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

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 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.