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

Autor:	Mitsumatsu, Yoshihiko, Peralta-Salas, Daniel, Slobodeanu, Radu
Rok vydání:	2023
Předmět:	Mathematics - Differential Geometry Mathematics - Dynamical Systems Mathematics - Symplectic Geometry
Druh dokumentu:	Working Paper
Popis:	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. Comment: 12 pages; new section 3.3 about global minima of the energy
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2311.15833 Zobrazit plný text záznamu View this record from Arxiv