Event Details
Date: March 12, 2021
Time: 12h-13h
Author: Franc Forstneric (Universidad de Liubliana).
Title: Schwarz-Pick lemma for harmonic maps which are conformal at a point
Summary: We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc \mathbb D in \mathbb C to the unit ball \mathbb B^n in \mathbb R^n, n\geq 2, at any point where the map is conformal. In dimension n=2 this generalizes the classical Schwarz-Pick lemma to harmonic maps \mathbb D \to \mathbb D which are conformal only at the reference point. In dimensions n\geq 3 it gives the optimal Schwarz-Pick lemma for conformal minimal discs \mathbb D \to \mathbb B^n. Let \mathcal M denote the restriction of the Bergman metric on the complex n-ball to the real n-ball \mathbb B^n. We show that conformal harmonic immersions M\to (\mathbb B^n,\mathcal 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 \mathbb D onto affine discs in \mathbb B^n. (Joint work with David Kalaj.)
Where: https://oficinavirtual.ugr.es/redes/SOR/SALVEUGR/accesosala.jsp?IDSALA=22960034
Password: 215111
Time: 12h-13h
Author: Franc Forstneric (Universidad de Liubliana).
Title: Schwarz-Pick lemma for harmonic maps which are conformal at a point
Summary: We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc \mathbb D in \mathbb C to the unit ball \mathbb B^n in \mathbb R^n, n\geq 2, at any point where the map is conformal. In dimension n=2 this generalizes the classical Schwarz-Pick lemma to harmonic maps \mathbb D \to \mathbb D which are conformal only at the reference point. In dimensions n\geq 3 it gives the optimal Schwarz-Pick lemma for conformal minimal discs \mathbb D \to \mathbb B^n. Let \mathcal M denote the restriction of the Bergman metric on the complex n-ball to the real n-ball \mathbb B^n. We show that conformal harmonic immersions M\to (\mathbb B^n,\mathcal 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 \mathbb D onto affine discs in \mathbb B^n. (Joint work with David Kalaj.)
Where: https://oficinavirtual.ugr.es/redes/SOR/SALVEUGR/accesosala.jsp?IDSALA=22960034
Password: 215111
