Annals of Global Analysis and Geometry

, Volume 55, Issue 2, pp 281–298

# Capacity inequalities and rigidity of cornered/conical manifolds

• Tiarlos Cruz
Article

## 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.

## Keywords

Capacity Inverse mean curvature flow Rigidity Riemannian penrose inequality Convex cone

## Notes

### 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.

## References

1. 1.
Almaraz, S., Barbosa, E., de Lima, L.: A positive mass theorem for asymptotically flat manifolds with a non-compact boundary. Comm. Anal. Geom. 24, 673–715 (2016)
2. 2.
Bray, H.L.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differential Geom. 59(2), 177–267 (2001)
3. 3.
Bray, H.L., Miao, P.: On the capacity of surfaces in manifolds with nonnegative scalar curvature. Invent. Math. 172, 459–475 (2008)
4. 4.
Choe, J.: The isoperimetric inequality for minimal surfaces in a Riemannian manifold. J. Reine Angew. Math. 506, 205–214 (1999)
5. 5.
Courant, R.: On Plateau’s problem with free boundaries. Proc. Natl. Acad. Sci. USA 31(8), 242–246 (1945)
6. 6.
Figalli, A., Indrei, E.: A sharp stability result for the relative isoperimetric inequality inside convex cones. J. Geom. Anal. 23(2), 938–969 (2013)
7. 7.
Freire, A., Schwartz, F.: Mass-capacity inequalities for conformally flat manifolds with boundary. Comm. PDE 39, 98–119 (2014)
8. 8.
Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differential Geom. 32(1), 299–314 (1990)
9. 9.
Guan, P., Li, J.: The quermassintegral inequalities for k-convex starshaped domains. Adv. Math. 221(5), 1725–1732 (2009)
10. 10.
Gromov, M.: Dirac and Plateau billiards in domains with corners. Cent. Eur. J. Math. 12(8), 1109–1156 (2014)
11. 11.
Grüter, M., Jost, J.: Allard type regularity results for varifolds with free boundaries. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e serie 13(1), 129–169 (1986)
12. 12.
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom. 59, 353–437 (2001)
13. 13.
Impera, D. Pigola, S., Setti, A.G.: Global maximum principles and divergence theorems on complete manifolds with boundary. J. Reine Angew. Math, to appearGoogle Scholar
14. 14.
Jauregui, J.L.: Penrose-type inequalities with a Euclidean background. arXiv:1108.4042v1
15. 15.
Lambert, B., Scheuer, J.: The inverse mean curvature flow perpendicular to the sphere. Math. Ann. 364(3), 1069–1093 (2016)
16. 16.
Lambert, B., Scheuer, J.: A geometric inequality for convex free boundary hypersurfaces in the unit ball. Proc. Amer. Math, to appearGoogle Scholar
17. 17.
Lions, P.L., Pacella, F.: Symmetrization for a class of elliptic equations with mixed boundary conditions. Atti Sem. Mat. Fis. Univ. Modena XXXIV, 75–94 (1985–1986)Google Scholar
18. 18.
Lions, P.L., Pacella, F.: Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions. Indiana Univ. Math. J. 37, 301–324 (1988)
19. 19.
Lions, P.L., Pacella, F.: Isoperimetric inequalities for convex cones. Proc. Amer. Math. Soc. 109, 477–485 (1990)
20. 20.
Marquardt, T.: The inverse mean curvature flow for hypersurfaces with boundary. Dissertation, Freie Universität Berlin (2012)Google Scholar
21. 21.
Marquardt, T.: Inverse mean curvature flow for star-shaped hypersurfaces evolving in a cone. J. Geom. Anal. 23, 1303–1313 (2013)
22. 22.
Marquardt, T.: Weak solutions of inverse mean curvature flow for hypersurfaces with boundary, to appear in J. Reine Angew. MathGoogle Scholar
23. 23.
Meeks III, W.H., Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179, 151–168 (1982)
24. 24.
Morgan, F., Ritoré, M.: Isoperimetric regions in cones. Trans. Amer. Math. Soc. 354(6), 2327–2339 (2002)
25. 25.
Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, NJ (1951)Google Scholar
26. 26.
Pacella, F., Tricarico. M.: Symmetrization for a class of eliptic equations with mixed boundary conditions. Atti Sem. Mat. Fis. Univ. Modena XXXIV, 75–94 (1985–1986)Google Scholar
27. 27.
Ritoré, M., Rosales, C.: Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones. Trans. Amer. Math. Soc. 356(11), 4601–4622 (2004)
28. 28.
Ritoré, M., Vernadakis, E.: Isoperimetric inequalities in convex cylinders and cylindrically bounded convex bodies. Calc. Var. Par. Differ. Equ. 54(1), 643–663 (2015)
29. 29.
Schwartz, F.: A Volumetric Penrose Inequality for Conformally Flat Manifolds. Annales Henri Poincaré. vol. 12. no. 1. SP Birkhäuser Verlag Basel (2011)Google Scholar
30. 30.
Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65(1), 45–76 (1979)
31. 31.
Schoen, R., Yau, S.T.: The energy and the linear momentum of space–times in general relativity. Comm. Math. Phys. 79(1), 47–51 (1981)
32. 32.
Szegö, G.: Über einige Extremalaufgaben der Potentialtheorie. Math. Z. 31(1), 583–593 (1930). (MR MR1545137)
33. 33.
Szegö, G.: Über einige neue Extremaleigenschaften der Kugel. Math. Z. 33, 419–425 (1931). (German)
34. 34.
Sternberg, P., Williams, G., Ziemer, W.P.: $$C^{1,1}$$-Regularity of constrained area minimizing hypersurfaces. J. Differ. Equ. 94, 83–94 (1991)
35. 35.
Volkmann, A.: A monotonicity formula for free boundary surfaces with respect to the unit ball. Comm. Anal. Geom. 24(1), 195–221 (2016)
36. 36.
Witten, E.: A new proof of the positive energy theorem. Comm. Math. Phys. 80(3), 381–402 (1981)