Finsler metric F to the universal covering space defines an asymmetric distance [(d)\tilde]{\widetilde d} on [(M)\tilde]{\widetilde M}. It is well-known that the classical comparison theorem of Aleksandrov does not exist in the Finsler setting. Therefore, it is necessary to introduce new Finsler tools for the study of the asymmetric metric space ([(M)\tilde], [(d)\tilde]){(\widetilde M, \widetilde d)}. In this paper, by using the geometric flip map and the unstable-stable angle introduced in [2], we prove that if (M, F) is a closed Finsler manifold of negative flag curvature, then ([(M)\tilde], [(d)\tilde]){(\widetilde M, \widetilde d)} is an asymmetric δ-hyperbolic space in the sense of Gromov.     

RC-positivity and rigidity of harmonic maps into Riemannian manifolds Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.     

Conjugate and conformally conjugate parallelisms on Finsler manifolds

In this paper we study conjugate parallelisms and their conformal changes on Finsler manifolds. We provide sufficient conditions for a Finsler manifold endowed with two conjugate (resp. conformally conjugate) covering parallelisms to become a Berwald (resp. Wagner) manifold. As an application for Lie groups, we give a new proof for a theorem of Latifi and Razavi about bi-invariant Finsler functions being Berwald. By introducing the concept of a conformal change of a parallelism, we also obtain a conceptual proof of a theorem of Hashiguchi and Ichijyō: the class of generalized Berwald manifolds is closed under conformal change.     

A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the \(L^p\)-norm of the tension field is bounded and the n-energy of the map is sufficiently small, then every biharmonic map must be harmonic, where \(2<p<n\).     

Bounded harmonic maps on a class of manifolds

Without imposing any curvature assumptions, we show that bounded harmonic maps with image contained in a regular geodesic ball share similar behaviour at infinity with the bounded harmonic functions on the domain manifold.

     

Pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature

In this paper, we use heat flow method to prove the existence of pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature, which is a generalization of Eells–Sampson’s existence theorem. Furthermore, when the target manifold has negative sectional curvature, we analyze horizontal energy of geometric homotopy of two pseudo-harmonic maps and obtain that if the image of a pseudo-harmonic map is neither a point nor a closed geodesic, then it is the unique pseudo-harmonic map in the given homotopic class. This is a generalization of Hartman’s theorem.