Critical metrics of the volume functional on three-dimensional manifolds

TL;DR

This paper proves the three-dimensional CPE conjecture with non-negative Ricci curvature, classifies vacuum static spaces, and characterizes certain critical metrics as geodesic balls in space forms.

Contribution

It provides the first proof of the 3D CPE conjecture under non-negative Ricci curvature and classifies related static spaces and critical metrics.

Findings

Proved the 3D CPE conjecture with non-negative Ricci curvature.

Classified 3D vacuum static spaces with non-negative Ricci curvature.

Showed certain critical metrics are isometric to geodesic balls in space forms.

Abstract

In this paper, we prove the three-dimensional $CPE$ conjecture with non-negative Ricci curvature. Moreover, we establish a classification result on three-dimensional vacuum static space with non-negative Ricci curvature. Finally, we show that a three-dimensional compact, oriented, connected Miao-Tam critical metric with smooth boundary, non-negative Ricci curvature and non-negative potential function is isometric to a geodesic ball in a simply connected space form $\mathbb{R}^3$ or $\mathbb{S}^3$.