Laplace Operator On Finsler Manifolds

The content of this paper is consisted of six sections. The first one is introduction. In chapter one, we introduce many important geometric quantities on Finsler manifolds. chapter two define two kinds of important Laplacian in Finsler space. In chapter three, we derive Laplace comparison theorem of distance function on the condition that Ricci curvature has function lower bound. As its applications, we obtain bound estimates for the first eigenvalue of Laplacian on Finsler manifolds. Our theorem generalize results of S.Y.Cheng ([21]) and P.Li ([28]) in Riemannian case. In chapter four, we derive Laplace comparison theorem of distance function on the condition that Ricci curvature has function upper bound. As its geometric applications, we obtain lower bound estimates for the first eigenvalue. Lastly, we introduce the second kind of Laplace operator . Using heat equation method, we prove existence theorem of harmonic maps from compact Finsler manifolds.Firstly, we introduce a Laplacian which has close relationship with curvature vector. Let f:M→R be a smooth function on Finsler manifold(M, F), the Laplacian of f is defined:Δf := div gardfwhere the gradient of f is defined by Legendre transformationι:(?)f=ι^-1(df).By the Euler-Lagrange equation, we obtain Laplace comparison theorem on different curvature conditions.When Ricci curvature of manifold has function lower bound, we have:As its applications, we obtain bound estimates of the first eigenvalue on Finsler manifolds. Especially, the upper bound of the first eigenvalue is a constant which only depends on n, k and d.When Ricci curvature of manifold has negative function upper bound, we have:Hence, we obtain lower bound estimates of the first eigenvalue on geodesic balls. Lastly, we introduce another kind of linear operator which is mean value Laplacian. In the local coordinate, A/ is expressed by:This operator has important character which likes Riemann Laplacian: "self-adjoint". Using Bochner formula on Finsler manifold, we prove that any differential map from compact Finsler manifold can deform into harmonic map between two manifolds when Riemannian curvature of tangent manifold is negative.