Counting Minimal Surfaces in Quasi-Fuchsian three-Manifolds

TL;DR

This paper constructs quasi-Fuchsian 3-manifolds with exponentially many minimal surfaces, showing the potential for complex minimal surface configurations sharing the same boundary.

Contribution

It introduces a method to generate quasi-Fuchsian manifolds containing an exponential number of minimal surfaces, advancing understanding of their geometric complexity.

Findings

Existence of quasi-Fuchsian manifolds with at least 2^N minimal surfaces

Construction of minimal surfaces sharing the same asymptotic boundary

Demonstration of exponential growth in minimal surface count

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 $S^2_\infty$ 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.