On the existence of critical compatible metrics on contact $3$-manifolds
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908330100785152 |
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| author | Mitsumatsu, Yoshihiko Peralta-Salas, Daniel Slobodeanu, Radu |
| author_facet | Mitsumatsu, Yoshihiko Peralta-Salas, Daniel Slobodeanu, Radu |
| contents | 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 $\widetilde{SL}(2, \mathbb R)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15833 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the existence of critical compatible metrics on contact $3$-manifolds Mitsumatsu, Yoshihiko Peralta-Salas, Daniel Slobodeanu, Radu Differential Geometry Dynamical Systems Symplectic Geometry 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 $\widetilde{SL}(2, \mathbb R)$. 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. |
| title | On the existence of critical compatible metrics on contact $3$-manifolds |
| topic | Differential Geometry Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2311.15833 |