Einstein-type structures, Besse's conjecture and a uniqueness result for a $\phi$-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 $\phi :M\to N$. The resulting system, denoted by $(\phi -\mathrm{CPE})$, is natural from the variational viewpoint and describes stationary points for the integrated $\phi$-scalar curvature functional restricted to metrics with unit volume and constant $\phi$-scalar curvature. We prove both a rigidity statement for solutions to $(\phi -\mathrm{CPE})$ in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting $(\phi -\mathrm{CPE})$ with $\phi$ a harmonic map.

26 pages. Package axessibility included to make the paper available to visually impaired people. Revised version: the rigidity theorem was improved in the light of a result by Barbosa, Mirandola and Vitorio. We included comments and corrected a mistake in the variational formulation