Abstract
AbstractWe 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 admits a critical compatible metric for the Chern–Hamilton energy functional if and only if it is Sasakian or its associated Reeb flow is ‐conjugate to an algebraic Anosov flow modeled on . 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 admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is tight.
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