Finite topology minimal surfaces in homogeneous three-manifolds

TL;DR

This paper proves that complete, embedded minimal surfaces with finite topology in homogeneous three-manifolds have positive injectivity radius and explores their structure in locally homogeneous spaces, with applications to surfaces in spheres.

Contribution

It establishes positive injectivity radius for finite topology minimal surfaces in homogeneous three-manifolds and describes their lamination structure in locally homogeneous spaces.

Findings

Finite topology minimal surfaces in homogeneous spaces have positive injectivity radius.

Closure of such surfaces in locally homogeneous spaces forms a minimal lamination.

Complete, finite genus minimal surfaces in spheres with nonnegative scalar curvature are compact.

Abstract

We prove that any complete, embedded minimal surface $M$ with finite topology in a homogeneous three-manifold $N$ has positive injectivity radius. When one relaxes the condition that $N$ be homogeneous to that of being locally homogeneous, then we show that the closure of $M$ has the structure of a minimal lamination of $N$. As an application of this general result we prove that any complete, embedded minimal surface with finite genus and a countable number of ends is compact when the ambient space is $\mathbb{S}^3$ equipped with a homogeneous metric of nonnegative scalar curvature.