2021/10/24 by Zhong, Chunping
#32Q99 #53C60 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2110.12436
Let Bn and Pn be the unit ball and the unit polydisk in ℂn with n≥ 2 respectively. Denote Aut(Bn) and Aut(Pn) the holomorphic automorphism group of Bn and Pn respectively. In this paper, we prove that Bn admits no Aut(Bn)-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Poincar\acutee-Bergman metric, while Pn admits infinite many Aut(Pn)-invariant complete strongly convex complex Finsler metrics other than the Bergman metric. The Aut(Pn)-invariant complex Finsler metrics are explicitly constructed which depend on a real parameter t∈ [0,+∞) and integer k≥ 2. These metrics are proved to be strongly convex Kähler-Berwald metrics, and they posses very similar properties as that of the Bergman metric on Pn. As applications, the existence of Aut(M)-invariant strongly convex complex Finsler metrics is also investigated on some Siegel domains of the first and the second kind which are biholomorphic equivalently to the unit polydisc in ℂn. We also give a characterization of strongly convex Kähler-Berwald spaces and give a de Rahm type decomposition theorem for strongly convex Kähler-Berwald spaces.