Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 099, 8 pages      arXiv:2007.12563      https://doi.org/10.3842/SIGMA.2020.099
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

Chao Li
Department of Mathematics, Princeton University, Fine Hall, 304 Washington Rd, Princeton, NJ 08544, USA

Received July 27, 2020, in final form September 30, 2020; Published online October 06, 2020

Abstract
In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds. Our result is a localization of the positive mass theorem for asymptotically hyperbolic manifolds. We also motivate and formulate some open questions concerning related rigidity phenomenon and convergence of metrics with scalar curvature lower bounds.

Key words: dihedral rigidity; scalar curvature; comparison theorem; hyperbolic manifolds.

pdf (366 kb)   tex (18 kb)  

References

  1. Aleksandrov A.D., Berestovskii V.N., Nikolaev I.G., Generalized Riemannian spaces, Russian Math. Surveys 41 (1986), 1-54.
  2. Andersson L., Cai M., Galloway G.J., Rigidity and positivity of mass for asymptotically hyperbolic manifolds, Ann. Henri Poincaré 9 (2008), 1-33, arXiv:math.DG/0703259.
  3. Bamler R.H., A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature, Math. Res. Lett. 23 (2016), 325-337, arXiv:1505.00088.
  4. Brendle S., Marques F.C., Scalar curvature rigidity of geodesic balls in $S^n$, J. Differential Geom. 88 (2011), 379-394, arXiv:1005.2782.
  5. Brendle S., Marques F.C., Neves A., Deformations of the hemisphere that increase scalar curvature, Invent. Math. 185 (2011), 175-197, arXiv:1004.3088.
  6. Burkhardt-Guim P., Pointwise lower scalar curvature bounds for $C^0$ metrics via regularizing Ricci flow, Geom. Funct. Anal. 29 (2019), 1703-1772, arXiv:1907.13116.
  7. Carlotto A., Chodosh O., Eichmair M., Effective versions of the positive mass theorem, Invent. Math. 206 (2016), 975-1016, arXiv:1503.05910.
  8. Chodosh O., Eichmair M., Moraru V., A splitting theorem for scalar curvature, Comm. Pure Appl. Math. 72 (2019), 1231-1242, arXiv:1804.01751.
  9. Chruściel P.T., Delay E., The hyperbolic positive energy theorem, arXiv:1901.05263.
  10. Chruściel P.T., Herzlich M., The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), 231-264, arXiv:math.DG/0110035.
  11. Chruściel P.T., Nagy G., The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times, Adv. Theor. Math. Phys. 5 (2001), 697-754, arXiv:gr-qc/0110014.
  12. Cox G., Miao P., Tam L.-F., Remarks on a scalar curvature rigidity theorem of Brendle and Marques, Asian J. Math. 17 (2013), 457-469, arXiv:1109.3942.
  13. Edelen N., Li C., Regularity of free boundary minimal surfaces in locally polyhedral domains, arXiv:2006.15441.
  14. Gromov M., Dirac and Plateau billiards in domains with corners, Cent. Eur. J. Math. 12 (2014), 1109-1156, arXiv:1811.04318.
  15. Gromov M., A dozen problems, questions and conjectures about positive scalar curvature, in Foundations of Mathematics and Physics One Century after Hilbert, Springer, Cham, 2018, 135-158, arXiv:1710.05946.
  16. Gromov M., Metric inequalities with scalar curvature, Geom. Funct. Anal. 28 (2018), 645-726, arXiv:1710.04655.
  17. Gromov M., Scalar curvature of manifolds with boundaries: natural questions and artificial constructions, arXiv:1811.04311.
  18. Gromov M., Four lectures on scalar curvature, arXiv:1908.10612.
  19. Gromov M., Lawson Jr. H.B., Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. 111 (1980), 209-230.
  20. Grüter M., Jost J., Allard type regularity results for varifolds with free boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), 129-169.
  21. Huang L.-H., Jang H.C., Martin D., Mass rigidity for hyperbolic manifolds, Comm. Math. Phys. 376 (2020), 2329-2349, arXiv:1904.12010.
  22. Li C., A polyhedron comparison theorem for 3-manifolds with positive scalar curvature, Invent. Math. 219 (2020), 1-37, arXiv:1710.08067.
  23. Li C., The dihedral rigidity conjecture for $n$-prisms, arXiv:1907.03855.
  24. Li C., Mantoulidis C., Positive scalar curvature with skeleton singularities, Math. Ann. 374 (2019), 99-131, arXiv:1708.08211.
  25. Miao P., Measuring mass via coordinate cubes, Comm. Math. Phys. 379 (2020), 773-783, arXiv:1911.11757.
  26. Min-Oo M., Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285 (1989), 527-539.
  27. Schoen R., Yau S.-T., On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45-76.
  28. Schoen R., Yau S.-T., On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159-183.
  29. Schoen R., Yau S.-T., Positive scalar curvature and minimal hypersurface singularities, arXiv:1704.05490.
  30. Simons J., Minimal varieties in riemannian manifolds, Ann. of Math. 88 (1968), 62-105.
  31. Toponogov V.A., Evaluation of the length of a closed geodesic on a convex surface, Dokl. Akad. Nauk SSSR 124 (1959), 282-284.
  32. Wang X., The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), 273-299.
  33. Witten E., A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381-402.
  34. Witten E., Yau S.-T., Connectedness of the boundary in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999), 1635-1655, arXiv:hep-th/9910245.

Previous article  Next article  Contents of Volume 16 (2020)