Hyperbolicity theory for minimal surfaces in Euclidean spaces

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 $mathbb R^n$, $nge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $nge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $mathbb Dto mathbb B^n$. This implies that conformal harmonic immersions $M to mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$mathrm{e}$ 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.