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 $ℂ$ into the unit ball ${\mathbb{𝔹}}^n$ of ${ℝ}^n$, $n\ge 2$, at any point where the map is conformal. For $n=2$ 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 maps $M\to {\mathbb{𝔹}}^n$ from any hyperbolic conformal surface are distance decreasing in the Poincaré metric on $M$ and the Cayley–Klein metric on the ball ${\mathbb{𝔹}}^n$, and the extremal maps are the conformal embeddings of the disc $\mathbb{𝔻}$ onto affine discs in ${\mathbb{𝔹}}^n$. Motivated by these results, we introduce an intrinsic pseudometric on any Riemannian manifold of dimension at least three by using conformal minimal discs, and we lay foundations of the corresponding hyperbolicity theory.