Hypergeometric solutions of the closed eigenvalue problem on Heisenberg Isoperimetric Profiles

TL;DR

This paper investigates a specific eigenvalue problem on isoperimetric hypersurfaces in the Heisenberg group, revealing solutions in terms of hypergeometric functions and exploring their properties within sub-Riemannian geometry.

Contribution

It introduces hypergeometric solutions to the eigenvalue problem on Heisenberg isoperimetric profiles, expanding understanding of spectral properties in sub-Riemannian geometry.

Findings

Eigenfunctions are hypergeometric functions

Radial symmetry simplifies the eigenvalue problem

Initial steps towards general case analysis

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 \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.