2011/10/04 by Francescopaolo Montefalcone, Montefalcone, Francescopaolo
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Dermatological and Skeletal Disorders #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.DG #math.MG
paper · pdf · doi:10.48550/arxiv.1110.0703
27 pages
openalex publication_date 2011/10/04 · arxiv created 2011/11/17 · arxiv updated 2011/11/18 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
After introducing the sub-Riemannian geometry of the Heisenberg group Hn, n ≥ 1, we recall some basics about hypersurfaces endowed with the H-perimeter measure and horizontal Green's formulas. Then, we describe a class of compact closed hypersurfaces of constant horizontal mean curvature called "Isoperimetric Profiles"(they are not CC-balls!); see Section 2.1. Our main purpose is to study a closed eigenvalue problem on Isoperimetric Profiles, i.e. LHS ϕ+ λϕ= 0, where LHS is a 2nd order horizontal tangential operator analogous to the Laplace-Beltrami operator; see Section 1.5. This is done starting from the radial symmetry of Isoperimetric Profiles with respect to a barycentric axis parallel to the center T of the Lie algebra hn. An interesting feature of radial eigenfunctions is in that they are hypergeometric functions; see Theorem 2.10. Finally, in Section 2.3 we shall begin the study of the general case.