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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: lftam@math.cuhk.edu.hk
*
e-mail: mclee@math.northwestern.edu
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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.

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