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Schwarz–Pick lemma
for harmonic maps which are conformal at a point
Franc Forstnerič and David Kalaj |
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Vol. 17 (2024), No. 3, 981–1003
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Abstract |
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We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc in into the unit ball of , , at any point where the map is conformal. For 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 maps from any hyperbolic conformal surface are distance decreasing in the Poincaré metric on and the Cayley–Klein metric on the ball , and the extremal maps are the conformal embeddings of the disc onto affine discs in . 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
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Mathematical Subject Classification
Primary: 53A10
Secondary: 30C80, 31A05, 32Q45
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Milestones
Received: 1 December 2021
Revised: 5 July 2022
Accepted: 11 August 2022
Published: 24 April 2024
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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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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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