A conformally Anosov Reeb flow on a contact 3-manifold forces universal tightness, irreducibility, and absence of exact cobordisms to the standard three-sphere, which therefore admits no such flow.
Riemannian geometry of contact and symplectic manifolds
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Dynamics and Topology of Conformally Anosov Contact 3-Manifolds
A conformally Anosov Reeb flow on a contact 3-manifold forces universal tightness, irreducibility, and absence of exact cobordisms to the standard three-sphere, which therefore admits no such flow.