COMPLEX P-LAPLACIAN ON KAHLER MANIFOLDS AND ITS APPLICATIONS

Abstract

From a geometric point of view, we use coordinates as the main tool to define the holomorphic gradient, the antiholomorphic gradient, and the complex gradient of a complex-valued function on Kahler manifolds. Then we define the holomorphic Laplacian, the antiholomorphic Laplacian, and the complex Laplacian of a real-valued function. For the first time, we introduce the holomorphic p-Laplacian, the antiholomorphic p-Laplacian, and the complex p-Laplacian, and we find the relationship among them. We also find a relationship between the complex p-Laplacian and the usual Riemannian p-Laplacian. Finally, based on this relationship, we make global integral estimates on complete noncompact Kahler manifolds as an application of the complex p-Laplacian, the holomorphic p-Laplacian, and the antiholomorphic p-Laplacian.