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

Abstract

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb{D}$ in $\mathbb{C}$ into 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, and for $n\ge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $\mathbb{D}\to {\mathbb{B}}^n$. This implies that conformal harmonic immersions $M\to {\mathbb{B}}^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$\acute{\mathrm{e}}$ metric on $M$ and the Cayley-Klein metric on the ball ${\mathbb{B}}^n$, and the extremal maps are precisely the conformal embeddings of the disc $\mathbb{D}$ onto affine discs in ${\mathbb{B}}^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in ${\mathbb{R}}^n$ based on minimal surfaces.

References in corpus (2)

• Hyperbolic domains in real Euclidean spaces
• A converse to the Schwarz lemma for planar harmonic maps

Cited by in corpus (3)

• Hyperbolic domains in real Euclidean spaces
• Flexible domains for minimal surfaces in Euclidean spaces
• On the Gromov hyperbolicity of the minimal metric