Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

TL;DR

This paper classifies weakly Einstein critical metrics of the volume functional on compact manifolds with boundary, providing complete results in low dimensions and extending to higher dimensions under curvature constraints.

Contribution

It offers a complete classification of such metrics in dimensions 3 and 4, and extends the classification to higher dimensions with Weyl tensor constraints.

Findings

Complete classification in 3 and 4 dimensions

Extension to higher dimensions with Weyl tensor conditions

Results under nonnegative scalar curvature

Abstract

The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold $M$ with smooth boundary $\partial M$. Here, we will give the complete classification for an $n$-dimensional, $n=3$ or $4,$ weakly Einstein critical metric of the volume functional with nonnegative scalar curvature. Moreover, in the higher dimensional case ($n\geq5$), we will established a similar result for weakly Einstein critical metric under a suitable constraint on the Weyl tensor.