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

Forstnerič, Franc (avtor), Kalaj, David (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: harmonic maps, minimal surfaces, Schwarz–Pick lemma, Cayley–Klein metric
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.01.2024
• Leto izida: 2024
• Št. strani: Str. 981-1003
• Številčenje: Vol. 17, no. 3
• PID: 20.500.12556/RUL-155980
• UDK: 517.5
• ISSN pri članku: 2157-5045
• DOI: 10.2140/apde.2024.17.981
• COBISS.SI-ID: 193933059
• Datum objave v RUL: 25.04.2024

Gradivo je del revije

• Naslov: Analysis & PDE
• Skrajšan naslov: Anal. PDE
• Založnik: Mathematical Sciences Publishers
• ISSN: 2157-5045
• COBISS.SI-ID: 16644953

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: harmonične preslikave, minimalne ploskve, Schwarz-Pickov lema, Cayley-Kleinova metrika