Critical metrics of the volume functional on three‐dimensional manifolds

Abstract

AbstractIn this paper, we prove the three‐dimensional conjecture with nonnegative Ricci curvature. Moreover, we establish rigidity theorems for three‐dimensional compact, oriented, connected V‐static metrics with nonnegative Ricci curvature. Finally, we obtain classification results on three‐dimensional vacuum static space and Miao–Tam critical metric with nonnegative Ricci curvature.