  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 $\mathbb{S}^3$.
