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

Publication date: 24 February 2021

Abstract: We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc

m a t h b b D

in

m a t h b b C

into the unit ball

m a t h b b B^{n}

in

m a t h b b R^{n}

,

n g e 2

, at any point where the map is conformal. In dimension

n = 2

, this generalizes the classical Schwarz-Pick lemma, and for

n g e 3

it gives the optimal Schwarz-Pick lemma for conformal minimal discs

m a t h b b D o m a t h b b B^{n}

. This implies that conformal harmonic immersions

M o m a t h b b B^{n}

from any hyperbolic conformal surface are distance-decreasing in the Poincar

m a t h r m' e

metric on

M

and the Cayley-Klein metric on the ball

m a t h b b B^{n}

, and the extremal maps are precisely the conformal embeddings of the disc

m a t h b b D

onto affine discs in

m a t h b b B^{n}

. By using these results, we lay the foundations of the hyperbolicity theory for domains in

m a t h b b R^{n}

based on minimal surfaces.