Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents

TL;DR

This paper explores how strong pseudoconvexity of bounded domains with smooth boundaries can be characterized through intrinsic complex geometric properties, using dynamical behavior of geodesics in the Kobayashi metric, with implications for biholomorphic equivalences.

Contribution

It provides new characterizations of strong pseudoconvexity in terms of boundary behavior of the squeezing function and Bergman metric curvature, especially for convex domains with low regularity.

Findings

Strong pseudoconvexity characterized by boundary behavior of geometric measures

Biholomorphic equivalence preserves strong pseudoconvexity under certain regularity conditions

Partial answer to a question of Forn{ a}ss and Wold regarding boundary characterizations

Abstract

In this paper we consider the following question: For bounded domains with smooth boundary, can strong pseudoconvexity be characterized in terms of the intrinsic complex geometry of the domain? Our approach to answering this question is based on understanding the dynamical behavior of real geodesics in the Kobayashi metric and allows us to prove a number of results for domains with low regularity. For instance, we show that for convex domains with $C^{2,\epsilon}$ boundary strong pseudoconvexity can be characterized in terms of the behavior of the squeezing function near the boundary, the behavior of the holomorphic sectional curvature of the Bergman metric near the boundary, or any other reasonable measure of the complex geometry near the boundary. The first characterization gives a partial answer to a question of Forn{\ae}ss and Wold. As an application of these characterizations, we show that a convex domain with $C^{2,\epsilon}$ boundary which is biholomorphic to a strongly pseudoconvex domain is also strongly pseudoconvex.