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Some local maximum principles along Ricci flows

Part of: Global differential geometry

Published online by Cambridge University Press:  04 November 2020

Man-Chun Lee  and
Luen-Fai Tam
Man-Chun Lee*
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL60208, USA
Luen-Fai Tam
Affiliation:
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t>0$ for some constant ${\alpha }>0$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ .

Keywords

Ricci flowmaximum principle

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 2 , April 2022 , pp. 329 - 348
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

Research partially supported by NSF grant DMS-1709894. Research partially supported by Hong Kong RGC General Research Fund #CUHK 14301517.

References

Bamler, R., Cabezas-Rivas, E., and Wilking, B., The Ricci flow under almost non-negative curvature conditions . Invent. Math. 217(2019), no. 1, 95–126.CrossRefGoogle Scholar
Cabezas-Rivas, E. and Wilking, B., How to produce a Ricci flow via Cheeger-Gromoll exhaustion . J. Eur. Math. Soc. (JEMS) 17(2015), no. 12, 3153–3194, MR3429162, Zbl 1351.53078.CrossRefGoogle Scholar
Chau, A. and Lee, M.-C., The Kähler Ricci flow around complete bounded curvature Kähler metrics . Trans. Amer. Math. Soc. 373(2020), no. 5, 3627–3647.CrossRefGoogle Scholar
Chau, A., Tam, L.-F., and Yu, C., Pseudolocality for the Ricci flow and applications . Can. J. Math. 63(2011), no. 1, 55–85.CrossRefGoogle Scholar
Chen, B.-L., Strong uniqueness of the Ricci flow . J. Differ. Geom. 82(2009), no. 2, 363–382, MR2520796, Zbl 1177.53036.CrossRefGoogle Scholar
Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., and Ni, L., The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects. Mathematical Surveys and Monographs, 163, American Mathematical Society, Providence, RI, 2010.CrossRefGoogle Scholar
Giesen, G. and Topping, P.-M., Existence of Ricci flows of incomplete surfaces . Comm. Partial Differ. Equat. 36(2011), no. 10, 1860–1880.CrossRefGoogle Scholar
Guenther, C.-M., The fundamental solution on manifolds with time-dependent metrics . J. Geom. Anal. 12(2002), no. 3, 425–436.CrossRefGoogle Scholar
Grigoryan, A., Gaussian upper bounds for the heat kernel on arbitrary manifolds . J. Differ. Geom. 45(1997), 33–52.Google Scholar
He, F., Existence and applications of Ricci flows via pseudolocality. Preprint, 2016. arXiv:1610.01735 Google Scholar
Huang, S.-C. and Tam, L.-F., Kähler-Ricci flow with unbounded curvature . Amer. J. Math. 140(2018), no. 1, 189–220.CrossRefGoogle Scholar
Hochard, R., Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below. Preprint, 2016. arXiv:1603.08726.Google Scholar
Hochard, R., Theórèmes d’existence en temps court du flot de Ricci pour des variétés non-complètes, non-éffondrées, àcourbure minorée. PhD thesis, Universite de Bordeaux, 2019.Google Scholar
Lai, Y., Ricci flow under local almost non-negative curvature conditions . Adv. Math. 343(2019), 353–392.CrossRefGoogle Scholar
Lee, M.-C. and Tam, L.-F., Chern-Ricci flows on noncompact manifolds . J. Differ. Geom. 115(2020), no. 3, 529–564.CrossRefGoogle Scholar
Lee, M.-C. and Tam, L.-F., Kähler manifolds with almost non-negative curvature. Preprint, 2020. arXiv:1910.02531 Google Scholar
Li, X. and Ni, L., Kähler-Ricci Shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature. J. Math. Pure Appl. 138(2020), 28–45.CrossRefGoogle Scholar
Liu, G., Gromov-Hausdorff limits of Kähler manifolds with bisectional curvature lower bound . Commun. Pure Appl. Math. 71(2018), no. 2, 267–303 CrossRefGoogle Scholar
Lott, J., Comparison geometry of holomorphic bisectional for Kähler manifolds and limit spaces. Preprint, 2020. arXiv:2005.02906 CrossRefGoogle Scholar
McLeod, A.-D. and Topping, P.-M., Global regularity of three-dimensional Ricci limit spaces. Preprint, 2020. arXiv:1803.00414 Google Scholar
McLeod, A.-D. and Topping, P.-M., Pyramid Ricci flow in higher dimensions . Math. Z. 296(2020), 511–523.CrossRefGoogle Scholar
Perelman, G., The entropy formula for the Ricci flow and its geometric applications. Preprint, 2020. arXiv:math.DG/0211159 Google Scholar
Shi, W.-X., Ricci flow and the uniformization on complete noncompact Kähler manifolds . J. Differ. Geom. 45(1997), no. 1, 94–220.CrossRefGoogle Scholar
Simon, M., Deformation of ${C}^0$ Riemannian metrics in the direction of their Ricci curvature . Comm. Anal. Geom. 10(2002), no. 5, 1033–1074.CrossRefGoogle Scholar
Simon, M. and Topping, P.-M., Local control on the geometry in 3D Ricci flow. Preprint, 2020. arXiv:1611.06137 Google Scholar
Simon, M. and Topping, P.-M., Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces. Preprint, 2017. arXiv:1706.09490 Google Scholar
Wilking, B., A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities . J. Reine Angew. Math. 679(2013), 223–247.CrossRefGoogle Scholar
Xu, G., Short-time existence of the Ricci flow on noncompact Riemannian manifolds . Trans. Amer. Math. Soc. 365(2013), no. 11, 5605–5654.CrossRefGoogle Scholar