N ov 2 02 1 WHEN IS THE KOBAYASHI METRIC A KÄHLER METRIC ?

We prove that if the Kobayashi metric on a strongly pseudoconvex domain with smooth boundary is a Kähler metric, then the universal cover of the domain is the unit ball.

[1]  B. Wong.  On the holomorphic curvature of some intrinsic metrics , 1977 .

[2]  Samir Khuller,et al.  Open problems , 1997, SIGACT News.

[3]  Chin-Huei Chang,et al.  Extremal analytic discs with prescribed boundary data , 1988 .

[4]  J. Fornæss,et al.  Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains , 2017, Mathematische Zeitschrift.

[5]  Marco Abate,et al.  Iteration theory of holomorphic maps on taut manifolds , 1989 .

[6]  A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in Cn , 2005, math/0512409.

[7]  Xiaojun Huang.  A preservation principle of extremal mappings near a strongly pseudoconvex point and its applications , 1994 .

[8]  Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries , 2020 .

[9]  A. Sukhov.  ON BOUNDARY REGULARITY OF HOLOMORPHIC MAPPINGS , 1995 .

[10]  S. Shnider,et al.  Spherical hypersurfaces in complex manifolds , 1976 .

[11]  J. Igusa.  On the Structure of a Certain Class of Kaehler Varieties , 1954 .

[12]  J. Eschenburg.  Comparison Theorems in Riemannian Geometry , 1994 .

[13]  M. Bonk,et al.  Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains , 2000 .

[14]  Tim Schmitz,et al.  The Foundations Of Differential Geometry , 2016 .

[15]  Charles Stanton.  A characterization of the ball by its intrinsic metrics , 1983 .

[16]  R. Shafikov,et al.  Uniformization of strictly pseudoconvex domains. I , 2004, math/0407316.