Characterizing domains by the limit set of their automorphism group

TL;DR

This paper characterizes convex domains by their automorphism groups' limit sets, linking geometric properties to biholomorphic equivalence with polynomial ellipsoids, and introduces new results in complex analysis and geometric function theory.

Contribution

It establishes a criterion for biholomorphic equivalence to polynomial ellipsoids based on automorphism group limit sets and advances understanding of domain geometry and automorphism dynamics.

Findings

Domains biholomorphic to polynomial ellipsoids characterized by automorphism limit sets

Proved Greene-Krantz conjecture for uniform non-tangential convergence

Developed new Wolff-Denjoy theorem for complex domains

Abstract

In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed complex faces of the set. The proof relies on a detailed study of the geometry of the Kobayashi metric and ideas from the theory of non-positively curved metric spaces. We also obtain a number of other results including the Greene-Krantz conjecture in the case of uniform non-tangential convergence, new results about continuous extensions (of biholomorphisms and complex geodesics), and a new Wolff-Denjoy theorem.