On the existence of critical compatible metrics on contact $3$-manifolds

Abstract

We disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact $3$-manifolds. More precisely, we show that a contact $3$-manifold $(M,\alpha )$ admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb flow is $C^{\infty }$-conjugate to an algebraic Anosov flow modeled on $\widetilde{SL}(2,\mathbb{R})$. In particular, this yields a complete topological classification of compact $3$-manifolds that admit critical compatible metrics. As a corollary we prove that no contact structure on ${\mathbb{T}}^3$ admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is tight.

12 pages; matches the published version, with the Corrigendum incorporated

References in corpus (2)

• On Anosovity, divergence and bi-contact surgery
• Tight Beltrami fields with symmetry

Cited by in corpus (3)

• Homogeneity of magnetic trajectories in the Berger sphere
• Towards a Fluid computer
• Cosymplectic Chern--Hamilton conjecture