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The strong form of the Ahlfors-Schwarz lemma at the boundary and a rigidity result for Liouville's equation

Filippo Bracci, Daniela Kraus, Oliver Roth

Abstract

We prove a boundary version of the strong form of the Ahlfors-Schwarz lemma with optimal error term. This result provides nonlinear extensions of the boundary Schwarz lemma of Burns and Krantz to the class of negatively curved conformal pseudometrics defined on arbitary hyperbolic domains in the complex plane. Based on a new boundary Harnack inequality for solutions of the Gauss curvature equation, we also establish a sharp rigidity result for conformal metrics with isolated singularities. In the particular case of constant negative curvature this strengthens classical results of Nitsche and Heins about Liouville's equation $Δu=e^u$.

The strong form of the Ahlfors-Schwarz lemma at the boundary and a rigidity result for Liouville's equation

Abstract

We prove a boundary version of the strong form of the Ahlfors-Schwarz lemma with optimal error term. This result provides nonlinear extensions of the boundary Schwarz lemma of Burns and Krantz to the class of negatively curved conformal pseudometrics defined on arbitary hyperbolic domains in the complex plane. Based on a new boundary Harnack inequality for solutions of the Gauss curvature equation, we also establish a sharp rigidity result for conformal metrics with isolated singularities. In the particular case of constant negative curvature this strengthens classical results of Nitsche and Heins about Liouville's equation .
Paper Structure (12 sections, 12 theorems, 120 equations)

This paper contains 12 sections, 12 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Proofs
  5. Proof of Theorem \ref{['thm:main1new']}
  6. Theorem \ref{['thm:main1new']} and the Schwarz--Pick lemma of A. Beardon and D. Minda
  7. Proof of Theorem \ref{['thm:main2new']}
  8. Proof of Theorem \ref{['thm:3']}
  9. Sharpness of Theorem \ref{['thm:3']}
  10. Proof of Theorem \ref{['thm:gg']}
  11. Proof of Theorem \ref{['thm:4']}
  12. Some questions

Key Result

Theorem 2.1

Let $\Omega$ be a hyperbolic subdomain of $\hat{\mathbb C}$ and let $\lambda(z)\, |dz|$ be a conformal pseudometric on $\Omega$ with curvature $\le -4$. Suppose that $\{z_n\}$ is a sequence of points in $\Omega$ tending to a boundary point $p \in \partial \Omega$. If for one -- and hence for any -- $q \in \Omega$, then

Theorems & Definitions (32)

  • Theorem 2.1: The strong form of the Ahlfors--Schwarz lemma at the boundary
  • Remark 1
  • Remark 2: The Schwarz--Pick lemma for hyperbolic derivatives of Beardon and Minda BeardonMinda2007
  • Remark 3: Theorem \ref{['thm:main1new']} and the boundary Schwarz Lemma of Burns--Krantz
  • Theorem 2.2: Sharpness of the strong form of Ahlfors' lemma at the boundary
  • Theorem 2.3: The strong boundary Ahlfors lemma for hyperbolic domains with punctures
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 2.4
  • ...and 22 more