arXiv subjects
Publications and source records attributed to C. Tiarlos Cruz.
We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.
Building upon the work of Brendle, Marques and Neves on the construction of counterexamples to Min-Oo's conjecture, we exhibit deformations of the de Sitter-Schwarzschild space of dimension $n\geq 3$ satisfying the dominant energy condition and agreeing with the standard metric along the event and cosmological horizons, which remain totally geodesic. Our results actually hold for generalized Kottler-de Sitter-Schwarzschild spaces whose cross sections are compact rank one symmetric spaces and indicate that there exists no analogue of the Penrose inequality in the case of positive cosmological constant. As an application we construct solutions of Einstein field equations satisfying the dominant energy condition and being asymptotic to (or agreeing with) the de Sitter-Schwarzschild space-time both at the event horizon and at spatial infinity.
We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $Σ$ with nonpositive Yamabe invariant in a Riemannian $n$-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that $Σ$ is locally volume-minimizing in a manifold $M^n$ with scalar curvature bounded below by a nonpositive constant and mean convex boundary, we conclude that locally $M$ splits along $Σ$. In the case that the scalar curvature of $M$ is at least $-n(n-1)$ and $Σ$ locally minimizes a certain functional inspired by [30], a neighborhood of $Σ$ in $M$ is isometric to $((-\varepsilon,\varepsilon)\timesΣ,dt^2+e^{2t}g)$, where $g$ is Ricci flat.