Integral formulas for differential forms of type (p,q) on complex Finsler manifolds

Using the invariant integral kernel introduced by Demailly and Laurent-Thiebaut, complex Finsler metric and nonlinear connection associating with Chern-Finsler connection, we research the integral representation theory on complex Finsler manifolds. The Koppelman and Koppelman-Leray formulas are obtained, and the -equations are solved.     

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

Hodge decomposition theorem on strongly Kahler Finsler manifolds Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the (?)-Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M,F) is a compact strongly Kahler Finsler manifold, we define a (?)-Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in Kahler manifolds is still true in the more general compact strongly Kahler Finsler manifolds.     

The Poincaré-Bertrand Formula for the Bochner-Martinelli Integral

The Poincaré-Bertrand formula and the composition formula for the Bochner-Martinelli integral on piecewise smooth manifolds are obtained. As an application, the regularization problem for linear singular integral equation with Bochner-Martinelli kernel and variable coefficients is discussed.     

The Koppelman-Leray formula on complex Finsler manifolds

By means of the invariant integral kernel (the Berndtsson kernel), the complex Finsler metric and the non-linear connection associated with the Chern-Finsler connection to research into the integral representation theory on complex Finsler manifolds, the Koppelman and Koppelman-Leray formulas are obtained, and the (?)--equations are solved.     

Hodge decomposition theorem on strongly K(a)hler Finsler manifolds

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the (θ)-Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M, F) is a compact strongly K(a)hler Finsler manifold, we define a (θ)-Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in K(a)hler manifolds is still true in the more general compact strongly K(a)hler Finsler manifolds.     

BOCHNER TECHNIQUE ON STRONG KHLER-FINSLER MANIFOLDS

By using the Chern-Finsler connection and complex Finsler metric,the Bochner technique on strong Khler-Finsler manifolds is studied.For a strong Khler-Finsler manifold M,the authors first prove that there exists a system of local coordinate which is normalized at a point v ∈ M-=T 1,0M\o(M),and then the horizontal Laplace operator H for diffierential forms on PTM is defined by the horizontal part of the Chern-Finsler connection and its curvature tensor,and the horizontal Laplace operator H on holomorphic vector bundle over PTM is also defined.Finally,we get a Bochner vanishing theorem for diffierential forms on PTM.Moreover,the Bochner vanishing theorem on a holomorphic line bundle over PTM is also obtained     

Horizontal Laplace Operator in Real Finsler Vector Bundles

A vector bundle F over the tangent bundle TM of a manifold M is said to be a Finsler vector bundle if it is isomorphic to the pull-back π^＊E of a vector bundle E over M（[1]）. In this article the authors study the h-Laplace operator in Finsler vector bundles. An h-Laplace operator is defined, first for functions and then for horizontal Finsler forms on E. Using the h-Laplace operator, the authors define the h-harmonic function and ho harmonic horizontal Finsler vector fields, and furthermore prove some integral formulas for the h-Laplace operator, horizontal Finsler vector fields, and scalar fields on E.     

Horizontal (-δ)-Laplacian on complex Finsler manifolds

A horizontal (-δ)-Laplacian is defined on strongly pseudoconvex complex Finsler manifolds, first for functions and then for horizontal differential forms of type (p,q). The principal part of the (-δ)-Laplacian is computed in local coordinates. As an application, the (-δ)-Laplacian on strongly Kahler Finsler manifold is obtained explicitly in terms of the horizontal covariant derivatives of the Chern-Finsler conncetion.