Constructions of H_r-hypersurfaces, barriers and Alexandrov Theorem in Hⁿ × R

2014/02/27 by Maria Fernanda Elbert, Elbert, Maria Fernanda, Ricardo Sá Earp +2
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #math.DG

paper · pdf · doi:10.48550/arxiv.1402.6886

arxiv created 2014/02/27 · openalex publication_date 2014/02/27 · arxiv updated 2014/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we are concerned with hypersurfaces in Hⁿ× R with constant r-mean curvature, to be called H_r-hypersurfaces. We construct examples of complete H_r-hypersurfaces which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire H_r-graph for each value 0<H_r≤(n-r)/(n). Also, for each value H_r>(n-r)/(n), there is a unique embedded compact strictly convex rotational H_r-hypersurface. By using them as barriers, we obtain some interesting geometric results, including height estimates and an Alexandrov-type Theorem. Namely, we prove that an embedded compact H_r-hypersurface in Hⁿ× R is rotational (H_r>0).

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