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Li-Yau gradient bound for collapsing manifolds under integral curvature condition


Authors: Qi S. Zhang and Meng Zhu
Journal: Proc. Amer. Math. Soc. 145 (2017), 3117-3126
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/proc/13418
Published electronically: January 6, 2017
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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)}\vert Ric^-\vert^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 $ \vert Ric^-\vert\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 $ \vert Ric^-\vert$ 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
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
Email: mzhu@math.ucr.edu

DOI: https://doi.org/10.1090/proc/13418
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
Article copyright: © Copyright 2017 American Mathematical Society