Schwarz Lemma On Complex Finsler Manifolds And Its Applications

Complex Finsler geometry is just complex differential geometry without Hermitian quadratic restrictions,thus complex Finsler geometry is more general than Hermitian geometry.There are two intrinsic complex Finsler metrics on every complex manifold,namely,the Kobayashi-Royden pseudo-metric and the Caratheodory pseudo-metric.If the Kobayashi pseudo-distance on a complex manifold is a distance,then the complex manifold is a Kobayashi hyperbolic space.It is known from Schwarz lemma that a complex manifold is a Kobayashi hyperbolic space if there exists a complex Finsler metric whose holomorphic sectional curvature is bounded from above by a negative constant.It follows that complex Finsler geometry and Schwarz lemma play an important role in complex differential geometry.In this thesis,we are devoted to study Schwarz lemmas for holomorphic mappings between two complex manifolds endowed with strongly pseudoconvex complex Finsler metrics,respectively.Especially,to establish a Schwarz lemma from a complete K(?)hler manifold into a complex Finsler manifold,a Schwarz lemma from a complete strongly convex weakly K(?)hler-Finsler manifold into a complex Finsler manifold and give some applications of these two main theorems.We divide this thesis into five chapters.In Chapter one,we give some backgrounds of our study and outline the main results of this thesis.In Chapter two,we introduce the relevant knowledge of Riemannian and Hermitian geometry,especially K(?)hler geometry,real and complex Finsler geometry which are used in this thesis.In Chapter three,we firstly establish the relation between the distance function and the second variational formula on a Riemannian manifold(M,g)with a pole p,so that we can obtain the second order estimation of the square of the distance function.Next,using Gaussian lemma on a Riemannian manifold(M,g)with a pole p,we obtain the first order estimation of the square of the distance function.Using the relation between a Riemannian metric and a K(?)hler metric,we obtain the first,and the second order estimations of the square of the distance function on a K(?)hler manifold(M,h)with a pole p.The holomorphic sectional curvature at a point(z,v)of Hermitian metric h or complex Finsler metric G can be achieved by the Gaussian curvature of the pull-back metric φ*h or φ*G on the unit disk Δ in C,where φ is a holomorphic mapping of the unit disk Δ into the complex manifold M.Then we construct the auxiliary function on the unit disk Δ and apply the maximum principle to the auxiliary function.By using the first and second order estimations of the square of the distance function on a K(?)hler manifold,we obtain the Schwarz lemma from K(?)hler manifold with a pole p into a complex Finsler manifold.Finally,we are able to establish a Schwarz lemma from a complete K(?)hler manifold into a complex Finsler manifold by means of a careful handling of the cut points.In Chapter four,similar to Chapter three,we firstly obtain the first and second order estimations of the square of the distance function on a real Finsler manifold(M,G)with a pole p.On account of a relation between a real Finsler metric and a strongly convex complex Finsler metric,we obain the first,second order estimations of the square of the distance function on a strongly convex weakly K(?)hler-Finsler manifold with a pole p.Since the square of the distance function on a real Finsler manifold with a pole p is not a smooth function.Before using the first,second order estimations of the square of the distance function,it is necessary to polish the square of the distance function and obtain a smooth function(called polish function)satisfying the first,second order estimation conditions,so as to replace the square of the distance function with the polish function.In the same way,we construct the auxiliary function on the unit disk Δ and apply the maximum principle to the auxiliary function.Therefore,we obtain the Schwarz lemma from a strongly convex K(?)hler-Finsler manifold with a pole p into a complex Finsler manifold.Finally,using the technique which deals with cut points,we obain a Schwarz lemma from a complete strongly convex K(?)hler-Finsler manifold into a complex Finsler manifold.In Chapter five,using the Schwarz lemmas established in previous chapters,we are able to obtain some rigidity theorems of holomorphic mappings between complex Finsler manifolds,that is,under what condition does a holomorphic mapping between two complex Finsler manifolds reduce to a constant.Accordingly,we give some examples of rigid results.Finally,using the Schwarz lemma in Chapter three,we obtain some results on the Kobayashi-Royden metric and Caratheodory metric on a strongly bounded convex domain.