Abstract
In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon (2002) that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in the distributional sense provided that the initial metric is W1,p for some n < p ⩽ ∞. As an application, we use this result to study the relation between the Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in the distributional sense, then the manifold is isometric to a Ricci flat manifold.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Aleksandrov A D, Berestovskii V N, Nikolaev I G. Generalized Riemannian spaces. Russian Math Surveys, 1986, 41: 1–54
Bamler R H. A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature. Math Res Lett, 2016, 23: 325–337
Bamler R H, Zhang Q S. Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature. Adv Math, 2017, 319: 396–450
Burago D, Burago Y, Ivanov S. A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. Providence: Amer Math Soc, 2001
Burkhardt-Guim P. Pointwise lower scalar curvature bounds for C0 metrics via regularizing Ricci flow. Geom Funct Anal, 2019, 29: 1703–1772
Cao H D, Zhu X P. A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J Math, 2006, 10: 165–492
Cheeger J. Integral bounds on curvature, elliptic estimates, and rectifiability of singular sets. Geom Funct Anal, 2003, 13: 20–72
Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. I. J Differential Geom, 1997, 45: 406–480
Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. II. J Differential Geom, 2000, 54: 13–35
Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. III. J Differential Geom, 2000, 54: 37–74
Cheeger J, Jiang W S, Naber A. Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below. Ann of Math (2), 2021, 193: 407–538
Cheeger J, Naber A. Lower bounds on Ricci curvature and quantitative behavior of singular sets. Invent Math, 2013, 191: 321–339
Colding T H, Naber A. Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann of Math (2), 2012, 176: 1173–1229
Grant J D E, Tassotti N. A positive mass theorem for low-regularity Riemannian metrics. arXiv:1408.6425, 2014
Gromov M. Dirac and Plateau billiards in domains with corners. Cent Eur J Math, 2014, 12: 1109–1156
Gromov M, Lawson Jr H B. Spin and scalar curvature in the presence of a fundamental group. I. Ann of Math (2), 1980, 111: 209–230
Hamilton R S. Three-manifolds with positive Ricci curvature. J Differential Geom, 1982, 17: 255–306
Jiang W S. Bergman kernel along the Kähler-Ricci flow and Tian’s conjecture. J Reine Angew Math, 2016, 717: 195–226
Jiang W S, Naber A. L2 curvature bounds on manifolds with bounded Ricci curvature. Ann of Math (2), 2021, 193: 107–222
Jiang W S, Sheng W M, Zhang H Y. Removable singularity of positive mass theorem with continuous metrics. Math Z, 2022, 302: 839–874
Jiang W S, Wang F, Zhu X H. Bergman kernels for a sequence of almost Kähler-Ricci solitons. Ann Inst Fourier (Grenoble), 2017, 67: 1279–1320
Kazdan J L, Warner F W. Prescribing curvatures. In: Differential Geometry. Proceedings of Symposia in Pure Mathematics, vol. 27. Part 2. Providence: Amer Math Soc, 1975, 309–319
Kleiner B, Lott J. Notes on Perelman’s papers. Geom Topol, 2008, 12: 2587–2855
Lamm T, Simon M. Ricci flow of W2,2-metrics in four dimensions. arXiv:2109.08541, 2021
Lee D A, LeFloch P G. The positive mass theorem for manifolds with distributional curvature. Comm Math Phys, 2015, 339: 99–120
LeFloch P G, Mardare C. Definition and stability of Lorentzian manifolds with distributional curvature. Port Math, 2007, 64: 535–573
LeFloch P G, Sormani C. The nonlinear stability of rotationally symmetric spaces with low regularity. J Funct Anal, 2015, 268: 2005–2065
Li C, Mantoulidis C. Positive scalar curvature with skeleton singularities. Math Ann, 2019, 374: 99–131
Lott J, Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann of Math (2), 2009, 169: 903–991
McFeron D, Székelyhidi G. On the positive mass theorem for manifolds with corners. Comm Math Phys, 2012, 313: 425–443
Miao P. Positive mass theorem on manifolds admitting corners along a hypersurface. Adv Theor Math Phys, 2002, 6: 1163–1182
Morgan J W, Tian G. Ricci Flow and the Poincaré Conjecture. Providence: Amer Math Soc, 2007
Perelman G. The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1, 2002
Perelman G. Ricci flow with surgery on three-manifolds. arXiv:math/0303109v1, 2003
Perelman G. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245v1, 2003
Schoen R, Yau S T. On the structure of manifolds with positive scalar curvature. Manuscripta Math, 1997, 28: 159–183
Shi W X. Deforming the metric on complete Riemannian manifolds. J Differential Geom, 1989, 30: 223–301
Shi Y G, Tam L F. Scalar curvature and singular metrics. Pacific J Math, 2017, 293: 427–470
Simon M. Deformation of C0 Riemannian metrics in the direction of their Ricci curvature. Comm Anal Geom, 2002, 10: 1033–1074
Sormani C. Conjectures on convergence and scalar curvature. arXiv:2103.10093, 2021
Sturm K-T. On the geometry of metric measure spaces. I. Acta Math, 2006, 196: 65–131
Sturm K-T. On the geometry of metric measure spaces. II. Acta Math, 2006, 196: 133–177
Sturm K-T. A curvature-dimension condition for metric measure spaces. C R Math Acad Sci Paris, 2006, 342: 197–200
Acknowledgements
The first author was supported by National Natural Science Foundation of China (Grant Nos. 12125105 and 12071425) and the Fundamental Research Funds for the Central Universities. The second author was supported by National Natural Science Foundation of China (Grant Nos. 11971424 and 12031017). The third author was supported by National Natural Science Foundation of China (Grant No. 11971424). The authors thank Professor Dan Lee and Professor Christina Sormani for many helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, W., Sheng, W. & Zhang, H. Weak scalar curvature lower bounds along Ricci flow. Sci. China Math. 66, 1141–1160 (2023). https://doi.org/10.1007/s11425-021-2037-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-021-2037-7