Minimal discs in hyperbolic space bounded by a quasicircle at infinity

作者：Seppi Andrea

来源：Commentarii Mathematici Helvetici, 2016, 91(4): 807-839.

DOI：10.4171/CMH/403

• Minimal surfaces
• Teichmuller theory

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 Teichmuller 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 Teichmuller 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.

• 出版日期2016