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Almost Non-negative Scalar Curvature on Riemannian Manifolds Conformal to Tori

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Abstract

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in \(C^1\). The full conjecture from the conformal perspective is also discussed as a possible approach to resolving the conjecture.

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References

  1. Allen, B., Hernandez-Vazquez, L., Parise, D., Payne, A., Wang, S.: Warped tori with almost non-negative scalar curvature. Geom. Dedic. (2018)

  2. Allen, B., Sormani, C.: Contrasting various notions of convergence in geometric analysis. Pac. J. Math. 303(1), 1–46 (2019)

    Article  MathSciNet  Google Scholar 

  3. Allen, B., Sormani, C.: Relating notions of convergence in geometric analysis. Nonlinear Anal. 200 (2020)

  4. Allen, B., Perales, R., Sormani, C.: Volume above distance below. arXiv:2003.01172 (2020)

  5. Anderson, M.: Scalar curvature and geometrization conjectures for 3-manifolds. MSRI Publ. 30, 49–82 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Anderson, M.T.: Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds. I. Geom. Funct. Anal. 9(5), 855–967 (1999)

    Article  MathSciNet  Google Scholar 

  7. Aubin, T.: Équations différentielles non linéaires et probléme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. (9) 55(3), 269–296 (1976)

    MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  Google Scholar 

  9. Bergery, B.: La coubure scalire des variétés riemanniennes, Lecture Notes in Mathematics, Springer, New York, 1981.22 BRIAN ALLEN

  10. Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)

    Book  Google Scholar 

  11. Gromov, M.: Dirac and Plateau billiards in domains with corners. Cent. Eur. J. Math. 12(8), 1109–1156 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Gromov, M., Lawson Jr., H.B.: Spin and scalar curvature in the presence of a fundamental group. I. Ann. Math. (2) 111(2), 209–230 (1980)

    Article  MathSciNet  Google Scholar 

  13. Huang, L.-H., Lee, D.A.: Stability of the positive mass theorem for graphical hypersurfaces of Euclidean space. Commun. Math. Phys. 337(1), 151–169 (2015)

    Article  MathSciNet  Google Scholar 

  14. Huang, L.-H., Lee, D.A., Sormani, C.: Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space. J. Reine Angew. Math. 727, 269–299 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Lee, J.M., Parker, T.H.: The yamabe problem. Bull. Am. Math. Soc. 17(1), 37 (1987)

    Article  MathSciNet  Google Scholar 

  16. Lee, M.-C., Naber, A., Neumayer, R.: dp convergence and \(\varepsilon \)-regularity theorems for entropy and scalar curvature lower bounds (2021). arXiv:2010.15663

  17. Pacheco, A.J.C., Ketterer, C., Perales, R.: Stability of graphical tori with almost nonnegative scalar curvature. Calc. Var. 59(134), 1 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20(2), 479–495 (1984)

    Article  MathSciNet  Google Scholar 

  19. Schoen, R.: Variational Theory for the Total Scalar Curvature Functional for Riemannian Metrics and Related Topics. Lecture Notes in Mathematics, vol. 1365, pp. 120–154. Springer, Berlin (1987)

    Google Scholar 

  20. 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)

    Article  MathSciNet  Google Scholar 

  21. Sormani, C.: Scalar Curvature and Intrinsic Flat Convergence. Measure Theory in Non-smooth Spaces, pp. 288–338. De Gruyter Press, Berlin (2017)

    Google Scholar 

  22. Sormani, C., Wenger, S.: Intrinsic flat convergence of manifolds and other integral current spaces. J. Differ. Geom. 87, 117 (2011)

    Article  Google Scholar 

  23. Trudinger, N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22, 265–274 (1968)

    MathSciNet  MATH  Google Scholar 

  24. Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was funded in part by NSF DMS - 1612049. The author would like to thank Christina Sormani for funding provided during the 2020 Virtual Workshop on Ricci and Scalar Curvature. I would also like to thank the organizers of this workshop Christina Sormani, Guofang Wei, Hang Chen, Lan-Hsuan Huang, Pengzi Miao, Paolo Piazza, Blaine Lawson, and Richard Schoen for the opportunity to give a plenary talk.

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  1. University of Hartford, Hartford, USA

    Brian Allen

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  1. Brian Allen
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Correspondence to Brian Allen.

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Allen, B. Almost Non-negative Scalar Curvature on Riemannian Manifolds Conformal to Tori. J Geom Anal 31, 11190–11213 (2021). https://doi.org/10.1007/s12220-021-00677-2

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