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arXiv · 1110.0703

Hypergeometric solutions of the closed eigenvalue problem on Heisenberg Isoperimetric Profiles

Francescopaolo Montefalcone
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Abstract

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 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.

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Francescopaolo Montefalcone. 2011-11-17. Hypergeometric solutions of the closed eigenvalue problem on Heisenberg Isoperimetric Profiles. https://arxiv.org/abs/1110.0703

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