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

## University of Ljubljana

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 \(\mathbb R^n\), \(n\ge 2\), 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 \(n\ge 3\) 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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