A Mean Value Laplacian for Strongly Kähler-Finsler Manifolds

Abstract

It is well known that the Laplace operator plays an important role in the theory of harmonic integrals and the Bochner technique both in Riemannian and Kähler manifolds. In recent years, under the initiation of S.S. Chern, the global differential geometry of real and complex Finsler manifolds has gained a great development ([1], [2], [3], [4]). A lot of results about the Laplacian and its applications have been obtained in a real Finsler manifold ([5], [6]). But up to now there are no results for the Laplacian and its applications in a complex Finsler manifold. The key point in the theory of the Bochner technique and harmonic integrals is to define a suitable Laplace operator. In the case of Finsler manifolds the difficulty is that the Finsler metric depends on the fibre coordinates. Using the idea that the Laplacian on Euclidean space or on a Riemannian manifold measures the average value of a function around a point, P. Centore ([7]) generalizes the Laplacian on a Riemannian manifold to a real Finsler manifold. Considering a complex manifold as a real manifold, there is a one-to-one correspondence between the real coordinates and the complex coordinates ([8]). In this paper, we use the mean value idea to define the Laplacian on a strongly Kähler-Finsler manifold, first for functions and then for forms, and we derive some remarkable properties for the Laplacian for functions and extend the Laplacian to arbitrary forms. Indeed our Laplacian on strongly Kähler-Finsler manifolds generalizes the Kählerian Laplacian. And it is worth to remark that using the osculating Kähler metric — which we obtain in the following to define the pointwise and global inner product when we define the Hodge-Laplace operator of (p, q)-forms — is more natural than using the fundamental tensor of the Finsler metric and can avoid many complicated calculations.

This project is supported by the Natural Science Foundation of China (No. 10271097).