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

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.