Critical metrics of the volume functional on three-dimensional manifolds

In this paper, we prove the three-dimensional $CPE$ 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.