Minimal discs in hyperbolic space bounded by a quasicircle at infinity

TL;DR

This paper establishes a relationship between the principal curvatures of minimal discs in hyperbolic space and the Teichmüller norm of their boundary quasicircles, with implications for the geometry of quasi-Fuchsian manifolds.

Contribution

It provides a new estimate linking principal curvatures of minimal surfaces to the boundary quasicircle's Teichmüller norm and proves a universal bound for almost-Fuchsian manifolds based on Teichmüller distance.

Findings

Principal curvatures are estimated in a sublinear way by the Teichmüller norm.

Existence of a universal constant C for almost-Fuchsian manifolds with bounded Teichmüller distance.

Estimates on the convex hull of minimal surfaces and curvature control via Schauder estimates.

Abstract

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichm\"uller space, if the quasicircle is sufficiently close to being the boundary of a totally geodesic plane. As a by-product we prove that there is a universal constant C independent of the genus such that if the Teichm\"uller distance between the ends of a quasi-Fuchsian manifold $M$ is at most C, then $M$ is almost-Fuchsian. The main ingredients of the proofs are estimates on the convex hull of a minimal surface and Schauder-type estimates to control principal curvatures.