Einstein-type structures, Besse's conjecture and a uniqueness result for a φ-CPE metric in its conformal class

In this paper, we study an extension of the CPE conjecture to manifolds M which support a structure relating curvature to the geometry of a smooth map φ:M→N. The resulting system, denoted by (φ−CPE), is natural from the variational viewpoint and describes stationary points for the integrated φ-scalar curvature functional restricted to metrics with unit volume and constant φ-scalar curvature. We prove both a rigidity statement for solutions to (φ−CPE) in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting (φ−CPE) with φ a harmonic map.