[Paper Review] Characterizing polynomial domains by their automorphism group

Andrew Zimmer|arXiv (Cornell University)|Jun 25, 2015

Holomorphic and Operator Theory15 references3 citations

TL;DR

This paper establishes that a bounded convex domain with smooth boundary is biholomorphic to a weighted homogeneous polynomial domain if and only if the limit set of its automorphism group intersects at least two closed complex faces. The proof employs rescaling techniques, Kobayashi metric geometry, and non-positive curvature methods, confirming a special case of the Greene-Krantz conjecture and yielding new results on holomorphic maps, boundary extensions, and complex geodesics.

ABSTRACT

In this paper we study the automorphism group of bounded convex domains with smooth boundary. In particular, we show that such a domain is biholomorphic to a weighted homogeneous polynomial domain if and only if the limit set of the automorphism group intersects at least two closed complex faces of the set. The proof combines rescaling arguments with a detailed study of the geometry of the Kobayashi metric. In particular a number of ideas from the theory of non-positively curved metric spaces are used. A key step in the argument is establishing the Greene-Krantz conjecture in the case of uniform non-tangential convergence. We also obtain new results about the behavior of holomorphic maps between convex domains, in particular new results about continuous extensions (of bi-holomorphisms and complex geodesics) and a new Denjoy-Wolff theorem.

Motivation & Objective

• To characterize bounded convex domains with smooth boundary that are biholomorphic to weighted homogeneous polynomial domains.
• To determine the geometric conditions under which the automorphism group's limit set intersects multiple closed complex faces.
• To establish a special case of the Greene-Krantz conjecture concerning uniform non-tangential convergence of automorphisms.
• To extend holomorphic maps and complex geodesics continuously to the boundary of convex domains.
• To prove a new Denjoy-Wolff-type theorem for holomorphic self-maps of convex domains.

Proposed method

• Rescaling arguments are used to analyze the asymptotic behavior of automorphism sequences near boundary points.
• The geometry of the Kobayashi metric is studied in detail to understand the structure of the automorphism group's limit set.
• Techniques from non-positively curved metric spaces are applied to analyze convergence and curvature properties of the Kobayashi metric.
• The paper investigates the interaction between complex faces of the boundary and the action of the automorphism group.
• A detailed analysis of holomorphic maps between convex domains is conducted to establish continuous boundary extensions.
• A new Denjoy-Wolff theorem is derived using the interplay between automorphism dynamics and boundary geometry.

Experimental results

Research questions

• RQ1 Under what conditions is a bounded convex domain with smooth boundary biholomorphic to a weighted homogeneous polynomial domain?
• RQ2 How does the intersection of the automorphism group's limit set with multiple closed complex faces relate to the domain's algebraic structure?
• RQ3 Does the Greene-Krantz conjecture hold under uniform non-tangential convergence of automorphisms?
• RQ4 Can biholomorphisms and complex geodesics between convex domains be continuously extended to the boundary?
• RQ5 What dynamical properties do holomorphic self-maps of convex domains exhibit near the boundary, and how do they generalize the Denjoy-Wolff theorem?

Key findings

• A bounded convex domain with smooth boundary is biholomorphic to a weighted homogeneous polynomial domain if and only if the limit set of its automorphism group intersects at least two closed complex faces.
• The Greene-Krantz conjecture is confirmed in the case of uniform non-tangential convergence of automorphism sequences.
• New results are obtained on the continuous extension of biholomorphisms and complex geodesics from the interior to the boundary of convex domains.
• A new Denjoy-Wolff theorem is established for holomorphic self-maps of convex domains, generalizing the classical result to this geometric setting.
• The Kobayashi metric's geometry, particularly in relation to non-positive curvature, plays a central role in characterizing the automorphism group's limit set.