A Mean Value Laplacian for Strongly Kähler-Finsler Manifolds

TL;DRAbstract

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

Chat with Paper

AI Agents for this Paper