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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Li-Yau gradient bound for collapsing manifolds under integral curvature condition
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by Qi S. Zhang and Meng Zhu PDF
Proc. Amer. Math. Soc. 145 (2017), 3117-3126 Request permission

Abstract:

Let $(\mathbf {M}^n, g_{ij})$ be a complete Riemannian manifold. For any constants $p,\ r>0$, define $\displaystyle k(p,r)=\sup _{x\in M}r^2\left (\oint _{B(x,r)}|Ric^-|^p dV\right )^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\frac {n}{2}$, when $k(p,1)$ is small enough, a certain Li-Yau type gradient bound holds for the positive solutions of the heat equation on geodesic balls $B(O,r)$ in $\mathbf {M}$ with $0<r\leq 1$. Here the assumption that $k(p,1)$ is small allows the situation where the manifold is collapsing. Recall that in an earlier paper by Zhang and Zhu, a certain Li-Yau gradient bound was also obtained by the authors, assuming that $|Ric^-|\in L^p(\mathbf {M})$ and the manifold is noncollapsed. Therefore, to some extent, the results in this paper, as well as the earlier one complete the picture of Li-Yau gradient bounds for the heat equation on manifolds with $|Ric^-|$ being $L^p$ integrable, modulo the sharpness of constants.
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Additional Information
  • Qi S. Zhang
  • Affiliation: Department of Mathematics, University of California, Riverside, Riverside, California 92521
  • MR Author ID: 359866
  • Email: qizhang@math.ucr.edu
  • Meng Zhu
  • Affiliation: Department of Mathematics, East China Normal University, Shanghai 200241, People’s Republic of China — and — Department of Mathematics, University of California, Riverside, Riverside, California 92521
  • MR Author ID: 888985
  • Email: mzhu@math.ucr.edu
  • Received by editor(s): July 19, 2016
  • Received by editor(s) in revised form: August 4, 2016
  • Published electronically: January 6, 2017
  • Communicated by: Guofang Wei
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 3117-3126
  • MSC (2010): Primary 53C44
  • DOI: https://doi.org/10.1090/proc/13418
  • MathSciNet review: 3637958