Hypergeometric solutions of the closed eigenvalue problem on Heisenberg Isoperimetric Profiles

Details Hypergeometric solutions of the closed eigenvalue problem on Heisenberg Isoperimetric Profiles Hypergeometric solutions of the closed eigenvalue problem on Heisenberg Isoperimetric Profiles

2011-10-04

arxiv.org/abs/1110.0703v2

Mathematics

Metric Geometry

27 pages

Scientific paper

After introducing the sub-Riemannian geometry of the Heisenberg group Hn, n \geq 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 \phi + {\lambda}\phi = 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.

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