Two boundary rigidity results for holomorphic maps

TL;DR

This paper proves two new boundary Schwarz lemmas for holomorphic maps, one for convex domains with smooth boundaries and another for domains with invariant Kähler metrics, expanding boundary behavior understanding.

Contribution

It introduces two boundary Schwarz lemmas that do not require strong pseudoconvexity or boundary regularity, broadening the scope of boundary rigidity results for holomorphic maps.

Findings

First boundary Schwarz lemma for convex domains with $C^2$ boundary.

Second boundary Schwarz lemma for domains with invariant Kähler metrics.

Results apply to holomorphic homogeneous regular domains.

Abstract

In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with $C^2$ boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. The second is for biholomorphisms of domains who have an invariant K\"ahler metric with bounded sectional curvature. This second result applies to holomorphic homogeneous regular domains and appears to be the first boundary Schwarz lemma that makes no assumptions on the regularity of the boundary.