Abstract
In this paper, we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature, noncollasping volume and bounded diameter. Under such conditions, we can show that the manifold has finitely many diffeomorphism types, which generalizes Cheeger-Naber’s result to bounded scalar curvature along Ricci flow. In particular, this implies the manifold has uniform L2 Riemann curvature bound. As an application, we point out that four dimensional Ricci flow would not have uniform scalar curvature upper bound if the initial metric only satisfying lower Ricci curvature bound, lower volume bound and upper diameter bound.
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
Anderson, M. T.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc., 2, 455–490 (1989)
Anderson, M., Cheeger, J.: Diffeomorphism finiteness for manifolds with Ricci curvature and Ln/2-norm of curvature bounded. Geom. Funct. Anal., 1(3), 231–252 (1991)
Bamler, R.: Structure theory of singular spaces. J. Funct. Anal., 272(6), 2504–2627 (2017)
Bamler, R.: Convergence of Ricci flows with bounded scalar curvature. Ann. of Math. (2), 188(3), 753–831 (2018)
Bamler, R., Zhang, Q.: Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature. Adv. Math., 319, 396–450 (2017)
Bamler, R., Zhang, Q.: Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature Part II. Calc. Var. Partial Differential Equations, 58(2), Art. 49, 14 pp. (2019)
[7] Bando, S., Kasue, A., Nakajima, H.: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math., 97, 313–349 (1989)
Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. of Math. (2), 167(3), 1079–1097 (2008)
Brendle, S., Schoen, R.: Manifolds with 1/4-pinched curvature are space forms. J. Amer. Math. Soc., 22(1), 287–307 (2009)
Cao, H., Zhu, X.: A complete proof of the Poincare and geometrization conjectures-application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math., 10(2), 165–492 (2006)
Cheeger, J.: Finiteness theorems for Riemannian manifolds. Amer. Jour. Math., 92, 61–74 (1970)
Cheeger, J., Colding, T. H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math., 144(1), 189–237 (1996)
Cheeger, J., Colding, T. H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom., 46(3), 406–480 (1997)
Cheeger, J., Jiang, W., Naber, A.: Rectifiability of singular sets in noncollapsed spaces with Ricci curvature bounded below. Ann. of Math. (2), 193(2), 407–538 (2021)
Cheeger, J., Naber, A.: Lower bounds on Ricci curvature and quantitative behavior of singular sets. Invent. Math., 191, 321–339 (2013)
Cheeger, J., Naber, A.: Regularity of Einstein Manifolds and the Codimension 4 Conjecture. Ann. of Math. (2), 182(3), 1093–1165 (2015)
Chen, X., Wang, B.: Space of Ricci flows (II)-Part B: Weak compactness of the flows. J. Differential Geom., 116(1), 1–123 (2020)
Colding, T.: Ricci curvature and volume convergence. Ann. of Math. (2), 145(3), 477–501 (1997)
Ge, H., Jiang, W.: ϵ-regularity for shrinking Ricci solitons and Ricci flows. Geom. Funct. Anal., 27(5), 1231–1256 (2017)
R. Hamilton,: Three-manifolds with positive Ricci curvature. J. Differential Geometry, 17(2), 255–306 (1982)
Hebey, E.: Nonlinear analysis on manifolds: Sobolev Spaces and Inequalities, Courant Lect. Notes Math. 5, Courant Institute of Mathematical Sciences, New York, 2000
Hochard, R.: Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, arXiv:1603.08726v1 [math.DG], 2016
Jiang, W.: Bergman kernel along the Khler-Ricci flow and Tian’s conjecture. J. Reine Angew. Math., 717, 195–226 (2016)
Jiang, W., Naber, A.: L2 curvature bounds on manifolds with bounded Ricci curvature. Ann. of Math. (2), 193(1), 107–222 (2021)
Jiang, W., Wang, F., Zhu, X.: Bergman kernels for a sequence of almost Kähler-Ricci solitons. Ann. Inst. Fourier (Grenoble), 67(3), 1279–1320 (2017)
Koch, H., Lamm, T.: Geometric flows with rough initial data. Asian J. Math., 16(2), 209–235 (2012)
Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geometry & Topology, 12(5), 2587–2855 (2008)
Morgan, J., Tian, G.: Ricci Flow and the Poincare Conjecture. Clay Mathematics Monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. xlii+521 pp. ISBN: 978-0-8218-4328-4
Perelman, G.: Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers. In: Comparison Geometry (Berkeley, CA, 1993–94), 157–163, Math. Sci. Res. Inst. Publ., 30, Cambridge Univ. Press, Cambridge, 1997
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1 [math.DG], 2002
Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109, 2003
Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv: 0307245, 2003
Petersen, P., Wei, G. F.: Relative volume comparison with integral curvature bounds. Geom. Funct. Anal., 7(6), 1031–1045 (1997)
Petersen, P., Wei, G. F.: Analysis and geometry on manifolds with integral Ricci curvature bounds. II. Trans. Amer. Math. Soc., 353(2), 457–478 (2001)
Peters, S.: Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds. J. Reine Angew. Math., 349, 77–82 (1984)
Simon, M.: Deformation of C0 Riemannian metrics in the direction of their Ricci curvature. Comm. Anal. Geom., 10(5), 1033–1074 (2002)
Simon, M.: Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below. J. Reine Angew. Math., 662, 59–94 (2012)
Simon, M., Topping, P.: Local control on the geometry in 3D Ricci flow, arXiv:1611.06137 [math.DG], 2016
Simon, M., Topping, P.: Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces, arXiv:1706.09490v1 [math.DG], 2017
Song, J., Tian, G.: The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math., 170(3), 609–653 (2007)
Song, J., Tian, G.: Canonical measures and Kahler-Ricci flow. J. Amer. Math. Soc., 25(2), 303–353 (2012)
Song, J., Tian, G.: The Kahler-Ricci flow through singularities. Invent. Math., 207(2), 519–595 (2017)
Tian, G.: On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math., 101(1), 101–172 (1990)
Tian, G., Zhang, Z. L.: Regularity of Kahler-Ricci flows on Fano manifolds. Acta Math., 216(1), 127–176 (2016)
Tian, G., Zhang, Z.: On the Kähler-Ricci flow on projective manifolds of general type. Chinese Ann. Math. Ser. B, 27(2), 179–192 (2006)
Wang, B.: On the conditions to extend Ricci flow (II). Int. Math. Res. Not. IMRN, 14, 3192–3223 (2012)
Wei, G.: Manifolds with a lower Ricci curvature bound. In: Surveys in Differential Geometry. Vol. XI, 203–227, Surv. Differ. Geom., 11, Int. Press, Somerville, MA, 2007
Ye, R.: The logarithmic Sobolev inequality along the Ricci flow. Communications in Mathematics and Statistics, 3, 1–36 (2015)
Zhang, Q.: A uniform Sobolev inequality under Ricci flow. Int. Math. Res. Not. IMRN, 17, Art. ID rnm056, 17 pp. (2007)
Zhang, Q.: Bounds on volume growth of geodesic balls under Ricci flow. Math. Res. Lett., 19(1), 245–253 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (Grant No. 11701507) and the Fundamental Research Funds for the Central Universities 2019QNA3001 and DECRA 190101471
Rights and permissions
About this article
Cite this article
Jiang, W.S. Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature. Acta. Math. Sin.-English Ser. 37, 1751–1767 (2021). https://doi.org/10.1007/s10114-021-0149-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-021-0149-4