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

Eugenio Bellini

TL;DR

The paper classifies irregular K-contact forms on compact 3-manifolds, proving that such manifolds are lens spaces $L(p,q)$ with exactly two periodic Reeb orbits whose minimal periods have irrational ratio. It simultaneously links Reeb-period data to sub-Riemannian spectral invariants by deriving the small-time heat-kernel expansion of the sub-Laplacian, validating Colin de Verdière's conjecture in the irregular case. The authors provide a detailed geometric construction via tubular neighborhoods, toroidal charts, and Seifert-like momentum maps to classify irregular forms and construct forms with prescribed Reeb data on lens spaces. A complete record of the two-orbit phenomenon is given, together with an explicit correspondence between orbit periods, rotation numbers, and the resulting spectral invariants. The results illuminate how contact topology and sub-Riemannian analysis intertwine to yield concrete geometric and spectral fingerprints of irregular K-contact structures on lens spaces.

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 Verdière in the irregular K-contact case.

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

TL;DR

The paper classifies irregular K-contact forms on compact 3-manifolds, proving that such manifolds are lens spaces with exactly two periodic Reeb orbits whose minimal periods have irrational ratio. It simultaneously links Reeb-period data to sub-Riemannian spectral invariants by deriving the small-time heat-kernel expansion of the sub-Laplacian, validating Colin de Verdière's conjecture in the irregular case. The authors provide a detailed geometric construction via tubular neighborhoods, toroidal charts, and Seifert-like momentum maps to classify irregular forms and construct forms with prescribed Reeb data on lens spaces. A complete record of the two-orbit phenomenon is given, together with an explicit correspondence between orbit periods, rotation numbers, and the resulting spectral invariants. The results illuminate how contact topology and sub-Riemannian analysis intertwine to yield concrete geometric and spectral fingerprints of irregular K-contact structures on lens spaces.

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 Verdière in the irregular K-contact case.
Paper Structure (26 sections, 33 theorems, 278 equations)

This paper contains 26 sections, 33 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Application: Reeb periods are sub-Riemannian spectral invariants
  3. Structure of the paper
  4. K-contact Riemannian manifolds
  5. Contact forms and compatible metrics
  6. Rotation number
  7. Contact isometry group and momentum maps
  8. Lens spaces
  9. Toroidal atlases
  10. Tubular neighborhoods of Reeb orbits
  11. Tubes in Riemannian geometry
  12. Tubes around Reeb orbits
  13. Characterization of irregular K-contact forms
  14. Cut time and non existence of periodic orbits
  15. Volume function and cut locus
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(M,\alpha)$ be a compact three-dimensional manifold with a positive irregular K-contact form. Then $M$ is diffeomorphic to a lens space. Moreover, the Reeb vector field of $\alpha$ has precisely two periodic orbits, and the ratio of their minimal periods is irrational.

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 73 more