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.References
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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