Cosymplectic Chern--Hamilton conjecture

TL;DR

This paper classifies 3-dimensional compact cosymplectic manifolds with critical metrics for the Chern-Hamilton energy functional, identifying conditions under which such metrics exist and providing examples with specific topological properties.

Contribution

It fully classifies 3D compact cosymplectic manifolds admitting critical compatible metrics for the Chern-Hamilton energy functional, linking geometry with topological structures.

Findings

Critical metrics occur only on co-Kähler manifolds or certain mapping tori.

Critical metrics have minimal energy among compatible metrics.

Examples of manifolds with no critical cosymplectic structure are provided.

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-K\"ahler 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 $b_1 \geq 2$ admitting cosymplectic structures, but such that no cosymplectic structure admits a critical compatible metric.