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Positive scalar curvature with skeleton singularities

  • Chao Li
  • Christos Mantoulidis
Article

Abstract

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean (\(L^\infty \)) metrics that consolidate Gromov’s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles \(\le 2\pi \) along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles \(\le 2\pi \)) and arbitrary \(L^\infty \) isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.

Notes

Acknowledgements

The authors would like to thank Rick Schoen, Brian White, Rafe Mazzeo, Pengzi Miao, and Or Hershkovits for stimulating conversations on the subject of this paper, as well as Gerhard Huisken, Dan Lee, André Neves, Yuguang Shi, and Peter Topping for their interest in this work. The first author would like to thank ETH-FIM for their hospitality, during which part of this work was carried out. The second author would like to thank the Ric Weiland Graduate Fellowship at Stanford, which partially supported the early portion of this research.

References

  1. 1.
    Aleksandrov, A.D., Berestovskiĭ, V.N., Nikolaev, I.G.: Generalized Riemannian spaces. Uspekhi Mat. Nauk 41(3), 3–44 (1986)MathSciNetGoogle Scholar
  2. 2.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  3. 3.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46(3), 406–480 (1997)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom. 54(1), 13–35 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54(1), 37–74 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Colding, T.H., Naber, A.: Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. Math. 176(2), 1173–1229 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cheeger, J., Naber, A.: Lower bounds on Ricci curvature and quantitative behavior of singular sets. Invent. Math. 191(2), 321–339 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169(3), 903–991 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Sturm, K.-T.: A curvature-dimension condition for metric measure spaces. C. R. Math. Acad. Sci. Paris 342(3), 197–200 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gromov, M.: Dirac and Plateau billiards in domains with corners. Cent. Eur. J. Math. 12(8), 1109–1156 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Li, C.: A polyhedron comparison theorem for 3-manifolds with positive scalar curvature. arXiv:1710.08067. Accessed 21 Dec 2017
  14. 14.
    Gromov, M., Lawson Jr., H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. 58, 83–196 (1984)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kazdan, J.L., Warner, F.W.: Scalar curvature and conformal deformation of Riemannian structure. J. Differ. Geom. 10, 113–134 (1975)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Schoen, R.: Variational theory for the total scalar curvaturefunctional for Riemannian metrics and related topics, Topics incalculus of variations (Montecatini Terme, 1987), Lecture Notesin Math., vol. 1365, pp. 120–154, Springer, Berlin (1989)Google Scholar
  17. 17.
    Lohkamp, J.: Scalar curvature and hammocks. Math. Ann. 313(3), 385–407 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Schoen, R., Yau, S.-T.: Positive scalar curvature and minimal hypersurface singularities. arXiv:1704.05490. Accessed 17 Jan 2018
  19. 19.
    Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6, 1163–1182 (2002)MathSciNetGoogle Scholar
  22. 22.
    Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189 (2000)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Bray, H.L.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59(2), 177–267 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Shi, Y., Tam, L.-F.: Scalar curvature and singular metrics. Pac. J. Math. 293(2), 427–470 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    McFeron, D., Székelyhidi, G.: On the positive mass theorem for manifolds with corners. Commun. Math. Phys. 313(2), 425–443 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Shi, Y., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62(1), 79–125 (2002)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lee, D.A., LeFloch, P.G.: The positive mass theorem for manifolds with distributional curvature. Commun. Math. Phys. 339(1), 99–120 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Chruściel, P.T., Herzlich, M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math. 212(2), 231–264 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lee, D.A.: A positive mass theorem for Lipschitz metrics with small singular sets. Proc. Am. Math. Soc. 141(11), 3997–4004 (2013)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds. I: approximation of metrics with cone singularities. J. Am. Math. Soc. 28(1), 183–197 (2015)zbMATHGoogle Scholar
  31. 31.
    Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds. II: limits with cone angle less than \(2\pi \). J. Am. Math. Soc. 28(1), 199–234 (2015)zbMATHGoogle Scholar
  32. 32.
    Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds. III: limits as cone angle approaches \(2\pi \) and completion of the main proof. J. Am. Math. Soc. 28(1), 235–278 (2015)zbMATHGoogle Scholar
  33. 33.
    Tian, G.: K-stability and Kähler–Einstein metrics. Commun. Pure Appl. Math. 68(7), 1085–1156 (2015)zbMATHGoogle Scholar
  34. 34.
    Jeffres, T., Mazzeo, R., Rubinstein, Y.A.: Kähler–Einstein metrics with edge singularities. Ann. Math. 183(1), 95–176 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Atiyah, M., LeBrun, C.: Curvature, cones and characteristic numbers. Math. Proc. Camb. Philos. Soc. 155(1), 13–37 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Akutagawa, K., Carron, G., Mazzeo, R.: The Yamabe problem on stratified spaces. Geom. Funct. Anal. 24(4), 1039–1079 (2014)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Sormani, C., Wenger, S.: The intrinsic flat distance between Riemannian manifolds and other integral current spaces. J. Differ. Geom. 87(1), 117–199 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Sormani, C.: Scalar curvature and intrinsic flat convergence. https://www.degruyter.com/downloadpdf/books/9783110550832/9783110550832-008/9783110550832-008.pdf. Accessed 21 Dec 2017
  39. 39.
    Schoen, R., Yau, S.-T.: On the structure of manifolds with positive scalar curvature. Manuscr. Math. 28(1–3), 159–183 (1979)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Schoen, R., Yau, S.-T.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79(2), 231–260 (1981)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Mantoulidis, C., Miao, P.: Total mean curvature, scalar curvature, and a variational analog of Brown–York mass. Commun. Math. Phys. 352(2), 703–718 (2017)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (3) 17, 43–77 (1963)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Simon, L.: Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra (1983)Google Scholar
  44. 44.
    Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in \(3\)-manifolds of nonnegative scalar curvature. Commun. Pure Appl. Math. 33(2), 199–211 (1980)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Mantoulidis, C.: Geometric variational problems in mathematical physics, Ph.D. thesis, Stanford University, (2017)Google Scholar
  46. 46.
    Smith, P.D., Yang, D.: Removing point singularities of Riemannian manifolds. Trans. Am. Math. Soc. 333(1), 203–219 (1992)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Viaclovsky, J.A.: Monopole metrics and the orbifold Yamabe problem. Ann. Inst. Fourier (Grenoble) 60(7), 2503–2543 (2010)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Trudinger, N .S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Super. Pisa (3) 22, 265–274 (1968)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Aubin, T.: The scalar curvature. Math. Phys. Appl. Math. 3, 5–18 (1976)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20(2), 479–495 (1984)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Schoen, R., Yau, S.-T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math. 92(1), 47–71 (1988)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Ambrosio, L., Kirchheim, B.: Currents in metric spaces. Acta Math. 185(1), 1–80 (2000)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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