Schwarz lemma on polydiscs endowed with holomorphic invariant Kähler-Berwald metrics

2023/01/15 by Lin, Shuqing, Sun, Liling, Zhong, Chunping
#32H02 #53C56 #53C60 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2301.06102

Abstract

In this paper, we obtain a Schwarz lemma for holomorphic mappings from the unit polydisc P_m into the unit polydisc P_n, here P_m and P_n are endowed with Aut(P_m)-invariant Kähelr-Berwald metric F_t,k and Aut(P_n)-invariant Kähler-Berwald metric F_t,k respectively. Our result generalizes the Schwarz lemma for holomorphic mappings from P_m into P_n whenever P_m and P_n are endowed with the Bergman metrics respectively. We also obtain a distortion theorem on the unit polydisc P_m, where P_m is endowd with an Aut(P_m)-invariant Kähler-Berwald metric F_t,k, and show that for each fixed t∈[0,+∞) and integer k≥ 2, F_t,k is actually a Kähler Finsler-Einstein metric in the sense of T. Aikou.

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