A new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps

TL;DR

This paper develops new boundary versions of the Schwarz-Pick lemma for conformal pseudometrics and holomorphic maps, providing boundary rigidity results and extending classical inequalities to higher dimensions and sequences.

Contribution

It introduces novel boundary rigidity theorems for conformal pseudometrics and holomorphic maps, including sequential versions and boundary versions of classical lemmas.

Findings

New boundary Schwarz-Pick lemmas for the unit disk

Sequential boundary rigidity theorems for conformal pseudometrics

Boundary Schwarz lemma for holomorphic maps in higher dimensions

Abstract

In this paper we establish several invariant boundary versions of the (infinitesimal) Schwarz-Pick lemma for conformal pseudometrics on the unit disk and for holomorphic selfmaps of strongly convex domains in $\mathbb C^N$ in the spirit of the boundary Schwarz lemma of Burns-Krantz. Firstly, we focus on the case of the unit disk and prove a general boundary rigidity theorem for conformal pseudometrics with variable curvature. In its simplest cases this result already includes new types of boundary versions of the lemmas of Schwarz-Pick, Ahlfors-Schwarz and Nehari-Schwarz. The proof is based on a new Harnack-type inequality as well as a boundary Hopf lemma for conformal pseudometrics which extend earlier interior rigidity results of Golusin, Heins, Beardon, Minda and others. Secondly, we prove similar rigidity theorems for sequences of conformal pseudometrics, which even in the interior case appear to be new. For instance, a first sequential version of the strong form of Ahlfors' lemma is obtained. As an auxiliary tool we establish a Hurwitz-type result about preservation of zeros of sequences of conformal pseudometrics. Thirdly, we apply the one-dimensional sequential boundary rigidity results together with a variety of techniques from several complex variables to prove a boundary version of the Schwarz-Pick lemma for holomorphic maps of strongly convex domains in $\mathbb C^N$ for $N>1$.