Minimal discs in hyperbolic space bounded by a quasicircle at infinity

Andrea Seppi (1)

1 UNIPV - Dipartimento di Matematica “Felice Casorati” Department of Mathematics [Univ Pavia]

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 Teichmüller 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 Teichmüller 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.

Domaines: Mathématiques [math]

Andrea Seppi. Minimal discs in hyperbolic space bounded by a quasicircle at infinity. Commentarii Mathematici Helvetici, 2016, 91, pp.807-839. ⟨10.4171/CMH/403⟩. ⟨hal-01300367v2⟩