The Laplace Operators On Complex Finsler Manifolds And Their Applications

Laplacian plays an important role in harmonic integral and Bochner technique in dif-ferential geometry.In the past twenty years,with the advocacy of Mr.Shiing-shen Chern,a famous geometrist,the global differential geometry on the real and complex Finsler man-ifolds has been made notable progress.Laplacian also makes sense in Finsler manifolds.So far,there are a few results about the Laplacian on real Finsler manifolds and some on complex Finsler manifolds.The purpose of this thesis is to study the Laplace operators and their applications on a pseudoconvex complex Finsler manifold.This thesis contains four chapters.The first chapter gives some backgrounds and some knowledge of complex Finsler manifolds.The second chapter,we first introduce a family Hermitian metrics Ga,b with two param-eters a>0,b ? 0 on the holomorphic tangent bundle T1,0 M of a strongly pseudoconvex complex Finsler manifold(M,F).The family Hermitian metrics Ga,b are called g-nature metrics introduced by F,which contain Sasaki-Matumoto metric and Miron metric.Then we define the complex horizontal Laplacian ?h and the complex vertical Laplacian ?v as-sociated to the complex RunU connection on the holomorphic tangent bundle T1,0 M.We obtain a precise relationship among ?h,?v and the Hodge-Laplace operator ? associated to the complex connection on(T1,0 M,Ga,b),Furthermore,holomorphic Killing vector fields associated to Ga b are investigated.The third chapter,we define a global inner product of differential forms with(p,q)type on a compact pseudoconvex complex Finsler manifold,which leads to(?)'s dual operator(?)*and Laplace operator ?.The Laplace operator ? can be regarded as an extension of that on Hermitian manifolds.Next we derive the local coordinate expression of the Laplace operator.Finally,we prove that the Laplace operator is a self-adjoint elliptic operator and a Hodge decomposition theorem holds.The forth chapter,firstly we study the dual complex Finsler metric and the Legendre transformation of a strongly pseudoconvex complex Finsler metric.Secondly,we get a pre-cise relationship among the non-linear Laplace operator and the holomorphic S-curvature.Moreover,we prove that there exists a minimal geodesic joints any two points on a com-plete strongly pseudoconvex complex Finsler manifold(M,F)and the gradient ?1r of the distance functions r introduced by F satisfying F(?1r)= 1.At last,we obtain some Lapla-cian comparison theorems.