The cosymplectic Chern–Hamilton conjecture

Abstract In this paper, we study the Chern–Hamilton energy functional on compact cosymplectic manifolds, fully classifying in dimension 3 those manifolds admitting a critical compatible metric for this functional. This is the case if and only if either the manifold is co‐Kähler or if it is a mapping torus of the 2‐torus by a hyperbolic toral automorphism and equipped with a suspension cosymplectic structure. Moreover, any critical metric has minimal energy among all compatible metrics. We also exhibit examples of manifolds with first Betti number admitting cosymplectic structures, but such that no cosymplectic structure admits a critical compatible metric.