# Geometry Seminar

#### 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