In this paper, we prove the three-dimensional CP E conjecture with non-negative Ricci curvature. Moreover, 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 R 3 or S 3 . Finally, we establish a classification result on three-dimensional static space with non-negative Ricci curvature.

We give some rigidity theorems for an n(≥ 4)-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant σ 2 . Moreover, when n = 4, we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant σ 2 is isometric to a quotient of the round S 4 .

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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.

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