Classification of K-contact forms and spectral invariants of their sub-Laplacians

TL;DR

This paper classifies certain K-contact forms on three-dimensional manifolds with non-periodic Reeb orbits, identifies their topological and spectral properties, and confirms a conjecture relating orbit periods to spectral invariants.

Contribution

It provides a classification of non-regular K-contact forms on 3-manifolds and links their Reeb orbit periods to spectral invariants of the sub-Laplacian.

Findings

Compact 3-manifolds with such forms are lens spaces with exactly two irrationally related periodic orbits.

Classified these forms up to diffeomorphism based on orbit periods.

Confirmed a conjecture relating orbit periods to spectral invariants of the sub-Laplacian.

Abstract

A contact form is called K-contact if its Reeb vector field is Killing with respect to some Riemannian metric. In this paper we classify K-contact forms whose Reeb vector field admits at least one non-periodic orbit, on three-dimensional manifolds. We prove that if a compact three-manifold carries such a contact form, then it is diffeomorphic to a lens space and admits exactly two periodic Reeb orbits, whose periods have irrational ratio. We further classify, up to (global) diffeomorphism, these contact forms in terms of the periods of their closed Reeb orbits. We conclude by relating these periods to spectral invariants of the sub-Laplacian, confirming a conjecture of Y. Colin de Verdi\`ere in the irregular K-contact case.