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arXiv · 2201.00263

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

Giulio Colombo·Luciano Mari·Marco Rigoli
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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 $\varphi : M \to N$. The resulting system, denoted by $(\varphi-\mathrm{CPE})$, 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 $(\varphi-\mathrm{CPE})$ in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting $(\varphi-\mathrm{CPE})$ with $\varphi$ a harmonic map.

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Giulio Colombo, Luciano Mari, Marco Rigoli. 2022-01-01. Einstein-type structures, Besse's conjecture and a uniqueness result for a $\varphi$-CPE metric in its conformal class. https://doi.org/10.1007/s12220-022-01000-3

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