Warped tori with almost non-negative scalar curvature

  • Brian Allen
  • Lisandra Hernandez-Vazquez
  • Davide Parise
  • Alec Payne
  • Shengwen Wang
Original Paper
  • 10 Downloads

Abstract

For sequences of warped product metrics on a 3-torus satisfying the scalar curvature bound \(R_j \ge -\frac{1}{j}\), uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov–Hausdorff and the Sormani–Wenger intrinsic flat sense to a flat 3-torus.

Keywords

Scalar torus rigidity theorem Almost rigidity Stability Gromov–Hausdorff convergence Sormani–Wenger intrinsic flat convergence Warped products Uniform convergence Flat torus Convergence of Riemannian manifolds 

Mathematics Subject Classification

53 

Notes

Acknowledgements

This research began at the Summer School for Geometric Analysis, located at and supported by the Fields Institute for Research in Mathematical Sciences. The authors would like to thank the organizers of this summer school, Spyros Alexakis, Walter Craig, Robert Haslhofer, Spiro Karigiannis and McKenzie Wang. The authors would like to thank Christina Sormani for her direction, advice, and constant support. Specifically, Christina brought this team together during the Field’s institute to work on this project, organized workshops at the CUNY graduate center which the authors participated in, and provided travel support to the team members. Christina Sormani is supported by NSF grant DMS-1612049. Brian Allen is supported by the USMA. Davide Parise is partially supported by the Swiss National Foundation grant 200021L_175985.

References

  1. 1.
    Allen, B., Sormani, C.: Contrasting various notions of convergence in geometric analysis. arXiv:1803.06582 (2018)
  2. 2.
    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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bray, H., Brendle, S., Eichmair, M., Neves, A.: Area-minimizing projective planes in 3-manifolds. Commun. Pure Appl. Math. 63(9), 1237–1247 (2010)MathSciNetMATHGoogle Scholar
  4. 4.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001). (2002e:53053) MATHGoogle Scholar
  5. 5.
    Bray, H., Brendle, S., Neves, A.: Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom. 18(4), 821–830 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Basilio, J., Dodziuk, J., Sormani, C.: Sewing riemannian manifolds with positive scalar curvature. arXiv:1703.00984 (2017)
  7. 7.
    Basilio, J., Sormani, C.: Sequences of three-dimensional manifolds with positive scalar curvature. preprint to appear, see also Basilio’s doctoral dissertation at CUNYGC (2017)Google Scholar
  8. 8.
    Dobarro, F., Dozo, E.L.: Scalar curvature and warped products of riemann manifolds. Trans. Am. Math. Soc. 303, 161–168 (1987)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gromov, M., Jr Lawson, H.B.: Spin and scalar curvature in the presence of a fundamental group. I. Ann. Math. (2) 111(2), 209–230 (1980)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gromov, M.: Metric structures for riemannian and non-riemannian spaces, english ed.Modern Birkhäuser Classics, vol. 1, Birkäuser, 2007. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael BatesGoogle Scholar
  11. 11.
    Gromov, M.: Dirac and Plateau billiards in domains with corners. Cent. Eur. J. Math. 12(8), 1109–1156 (2014)MathSciNetMATHGoogle Scholar
  12. 12.
    Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differ. Geom. 80(3), 433–451 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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, 187 (2014)MathSciNetMATHGoogle Scholar
  14. 14.
    Petersen, P.: Riemannian geometry, Graduate Texts in Mathematics, vol. 171. Springer, Berlin (2016)CrossRefGoogle Scholar
  15. 15.
    Sormani, C.: Scalar curvature and intrinsic flat convergence, to appear as a chapter in Measure Theory in Non- Smooth Spaces, De Gruyter Press, edited by Nicola Gigli. arXiv:1606.08949 (2016)
  16. 16.
    Stampacchia, G.: Equations elliptiques au second ordre à coéfficients discontinues. Sem. Math. Sup. 16, 16 (1966)Google Scholar
  17. 17.
    Sormani C., Wenger, S.: Intrinsic flat convergence of manifolds and other integral current spaces. J. Differ. Geom. 87, 117–199 (2011)Google Scholar
  18. 18.
    Schoen, R., Yau, S.T.: Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Ann. Math. (2) 110(1), 127–142 (1979)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.United States Military AcademyWest PointUSA
  2. 2.Stony Brook UniversityStony BrookUSA
  3. 3.Ecole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland
  4. 4.Courant InstituteNew York UniversityNew YorkUSA
  5. 5.John Hopkins UniversityBaltimoreUSA

Personalised recommendations