Capacity inequalities and rigidity of cornered/conical manifolds
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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.
KeywordsCapacity Inverse mean curvature flow Rigidity Riemannian penrose inequality Convex cone
The author would like to thank Professor A. Neves for providing a wonderful scientific environment when he was visiting Imperial College London and where the first drafts of this work were written. Also, he thanks an anonymous referee for suggestions which helped substantially improve the presentation and L. Pessoa for bringing  to my attention. While at Imperial College, I was supported by CNPq/Brazil.
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