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

Franc Forstnerič and David Kalaj

Vol. 17 (2024), No. 3, 981–1003
Abstract

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc 𝔻 in into the unit ball 𝔹n of n, n 2, at any point where the map is conformal. For n = 2 this generalizes the classical Schwarz–Pick lemma, and for n 3 it gives the optimal Schwarz–Pick lemma for conformal minimal discs 𝔻 𝔹n. This implies that conformal harmonic maps M 𝔹n from any hyperbolic conformal surface are distance decreasing in the Poincaré metric on M and the Cayley–Klein metric on the ball 𝔹n, and the extremal maps are the conformal embeddings of the disc 𝔻 onto affine discs in 𝔹n. Motivated by these results, we introduce an intrinsic pseudometric on any Riemannian manifold of dimension at least three by using conformal minimal discs, and we lay foundations of the corresponding hyperbolicity theory.

Keywords
harmonic map, minimal surface, Schwarz–Pick lemma, Cayley–Klein metric
Mathematical Subject Classification
Primary: 53A10
Secondary: 30C80, 31A05, 32Q45
Milestones
Received: 1 December 2021
Revised: 5 July 2022
Accepted: 11 August 2022
Published: 24 April 2024
Authors
Franc Forstnerič
Faculty of Mathematics and Physics
University of Ljubljana
Ljubljana
Slovenia
Institute of Mathematics, Physics, and Mechanics
Ljubljana
Slovenia
David Kalaj
Faculty of Natural Sciences and Mathematics
University of Montenegro
Podgorica
Montenegro

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