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.     

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.     

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.