2010/12/09 by Benoit Daniel, Benôıt Daniel, William H. Meeks III +5
Mathematics · #53A35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Primary: 53A10. Secondary: 53C42 #math.DG #msc:53A10. #msc:53A35 #msc:53C42
paper · pdf · doi:10.48550/arxiv.1012.1986
17 pages
arxiv created 2010/12/09 · openalex publication_date 2010/12/09 · arxiv updated 2010/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 R2 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.