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

Forstnerič, Franc (author), Kalaj, David (author)

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 Poincaré 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 ${\mathbb B}^n$. 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.

• Language: English
• Keywords: harmonic maps, minimal surfaces, Schwarz–Pick lemma, Cayley–Klein metric
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2024
• Year: 2024
• Number of pages: Str. 981-1003
• Numbering: Vol. 17, no. 3
• PID: 20.500.12556/RUL-155980
• UDC: 517.5
• ISSN on article: 2157-5045
• DOI: 10.2140/apde.2024.17.981
• COBISS.SI-ID: 193933059
• Publication date in RUL: 25.04.2024

Record is a part of a journal

• Title: Analysis & PDE
• Shortened title: Anal. PDE
• Publisher: Mathematical Sciences Publishers
• ISSN: 2157-5045
• COBISS.SI-ID: 16644953

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: harmonične preslikave, minimalne ploskve, Schwarz-Pickov lema, Cayley-Kleinova metrika