Existence of Minimal Surfaces in Infinite Volume Hyperbolic 3-manifolds

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.