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

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 $\stackrel{\sim }{SL}(2,R)$. 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 $T^3$ admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is tight.