Finite topology minimal surfaces in homogeneous three-manifolds

William H. Meeks III; Joaquin Perez

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arXiv · 1505.06764

Finite topology minimal surfaces in homogeneous three-manifolds

William H. Meeks III·Joaquin Perez

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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.

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William H. Meeks III, Joaquin Perez. 2015-05-25. Finite topology minimal surfaces in homogeneous three-manifolds. https://arxiv.org/abs/1505.06764

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