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

Fecha: 12 de marzo de 2021.
Hora: 12:00 a 13:00.
Lugar: Online con password 215111.
Author: Franc Forstneric (Universidad de Liubliana).

Summary: We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc D in C to the unit ball Bn in Rn, n≥2, at any point where the map is conformal. In dimension n=2 this generalizes the classical Schwarz-Pick lemma to harmonic maps D→D which are conformal only at the reference point. In dimensions n≥3 it gives the optimal Schwarz-Pick lemma for conformal minimal discs D→Bn. Let M denote the restriction of the Bergman metric on the complex n-ball to the real n-ball Bn. We show that conformal harmonic immersions M→(Bn,M) from any hyperbolic open Riemann surface M with its natural Poincaré metric are distance-decreasing, and the isometries are precisely the conformal embeddings of D onto affine discs in Bn. (Joint work with David Kalaj.)