Counting Minimal Surfaces in Quasi-Fuchsian three-Manifolds

Abstract

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many of them. In this paper, for any prescribed integer $N>0$, we construct a quasi-Fuchsian manifold which contains at least $2^N$ such minimal surfaces. As a consequence, there exists some simple close Jordan curve on $S_{\infty }^2$ such that there are at least $2^N$ disk-type complete minimal surface in ${\mathbb{H}}^3$ sharing this Jordan curve as the asymptotic boundary.

22 pages, 9 figures, changes made following referee's many corrections and suggestions. Accepted by the Transactions of the AMS

References in corpus (3)

• Lindelöf's theorem for catenoids revisited
• Asymptotic Plateau Problem
• Least area Spherical Catenoids in Hyperbolic Three-Dimensional Space

Cited by in corpus (1)

• Computing discrete equivariant harmonic maps