Half-space theorems and the embedded Calabi-Yau problem in Lie groups
Abstract
We study the embedded Calabi-Yau problem for complete embedded constant mean curvature surfaces of finite topology or of positive injectivity radius in a simply-connected three-dimensional Lie group X endowed with a left-invariant Riemannian metric. We first prove a half-space theorem for constant mean curvature surfaces. This half-space theorem applies to certain properly immersed constant mean curvature surfaces of X contained in the complements of normal R^2 subgroups F of X. In the case X is a unimodular Lie group, our results imply that every minimal surface in X-F that is properly immersed in X is a left translate of F and that every complete embedded minimal surface of finite topology or of positive injectivity radius in X-F is also a left translate of F.
Cite
@article{arxiv.1012.1986,
title = {Half-space theorems and the embedded Calabi-Yau problem in Lie groups},
author = {Benoit Daniel and William H. Meeks and Harold Rosenberg},
journal= {arXiv preprint arXiv:1012.1986},
year = {2010}
}
Comments
17 pages