Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature

Abstract

Based on earlier work by Wiegerinck and Huang-Li, Hayashida-Kamimoto constructed unbounded non-smooth Reinhardt domains in \({\mathbb C}^n\), \(n\geq 3\), whose Bergman spaces are finite-dimensional and whose Bergman metrics have constant positive holomorphic sectional curvature \(2\). In this note, we prove a Sandwich Principle for the Hayashida-Kamimoto domains that allows us to construct, inside every prescribed sandwich, uncountably many pairwise biholomorphically inequivalent smooth non-Reinhardt domains whose Bergman metrics also have constant positive holomorphic sectional curvature \(2\). We also classify biholomorphisms between domains sandwiched by the domains of Hayashida-Kamimoto with the help of an affine rigidity theorem in the spirit of Cartan rigidity, which should be of independent interest. For explicit angular perturbation families, we determine the full automorphism group and obtain, in particular, an abundance of examples with trivial automorphism group.