Some Results On Harmonic Map Between Riemann-Finsler Manifolds

The content of this paper is divided into two parts.The first part contains chapter one,chapter two and chapter three. The second part consist of chapter four.In chapter one,we study the energy density of harmonic map from Finsler manifold and generalize classical result in[Se].In chapter two,we obtain lower estimates for the first eigenvalue of the Laplace operator on a compact Finsler manifold,and it generalize Lichnerowicz-Obata theorem[Li][Ob].In chapter three ,we derive the first and second variation formula for harmonic maps between Finsler manifolds.As an application,some nonexistence theorems of nonconstant stable harmonic maps from a Finsler manifold to a Riemannian manifold are given. In chapter four,we study the compact 2-harmonic submanifolds of Riemannian manifolds.Some intrinsic metrics in differential manifolds,such as cara-theodory metrics and kobayashi metrics in complex manifolds,are Finsler metrics.Finsler metrics is just Riemannian metrics without quadratic restriction,which was firstly introduced by B.Riemann in 1854.The geometry with Finsler metric is called Finsler geometry. The geometric methods developed in Finsler geometry are useful in studying some problems arising from biology,physics and other fields([AbPa][AnZal][AnZa2][AnInma][Asl][Bj][Mil][MiAn]).Recent studies on Finsler geometry have taken on a new look[BCS][Sh]. Harmonic maps between Riemannian manifolds are very important in both differential geometry and mathematical physics.Riemannian manifold and Finsler manifold are metric measure space,so we can study harmonic map between Finsler manifolds by the theory of harmonic map on general metric measure space, it will be hard to study harmonic map between Finsler manifolds by tensor analysis and it will be no distinctions between the theory of harmonic map on Finsler manifold and that of metric measure space.Harmonic map between Riemannian manifold also can be viewed as the harmonic map between tangent bundles of source manifold and target manifold. [Mo1]introduce other definition about harmonic map.In fact,it is just the harmonic map from projective sphere bundle of source manifold to tangent bundle of target manifold,[Mo1] also derive the first variation formula for harmonic maps from Finsler manifolds to Riemannian manifold :is called tension field of φ. By moving frame,we can get;Proposition. Let M is a compact Finsler manifold,N is a Riemannian manifold is a Riemannian manifoldBy an orthonormal basis, we can defineProm Proposition,we can getTheorem. Let a, b is positive constant , RiemN means sectional curvature of N.max then φ is constant map or totally geodesic map.Moreover,when e(φ)≤a/2b, φ must be a constant map .Let (M, F) is a Finsler manifold, (N,g) is a Riemannian manifold, φ : (M,F) → (N,g) .When r(φ) = trab dφ = 0, the first variation of energy vanish, φ is called harmonic map[Mol].In Riemannian geometry,harmonic map generalize harmonic function. When target manifold is R,.If u is a function of Finsler manifold,we can define Laplace operator ,it is well-defined .If u is called the eigenvalue of the laplacian A and u is called the corresponding eigenfunction. the nonzero least eigenvalue is called the first eigenvalue.Lemma. For a function u : (M, F) →R,we haveBy these formula, we can getTheorem. Let (M,F) be an n-dimensional compact Finsler manifold without boudary.Iffor some positive constant K,thenMoreover,the diameter of M is when λ1= mK.Some open problems in Finsler geometry have been proposed in [MS]. One of open problems is to study harmonic maps between Finsler manifolds and derive the first and second variation formula for harmonic maps between Finsler manifolds. Firstly, we define harmonic map between Finsler manifold.In fact,it is the harmonic map from projective sphere bundle of source manifold to the projective sphere bundle of target manifold. We haveTheorem. Let (M,F) be an n-dimensional compact Finsler manifold without boudary.φ : (M, F) →(M, F) b...