Minimal surfaces in finite volume non compact hyperbolic $3$-manifolds

TL;DR

This paper proves the existence of compact embedded minimal surfaces in finite volume hyperbolic 3-manifolds, characterizes their topology and asymptotic behavior, and establishes related rigidity results.

Contribution

It introduces new existence and classification results for minimal surfaces in finite volume hyperbolic 3-manifolds, including finiteness of topology and asymptotic analysis.

Findings

Existence of compact embedded minimal surfaces in finite volume hyperbolic 3-manifolds.

Finite topology and asymptotic behavior of bounded curvature minimal surfaces.

Rigidity theorems for minimal surfaces in this setting.

Abstract

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This determines its asymptotic behavior. Some rigidity theorems are obtained.