LAUSR

Let M be a strongly pseudoconvex compact complex Finsler manifold. We first introduce a class of g-nature metric Ga,b for the slit holomorphic tangent bundle M = T1,0M∖{0} on M. Then, we define the complex horizontal Laplacian □h, and complex vertical Laplacian □v, and obtain a precise relationship among □h, □v and the Hodge–Laplace operator □ on (M,Ga,b). As an application, we discuss the holomorphic Killing vector fields associated to Ga,b.