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 which support a structure relating curvature to the geometry of a smooth map . The resulting system, denoted by , is natural from the variational viewpoint and describes stationary points for the integrated -scalar curvature functional restricted to metrics with unit volume and constant -scalar curvature. We prove both a rigidity statement for solutions to in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting with a harmonic map.
Keywords
Cite
@article{arxiv.2201.00263,
title = {Einstein-type structures, Besse's conjecture and a uniqueness result for a $\varphi$-CPE metric in its conformal class},
author = {Giulio Colombo and Luciano Mari and Marco Rigoli},
journal= {arXiv preprint arXiv:2201.00263},
year = {2024}
}
Comments
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