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 stable ones. 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 closed Jordan curve on $S_{\infty }^2$ such that there are at least $2^N$ disk-type complete minimal surfaces in $\mathbb {H}^3$ sharing this Jordan curve as the asymptotic boundary.