An improvement of the sharp Li-Yau bound on closed manifolds. (English) Zbl 07921916
Summary: In this paper, we give a generalization of Zhang’s recent work about a sharp Li-Yau gradient bound on compact manifolds by extending Hamilton’s gradient estimates. In particular, we take a special auxiliary function to indicate that our estimate is a slight improvement of Zhang’s result.
MSC:
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 35K05 | Heat equation |
References:
| [1] | Aronson, DG; Bénilan, P., Régularité des solutions de l’équation des milieux poreux dans \(\textbf{R}^N\), C. R. Acad. Sci. Paris Sér. A-B, 288, 2, A103-A105, 1979 · Zbl 0397.35034 |
| [2] | Bakry, D.; Bolley, F.; Gentil, I., The Li-Yau inequality and applications under a curvature-dimension condition, Ann. Inst. Fourier (Grenoble), 67, 1, 397-421, 2017 · Zbl 1383.58017 · doi:10.5802/aif.3086 |
| [3] | Bakry, D.; Qian, Z-M, Harnack inequalities on a manifold with positive or negative Ricci curvature, Rev. Mat. Iberoam., 15, 1, 143-179, 1999 · Zbl 0924.58096 · doi:10.4171/rmi/253 |
| [4] | Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Graduate Studies in Mathematics, 77. American Mathematical Society, Providence, RI; Science Press Beijing, New York (2006) · Zbl 1118.53001 |
| [5] | Hamilton, R., A matrix Harnack estimate for the heat equation, Comm. Anal. Geom., 1, 1, 113-126, 1993 · Zbl 0799.53048 · doi:10.4310/CAG.1993.v1.n1.a6 |
| [6] | Kotschwar, B., Hamilton’s gradient estimate for the kernel on complete manifolds, Proc. Amer. Math. Soc., 135, 3013-3019, 2007 · Zbl 1127.58021 · doi:10.1090/S0002-9939-07-08837-5 |
| [7] | Li, J-F; Xu, X-J, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math., 226, 5, 4456-4491, 2011 · Zbl 1226.58009 · doi:10.1016/j.aim.2010.12.009 |
| [8] | Li, P.; Yau, S-T, On the parabolic kernel of the Schrödinger operator, Acta Math., 156, 3-4, 153-201, 1986 · Zbl 0611.58045 · doi:10.1007/BF02399203 |
| [9] | Li, Y., Li-Yau-Hamilton estimates and Bakry-Emery-Ricci curvature, Nonlinear Anal., 113, 1-32, 2015 · Zbl 1310.58015 · doi:10.1016/j.na.2014.09.014 |
| [10] | Qian, B., Remarks on differential Harnack inequalities, J. Math. Anal. Appl., 409, 1, 556-566, 2014 · Zbl 1310.58016 · doi:10.1016/j.jmaa.2013.07.043 |
| [11] | Song, X.-Y., Wu, L., Zhu, M.: A direct approach to sharp Li-Yau estimates on closed manifolds with negative Ricci lower bound. arXiv:2307.03879 (2023) |
| [12] | Yau, S-T, Harnack inequality for non-self-adjoint evolution equations, Math. Res. Lett., 2, 4, 387-399, 1995 · Zbl 0884.58091 · doi:10.4310/MRL.1995.v2.n4.a2 |
| [13] | Yu, C-J; Zhao, F-F, Li-Yau multiplier set and optimal Li-Yau gradient estimate on hyperbolic spaces, Potential Anal., 56, 2, 191-211, 2022 · Zbl 1483.35059 · doi:10.1007/s11118-020-09881-1 |
| [14] | Zhang, Q.S.: A sharp Li-Yau gradient bound on compact manifolds. arXiv:2110.08933 (2022) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
