Existence of Minimal Surfaces in Infinite Volume Hyperbolic 3-manifolds

TL;DR

This paper investigates the existence of minimal surfaces in infinite-volume hyperbolic 3-manifolds, providing new existence results, dichotomies, and explicit examples in various geometric contexts.

Contribution

It establishes new existence results and constructs explicit examples of minimal surfaces in infinite-volume hyperbolic 3-manifolds, addressing open cases in the field.

Findings

Dichotomy for doubly degenerate manifolds with bounded geometry: either all contain a closed minimal surface or admit a foliation by such surfaces.

First examples of Schottky manifolds with closed minimal surfaces.

Existence of finite-area, embedded, complete minimal surfaces in certain hyperbolic 3-manifolds with rank-1 cusps.

Abstract

The existence of embedded minimal surfaces in non-compact 3-manifolds remains a largely unresolved and challenging problem in geometry. In this paper, we address several open cases regarding the existence of finite-area, embedded, complete, minimal surfaces in infinite-volume hyperbolic 3-manifolds. Among other results, for doubly degenerate manifolds with bounded geometry, we prove a dichotomy: either every such manifold contains a closed minimal surface or there exists such a manifold admitting a foliation by closed minimal surfaces. We also construct the first examples of Schottky manifolds with closed minimal surfaces and demonstrate the existence of Schottky manifolds containing infinitely many closed minimal surfaces. Lastly, for hyperbolic 3-manifolds with rank-1 cusps, we show that a broad class of these manifolds must contain a finite-area, embedded, complete minimal surface.