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 ℂn, n≥ 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.
Related
- Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples
- Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature
- Constant Chern holomorphic sectional curvature on two-step solvmanifolds
- Hermitian manifolds with constant holomorphic sectional curvature or real bisectional curvature
- Hermitian connections with parallel torsion and constant holomorphic sectional curvature
BibTeX
@misc{heier2026biholomorphic,
title = {{Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature}},
author = {Gordon Heier and Ravi Mudaliar and Haotian Zuo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13831},
eprint = {2610.13831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13831},
}
RIS
TY - UNPB
TI - Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
AU - Gordon Heier
AU - Ravi Mudaliar
AU - Haotian Zuo
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13831
UR - https://arxiv.org/abs/2610.13831
AB - Based on earlier work by Wiegerinck and Huang-Li, Hayashida-Kamimoto constructed unbounded non-smooth Reinhardt domains in ℂ^n, n≥ 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.
ER -