Geometric and Functional Analysis

, Volume 29, Issue 6, pp 1703–1772 | Cite as

Pointwise lower scalar curvature bounds for \(C^0\) metrics via regularizing Ricci flow

  • Paula Burkhardt-GuimEmail author


In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for \(C^0\) metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from \(C^0\) initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from \(C^0\) initial data.



I would like to thank my advisor, Richard Bamler, for introducing me to this project, and for his help and encouragement. I would also like to thank Christina Sormani, for posing the problem of torus rigidity for Definition 1.2 to me, and for showing relevant references to me. Finally, I would like to thank Chao Li, for showing the work of Simon [Sim02] to me, and for many helpful comments on a previous version of this paper. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1752814. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.


  1. Bam14.
    R.H. Bamler. Stability of hyperbolic manifolds with cusps under Ricci flow. Advances in Mathematics 263 (2014), 412 –467. MR3239144.MathSciNetCrossRefGoogle Scholar
  2. Bam16.
    R.H. Bamler. A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature. Mathematical Research Letters (2)23 (2016), 325 –337. MR3512888.MathSciNetCrossRefGoogle Scholar
  3. BK18.
    R.H. Bamler and B. Kleiner. Uniqueness and stability of Ricci flow through singularities (2018). arXiv:1709.04122
  4. Bam11.
    R.H. Bamler. Stability of Einstein metrics of negative curvature. Ph.D. Thesis (2011).Google Scholar
    B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni. The Ricci low: techniques and applications. Part III : Geometric-analytic aspects. Mathematical Surveys and Monographs, vol. 163, American Mathematical Society, Providence (2010). MR2604955.Google Scholar
  6. DeT83.
    D.M. DeTurck. Deforming metrics in the direction of their Ricci tensors. The Journal of Differential Geometry (1)18 (1983), 157–162. MR0697987.MathSciNetCrossRefGoogle Scholar
  7. GL80.
    M. Gromov and H. Blaine Lawson, Jr. Spin and scalar curvature in the presence of a fundamental group. I. Annals of Mathematics. (2). (2)111 (1980), 209–230. MR0569070.MathSciNetCrossRefGoogle Scholar
  8. Gro14.
    M. Gromov, Dirac and Plateau billiards in domains with corners. Central European Journal of Mathematics (8)12 (2014), 1109–1156. MR3201312.MathSciNetzbMATHGoogle Scholar
  9. Kar77.
    H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics (1977), 509–541. MR0442975.MathSciNetCrossRefGoogle Scholar
  10. KL12.
    H. Koch and T. Lamm. Geometric flows with rough initial data. Asian Journal of Mathematics 16 (2012), 209–236. MR2916362.MathSciNetCrossRefGoogle Scholar
  11. KL15.
    H. Koch and T. Lamm. Parabolic equations with rough data. Mathematica Bohemica (4)140 (2015), 457–477. MR3432546.MathSciNetzbMATHGoogle Scholar
  12. Kry96.
    N.V. Krylov. Lectures on elliptic and parabolic equations in Hölder spaces. Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence (1996). MR1406091.Google Scholar
  13. Kry08.
    N.V. Krylov. Lectures on elliptic and parabolic equations in Sobolev spaces. Graduate Studies in Mathematics, vol. 96, American Mathematical Society, Providence (2008). MR2435520.Google Scholar
  14. SY79.
    R. Schoen and S.T. Yau. On the structure of manifolds with positive scalar curvature. Manuscripta Math. (1-3)28 (1979), 159–183.MathSciNetCrossRefGoogle Scholar
  15. Sim02.
    M. Simon. Deformation of \(C^{0}\) Riemannian metrics in the direction of their Ricci curvature. Communications in Analysis and Geometry (5)10 (2002), 1033–1074.MathSciNetCrossRefGoogle Scholar
  16. Sog17.
    C.D. Sogge. Fourier Integrals in Classical Analysis. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, (2017). MR3645429.Google Scholar
  17. Top06.
    P. Topping. Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, vol. 325, Cambridge University Press, Cambridge (2006). MR2265040.Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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