## 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

- Dominique Bakry, Francois Bolley, and Ivan Gentil,
*The Li-Yau inequality and applications under a curvature-dimension condition*, arXiv:1412.5165, 2014. - Dominique Bakry and Michel Ledoux,
*A logarithmic Sobolev form of the Li-Yau parabolic inequality*, Rev. Mat. Iberoam.**22**(2006), no. 2, 683–702. MR**2294794**, DOI 10.4171/RMI/470 - Huai-Dong Cao and Lei Ni,
*Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds*, Math. Ann.**331**(2005), no. 4, 795–807. MR**2148797**, DOI 10.1007/s00208-004-0605-3 - Huai-Dong Cao, Gang Tian, and Xiaohua Zhu,
*Kähler-Ricci solitons on compact complex manifolds with $C_1(M)>0$*, Geom. Funct. Anal.**15**(2005), no. 3, 697–719. MR**2221147**, DOI 10.1007/s00039-005-0522-y - Xianzhe Dai, Guofang Wei, and Zhenlei Zhang,
*Local Sobolev constant estimate for integral Ricci curvature bounds*, arXiv:1601.08191. - E. B. Davies,
*Heat kernels and spectral theory*, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR**990239**, DOI 10.1017/CBO9780511566158 - Nicola Garofalo and Andrea Mondino,
*Li-Yau and Harnack type inequalities in $\mathsf {RCD}^*(K,N)$ metric measure spaces*, Nonlinear Anal.**95**(2014), 721–734. MR**3130557**, DOI 10.1016/j.na.2013.10.002 - Richard S. Hamilton,
*A matrix Harnack estimate for the heat equation*, Comm. Anal. Geom.**1**(1993), no. 1, 113–126. MR**1230276**, DOI 10.4310/CAG.1993.v1.n1.a6 - Junfang Li and Xiangjin Xu,
*Differential Harnack inequalities on Riemannian manifolds I: linear heat equation*, Adv. Math.**226**(2011), no. 5, 4456–4491. MR**2770456**, DOI 10.1016/j.aim.2010.12.009 - Peter Li and Shing-Tung Yau,
*On the parabolic kernel of the Schrödinger operator*, Acta Math.**156**(1986), no. 3-4, 153–201. MR**834612**, DOI 10.1007/BF02399203 - Gary M. Lieberman,
*Second order parabolic differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR**1465184**, DOI 10.1142/3302 - P. Petersen and G. Wei,
*Relative volume comparison with integral curvature bounds*, Geom. Funct. Anal.**7**(1997), no. 6, 1031–1045. MR**1487753**, DOI 10.1007/s000390050036 - Zhongmin Qian, Hui-Chun Zhang, and Xi-Ping Zhu,
*Sharp spectral gap and Li-Yau’s estimate on Alexandrov spaces*, Math. Z.**273**(2013), no. 3-4, 1175–1195. MR**3030695**, DOI 10.1007/s00209-012-1049-1 - Laurent Saloff-Coste,
*Uniformly elliptic operators on Riemannian manifolds*, J. Differential Geom.**36**(1992), no. 2, 417–450. MR**1180389** - Feng-Yu Wang,
*Gradient and Harnack inequalities on noncompact manifolds with boundary*, Pacific J. Math.**245**(2010), no. 1, 185–200. MR**2602689**, DOI 10.2140/pjm.2010.245.185 - Jiaping Wang,
*Global heat kernel estimates*, Pacific J. Math.**178**(1997), no. 2, 377–398. MR**1447421**, DOI 10.2140/pjm.1997.178.377 - Hui-Chun Zhang and Xi-Ping Zhu,
*Local Li-Yau’s estimates on $RCD^*{(K,N)}$ metric measure spaces*, Calc. Var. Partial Differential Equations**55**(2016), no. 4, Art. 93, 30. MR**3523660**, DOI 10.1007/s00526-016-1040-5 - Qi S. Zhang and Meng Zhu,
*Li-Yau gradient bounds under nearly optimal curvature conditions*, arXiv:1511.00791.

## 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