Einstein and conformally flat critical metrics of the volume functional

TL;DR

This paper classifies Einstein and conformally flat metrics that are critical points of the volume functional on manifolds with fixed boundary metric and constant scalar curvature, advancing understanding of geometric variational problems.

Contribution

It provides a complete classification of Einstein and conformally flat critical metrics of the volume functional under specified boundary and scalar curvature conditions.

Findings

Identifies all Einstein critical metrics in the given setting.

Classifies conformally flat critical metrics with fixed boundary data.

Establishes conditions under which these metrics are characterized as critical points.

Abstract

Let $R$ be a constant. Let $\mathcal{M}^R_\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\Omega^n$ ($n\ge 3$) with smooth boundary $\Sigma $ such that $g$ has constant scalar curvature $R$ and $g|_{\Sigma}$ is a fixed metric $\gamma$ on $\Sigma$. Let $V(g)$ be the volume of $g\in\mathcal{M}^R_\gamma$. In this work, we classify all Einstein or conformally flat metrics which are critical points of $V(\cdot)$ in $\mathcal{M}^R_\gamma$.