On the Kähler-hyperbolicity of bounded symmetric domains

TL;DR

This work characterizes the Kähler-hyperbolicity length of bounded symmetric domains equipped with the Bergman metric, tying the constant gradient length of Bergman potentials to a universal invariant $\\mathsf{L}_\\Omega$ determined by rank and genus. By analyzing the gradient vector field of Kähler potentials and exploiting totally geodesic discs inside maximal polydiscs, the authors show that any local potential with constant gradient length must satisfy $|\\partial\\varphi|_\\omega^2=\\mathsf{L}_\\Omega^2/\\mathsf{K}$, while any global potential obeys a sharp lower bound $\sup|\\partial\\varphi|_\\omega^2 \ge\\mathsf{L}_\\Omega^2/\\mathsf{K}$. The results hold for irreducible and reducible domains, with the invariant $\\mathsf{L}_\\Omega^2$ given by $r c_\Omega$ (or the sum over factors in the reducible case), and the curvature of the relevant discs is explicitly computed as $\\kappa=-2\\mathsf{K}/\\mathsf{L}_\\Omega^2$. Collectively, these findings provide a precise, rigidity-based understanding of Bergman-potential gradients and their minimal possible magnitude under the Bergman-Kähler-Einstein setting, with implications for $L^\infty$ gradient bounds and $d$-boundedness in Kähler geometry.

Abstract

In this paper, we characterize the Kähler-hyperbolicity length of a bounded symmetric domain, defined by its rank and genus, as a unique constant determined by a constant gradient length of a special Bergman potential. Additionally, we establish a characterization of the lower bound of $L^\infty$ norm of the gradient length of any Bergman potential.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 2.1
• Corollary 2.2
• Corollary 2.3
• Proposition 2.4
• proof
• Proposition 3.1