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Commentarii Mathematici Helvetici

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Volume 91, Issue 4, 2016, pp. 807–839
DOI: 10.4171/CMH/403

Published online: 2016-10-24

Minimal discs in hyperbolic space bounded by a quasicircle at infinity

Andrea Seppi[1]

(1) Università degli Studi di Pavia, Italy

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.

Keywords: Minimal surfaces, Teichmüller theory

Seppi Andrea: Minimal discs in hyperbolic space bounded by a quasicircle at infinity. Comment. Math. Helv. 91 (2016), 807-839. doi: 10.4171/CMH/403