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.)
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