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

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-11-12 DOI:10.1112/blms.13183

Y. Mitsumatsu, D. Peralta-Salas, R. Slobodeanu

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引用次数: 0

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 $(M, α)$ 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 $\tilde{S L} (2, R)$ . 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 $T^{3}$ admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is tight.