Anosov contact metrics, Dirichlet optimization and entropy

The first main result of this paper classifies contact 3-manifolds admitting critical metrics, i.e. adapted metrics which are the critical points of the Dirichlet energy functional. This gives a complete answer to a question raised by Chern-Hamilton in 1984. Secondly, we show that in the case of Anosov contact metrics, the optimization of such energy functional is closely related to Reeb dynamics and can be described in terms of its entropy. We also study the consequences in the curvature realization problem for such contact metrics.