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Capacity inequalities and rigidity of cornered/conical manifolds

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Abstract

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.

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Notes

  1. For instance, for non-star-shaped initial hypersurfaces, singularities may occur in finite time.

  2. \(\partial ^*\) represents the reduced boundary in the sense of the set of locally finite perimeter in \({\mathbb {R}}^n\).

  3. The function \(\phi \) is sometimes called of electrostatic potential of \(\Sigma .\)

  4. In [13], the authors also provided a link between Neumann parabolicity and capacity of compact subsets.

  5. In fact, one cannot have any compact deformation which is a consequence of the positive mass theorem.

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Acknowledgements

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 [13] to my attention. While at Imperial College, I was supported by CNPq/Brazil.

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  1. Universidade Federal de Alagoas, Instituto de Matemática, Maceió, AL, 57072-970, Brazil

    Tiarlos Cruz

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  1. Tiarlos Cruz
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Correspondence to Tiarlos Cruz.

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This work was completed with the support of Cnpq/Brazil.

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Cruz, T. Capacity inequalities and rigidity of cornered/conical manifolds. Ann Glob Anal Geom 55, 281–298 (2019). https://doi.org/10.1007/s10455-018-9626-0

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