On the interplay between CPE metrics, vacuum static spaces and σ₂-singular spaces

2020/02/09 by Maria Andrade, Andrade, Maria
Mathematics · Physics and Astronomy · #53C21 #53C25 #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Primary: 53C24 #Secondary: 53C20

paper · pdf · doi:10.48550/arxiv.2002.03365

openalex publication_date 2020/02/09 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We call CPE metrics the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume. In this short note, we give a necessary and sufficient condition for a CPE metric to be Einstein in therms of σ₂-singular spaces. Such a result improves our understanding about CPE metrics and Besse's conjecture with a new geometric point of view. Moreover, we prove that the CPE condition can be replaced by the related vacuum static space condition to characterize closed Einstein manifolds in terms of σ₂-singular spaces.

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