Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Schwarz-Pick lemma for harmonic maps which are conformal at a point

Franc Forstneric, David Kalaj

Abstract

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb D$ in $\mathbb C$ into the unit ball $\mathbb B^n$ in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $n\ge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $\mathbb D\to \mathbb B^n$. This implies that conformal harmonic immersions $M \to \mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$\mathrm{é}$ metric on $M$ and the Cayley-Klein metric on the ball $\mathbb B^n$, and the extremal maps are precisely the conformal embeddings of the disc $\mathbb D$ onto affine discs in $\mathbb B^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in $\mathbb R^n$ based on minimal surfaces.

Schwarz-Pick lemma for harmonic maps which are conformal at a point

Abstract

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc in into the unit ball in , , at any point where the map is conformal. In dimension , this generalizes the classical Schwarz-Pick lemma, and for it gives the optimal Schwarz-Pick lemma for conformal minimal discs . This implies that conformal harmonic immersions from any hyperbolic conformal surface are distance-decreasing in the Poincar metric on and the Cayley-Klein metric on the ball , and the extremal maps are precisely the conformal embeddings of the disc onto affine discs in . By using these results, we lay the foundations of the hyperbolicity theory for domains in based on minimal surfaces.

Paper Structure

This paper contains 6 sections, 14 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Proof of Theorem \ref{['th:main']}
  4. Proof of Theorem \ref{['th:main2']}
  5. A Schwarz--Pick lemma for quasiconformal harmonic maps
  6. An intrinsic pseudometric defined by conformal harmonic discs

Key Result

Theorem \oldthetheorem

Let $\mathbb D=\{z\in\mathbb{C}:|z|<1\}$ denote the unit disc. If $f:\mathbb D\to \mathbb D$ is a harmonic map which is conformal at a point $z\in \mathbb D$, then at this point we have that with equality if and only if $f$ is a conformal diffeomorphism of the disc $\mathbb D$.

Theorems & Definitions (29)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Metric- and distance-decreasing property of conformal harmonic maps
  • proof
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 19 more