Half-space theorems and the embedded Calabi-Yau problem in Lie groups

TL;DR

This paper investigates the geometry of constant mean curvature surfaces in three-dimensional Lie groups, proving a half-space theorem and classifying minimal surfaces in certain subgroups, advancing understanding of the Calabi-Yau problem in this setting.

Contribution

It introduces a half-space theorem for constant mean curvature surfaces in Lie groups and classifies minimal surfaces in the complements of normal subgroups, extending geometric analysis in these spaces.

Findings

Proved a half-space theorem for constant mean curvature surfaces in Lie groups.

Classified minimal surfaces in the complements of normal subgroups in unimodular Lie groups.

Showed that properly immersed minimal surfaces are left translates of certain subgroups.

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.