Einstein-Type Structures, Besse’s Conjecture, and a Uniqueness Result for a $$\varphi $$-CPE Metric in Its Conformal Class

Abstract

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 \(\varphi : M \rightarrow N\). The resulting system, denoted by (\(\varphi \)-CPE), is natural from the variational viewpoint and describes stationary points for the integrated \(\varphi \)-scalar curvature functional restricted to metrics with unit volume and constant \(\varphi \)-scalar curvature. We prove both a rigidity statement for solutions to (\(\varphi \)-CPE) in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting (\(\varphi \)-CPE) with \(\varphi \) a harmonic map.