On the interplay between CPE metrics, vacuum static spaces and $$\sigma _2$$-singular spaces

A short overview of Besse’s conjecture

“…On the other hand, the first author together with Silva Santos obtained the following rigidity result for σ 2 -singular Einstein manifolds with positive σ 2 -curvature. The next lemma is crucial for the result in [2].…”

“…Motivated by these works and others, in [2] the first author provided a necessary and sufficient condition for a CPE metric to be Einstein for n ≥ 3, improving the understanding about CPE metrics and Besse's conjecture with a new geometric point of view that involves the potential function f inherited by the CPE condition. We recall some definitions and results.…”

“…This expository paper is based on a lecture given at XIV -ENAMA -Encontro Nacional de Análise Matemática e Aplicações, held in on-line format in Brazil in November 2021 and the main results was published in [2].…”