# American Institute of Mathematical Sciences

September  2014, 34(9): 3535-3554. doi: 10.3934/dcds.2014.34.3535

## Some results for the Perelman LYH-type inequality

 1 Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi 621, R.O.C, Taiwan

Received  December 2012 Revised  November 2013 Published  March 2014

Let $(M,g(t))$, $0\le t\le T$, $\partial M\ne\phi$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal of $\partial M$ is uniformly bounded below on $\partial M\times [0,T]$. We will prove a global Li-Yau gradient estimate for the solution of the generalized conjugate heat equation on $M\times [0,T]$. We will give another proof of Perelman's Li-Yau-Hamilton type inequality for the fundamental solution of the conjugate heat equation on closed manifolds without using the properties of the reduced distance. We will also prove various gradient estimates for the Dirichlet fundamental solution of the conjugate heat equation.
Citation: Shu-Yu Hsu. Some results for the Perelman LYH-type inequality. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3535-3554. doi: 10.3934/dcds.2014.34.3535
##### References:
 [1] A. Chau, L. F. Tam and C. Yu, Pseudolocality for the Ricci flow and applications, Canad. J. Math., 63 (2011), 55-85. doi: 10.4153/CJM-2010-076-2. [2] I. Chavel, Riemannian geometry: A modern introduction, Cambridge University Press, Cambridge, United Kingdom, 1995. doi: 10.1017/CBO9780511616822. [3] R. Chen, Neumann eigenvalue estimate on a compact Riemannian manifold, Proc. AMS, 108 (1990), 961-970. doi: 10.1090/S0002-9939-1990-0993745-X. [4] B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Graduate Studies in Mathematics, vol. 77, Amer. Math. Soc., Providence, R.I., U.S.A., 2006. [5] R. S. Hamilton, The formation of singularities in the Ricci flow, in Surveys in differential geometry, vol. 2 International Press, Cambridge, MA, 1995, 7-136. [6] S. Y. Hsu, Uniqueness of solutions of Ricci flow on complete noncompact manifolds, arXiv:0704.3468. [7] S. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008-1023. doi: 10.1016/j.jfa.2008.05.014. [8] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Mono. Vol 23, Amer. Math. Soc., Providence, R.I., 1968. [9] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201. doi: 10.1007/BF02399203. [10] L. Ni, The entropy formula for linear heat equation, J. Geometric Analysis, 14 (2004), 87-100. doi: 10.1007/BF02921867. [11] L. Ni, Addenda to "The entropy formula for linear heat equation'', J. Geometric Analysis, 14 (2004), 369-374. doi: 10.1007/BF02922078. [12] L. Ni, A note on Perelman's LYH-type inequality, Comm. Anal. and Geom., 14 (2006), 883-905. doi: 10.4310/CAG.2006.v14.n5.a3. [13] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159. [14] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1994. [15] P. Souplet and Q. S. Zhang, Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. London math. Soc., 38 (2006), 1045-1053. doi: 10.1112/S0024609306018947. [16] J. Wang, Global heat kernel estimates, Pacific J. Math., 178(2) (1997), 377-398. doi: 10.2140/pjm.1997.178.377. [17] F. W. Warner, Extension of the Rauch comparison theorem to submanifolds, Trans. Amer. Math. Soc., 122 (1966), 341-356. [18] Q. S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Notice, (2006), Art. ID 92314, 39 pp. doi: 10.1155/IMRN/2006/92314.

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##### References:
 [1] A. Chau, L. F. Tam and C. Yu, Pseudolocality for the Ricci flow and applications, Canad. J. Math., 63 (2011), 55-85. doi: 10.4153/CJM-2010-076-2. [2] I. Chavel, Riemannian geometry: A modern introduction, Cambridge University Press, Cambridge, United Kingdom, 1995. doi: 10.1017/CBO9780511616822. [3] R. Chen, Neumann eigenvalue estimate on a compact Riemannian manifold, Proc. AMS, 108 (1990), 961-970. doi: 10.1090/S0002-9939-1990-0993745-X. [4] B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Graduate Studies in Mathematics, vol. 77, Amer. Math. Soc., Providence, R.I., U.S.A., 2006. [5] R. S. Hamilton, The formation of singularities in the Ricci flow, in Surveys in differential geometry, vol. 2 International Press, Cambridge, MA, 1995, 7-136. [6] S. Y. Hsu, Uniqueness of solutions of Ricci flow on complete noncompact manifolds, arXiv:0704.3468. [7] S. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008-1023. doi: 10.1016/j.jfa.2008.05.014. [8] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Mono. Vol 23, Amer. Math. Soc., Providence, R.I., 1968. [9] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201. doi: 10.1007/BF02399203. [10] L. Ni, The entropy formula for linear heat equation, J. Geometric Analysis, 14 (2004), 87-100. doi: 10.1007/BF02921867. [11] L. Ni, Addenda to "The entropy formula for linear heat equation'', J. Geometric Analysis, 14 (2004), 369-374. doi: 10.1007/BF02922078. [12] L. Ni, A note on Perelman's LYH-type inequality, Comm. Anal. and Geom., 14 (2006), 883-905. doi: 10.4310/CAG.2006.v14.n5.a3. [13] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159. [14] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1994. [15] P. Souplet and Q. S. Zhang, Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. London math. Soc., 38 (2006), 1045-1053. doi: 10.1112/S0024609306018947. [16] J. Wang, Global heat kernel estimates, Pacific J. Math., 178(2) (1997), 377-398. doi: 10.2140/pjm.1997.178.377. [17] F. W. Warner, Extension of the Rauch comparison theorem to submanifolds, Trans. Amer. Math. Soc., 122 (1966), 341-356. [18] Q. S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Notice, (2006), Art. ID 92314, 39 pp. doi: 10.1155/IMRN/2006/92314.
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