Rigidez de semiespaços assintoticamente planos

Abstract

In the first part of this thesis, we explored a mass-capacity type inequality applied to threedimensional asymptotically flat half-spaces that are complete, smooth, and have non-negative scalar curvature, along with a convex mean boundary. In the case of equality in this inequality, we demonstrated that the manifold is isometric to the Schwarzschild half-space. In the second part, our focus shifted to static manifolds with boundary, emphasizing that the static potentials of these manifolds do not change sign, provided they are bounded and vanish at the horizon. Additionally, we derived an estimate linking the asymptotic expansion and the modified Hawking mass. In the case of equality, the manifold is isometric to R 3 +. Finally, in a reinterpretation of a result by Galloway-Cederbaum (CEDERBAUM; GALLOWAY, 2017) concerning photon spheres, and under certain assumptions on the Gaussian and mean curvatures, we demonstrated a result in which a compact and static manifold is isometric to ([3m, +∞) × S 2 , g), where g = 1 − 2m r −1 dr2 + r 2dσS 2.