Finite topology minimal surfaces in homogeneous three-manifolds

2015/05/25 by Meeks, William H., Perez, Joaquin
#49Q05 #53A10 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1505.06764

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 \mathbbS³ equipped with a homogeneous metric of nonnegative scalar curvature.

Related

• Embedded minimal surfaces of finite topology
• Compact Embedded Minimal Surfaces of Positive Genus Without Area Bounds
• Singular limit laminations, Morse index, and positive scalar curvature
• Compact complete minimal immersions in R³
• Minimal surfaces in finite volume non compact hyperbolic 3-manifolds