Advertisement

The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3553–3602 | Cite as

Sewing Riemannian Manifolds with Positive Scalar Curvature

  • J. Basilio
  • J. Dodziuk
  • C. SormaniEmail author
Article
  • 83 Downloads

Abstract

We explore to what extent one may hope to preserve geometric properties of three-dimensional manifolds with lower scalar curvature bounds under Gromov–Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three-dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.

Keywords

Scalar curvature Gromov-Hausdorff Intrinsic flat 

Mathematics Subject Classification

53C23 

Notes

Acknowledgements

J. Basilio was partially supported as a doctoral student by NSF DMS 1006059. C. Sormani was partially supported by NSF DMS 1006059.

References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163(7), 1405–1490 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Kirchheim, B.: Currents in metric spaces. Acta Math. 185(1), 1–80 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces, Volume 25 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  4. 4.
    Bamler, R.: A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature. Math. Res. Lett. 23(2), 325–337 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  6. 6.
    Burago, D., Ivanov, S.: Area spaces: first steps, with appendix by nigel higson. Geom. Funct. Anal. 19(3), 662–677 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Basilio, J., Sormani, C.: Sequences of three dimensional manifolds with positive scalar curvature. Preprint to appear (2017)Google Scholar
  8. 8.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46(3), 406–480 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Federer, H., Fleming, W.H.: Normal and integral currents. Ann. Math. 2(72), 458–520 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fukaya, K.: Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87(3), 517–547 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gromov, M., Blaine Lawson, H.: The classification of simply connected manifolds of positive scalar curvature. Ann. Math. 111(3), 423–434 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gromov, M., Blaine Lawson Jr., H.: Spin and scalar curvature in the presence of a fundamental group. I. Ann. Math. 111(2), 209–230 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd edn. CRC Press, Boca Raton (1998)zbMATHGoogle Scholar
  14. 14.
    Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces, volume 152 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael BatesGoogle Scholar
  15. 15.
    Gromov, M.: Dirac and Plateau billiards in domains with corners. Century Eur. J. Math. 12(8), 1109–1156 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gromov, M.: Plateau-Stein manifolds. Century Eur. J. Math. 12(7), 923–951 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Huang, L.-H., Lee, D., Sormani, C.: Stability of the positive mass theorem for graphical hypersurfaces of Euclidean space. Journal fur die Riene und Angewandte Mathematik (Crelle’s Journal) 727, 269–299 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lakzian, S., Sormani, C.: Smooth convergence away from singular sets. Commun. Anal. Geom. 21(1), 39–104 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lee, D.A., Sormani, C.: Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds. Journal fur die Riene und Angewandte Mathematik (Crelle’s Journal), 686 (2014)Google Scholar
  20. 20.
    LeFloch, P.G., Sormani, C.: The nonlinear stability of rotationally symmetric spaces with low regularity. J. Funct. Anal. 268(7), 2005–2065 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169(3), 903–991 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Matveev, R., Portegies, J.: Intrinsic flat and Gromov-Hausdorff convergence of manifolds with Ricci curvature bounded below. J. Geom. Anal. 27(3), 1855–1873 (2017)Google Scholar
  23. 23.
    Portegies, J.W.: Semicontinuity of eigenvalues under intrinsic flat convergence. Calc. Var. Partial Differ. Equ. 54(2), 1725–1766 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rosenberg, J., Stolz, S.: Metrics of positive scalar curvature and connections with surgery. Surveys on Surgery Theory, Number 149 in Annals of Mathematics Studies 2. Princeton University Press, Princeton (2001)Google Scholar
  25. 25.
    Sormani, C.: Intrinsic flat Arzela-Ascoli theorems. Commun. Anal. Geom. 27(1), (2019) (on arxiv since 2014)Google Scholar
  26. 26.
    Sormani, C.: Scalar curvature and intrinsic flat convergence. In: Gigli, N. (ed) Measure Theory in Non-Smooth Spaces, Chap. 9, pp. 288–338. De Gruyter Press (2017)Google Scholar
  27. 27.
    Sormani, C., Stavrov, I.: Geometrostatic manifolds of small ADM mass (2017). arXiv: 1707.03008
  28. 28.
    Sturm, K.-T.: A curvature-dimension condition for metric measure spaces. C. R. Math. Acad. Sci. Paris 342, 197?200 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sormani, C., Wenger, S.: Weak convergence and cancellation, appendix by Raanan Schul and Stefan Wenger. Calc. Var. Partial Differ. Equ., 38(1–2) (2010)Google Scholar
  31. 31.
    Sormani, C., Wenger, S.: Intrinsic flat convergence of manifolds and other integral current spaces. J. Differ. Geom., 87 (2011)Google Scholar
  32. 32.
    Schoen, R., Yau, S.T.: On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28(1–3), 159–183 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Villani, C.: Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2009). Old and newGoogle Scholar
  35. 35.
    Wenger, S.: Compactness for manifolds and integral currents with bounded diameter and volume. Calc. Var. Partial Differ. Equ. 40(3–4), 423–448 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.CUNY Graduate Center and Sarah Lawrence CollegeBronxvilleUSA
  2. 2.CUNY Graduate Center and Queens CollegeNew YorkUSA
  3. 3.CUNY Graduate Center and Lehman CollegeBronxUSA

Personalised recommendations