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

Some results for the Perelman LYH-type inequality

Shu-Yu Hsu 1,

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.
Keywords: Li-Yau gradient estimate, Perelman's Li-Yau-Hamilton type inequality., conjugate heat equation, Ricci flow, Dirichlet fundamental solutions.
Mathematics Subject Classification: Primary: 58J35, 58C99; Secondary: 35K0.
Citation: Shu-Yu Hsu. Some results for the Perelman LYH-type inequality. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.4153/CJM-2010-076-2.  Google Scholar

[2]

I. Chavel, Riemannian geometry: A modern introduction,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511616822.  Google Scholar

[3]

R. Chen, Neumann eigenvalue estimate on a compact Riemannian manifold,, Proc. AMS, 108 (1990), 961.  doi: 10.1090/S0002-9939-1990-0993745-X.  Google Scholar

[4]

B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow,, Graduate Studies in Mathematics, (2006).   Google Scholar

[5]

R. S. Hamilton, The formation of singularities in the Ricci flow,, in Surveys in differential geometry, (1995), 7.   Google Scholar

[6]

S. Y. Hsu, Uniqueness of solutions of Ricci flow on complete noncompact manifolds,, , ().   Google Scholar

[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.  doi: 10.1016/j.jfa.2008.05.014.  Google Scholar

[8]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type,, Transl. Math. Mono. Vol 23, (1968).   Google Scholar

[9]

P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator,, Acta Math., 156 (1986), 153.  doi: 10.1007/BF02399203.  Google Scholar

[10]

L. Ni, The entropy formula for linear heat equation,, J. Geometric Analysis, 14 (2004), 87.  doi: 10.1007/BF02921867.  Google Scholar

[11]

L. Ni, Addenda to "The entropy formula for linear heat equation'',, J. Geometric Analysis, 14 (2004), 369.  doi: 10.1007/BF02922078.  Google Scholar

[12]

L. Ni, A note on Perelman's LYH-type inequality,, Comm. Anal. and Geom., 14 (2006), 883.  doi: 10.4310/CAG.2006.v14.n5.a3.  Google Scholar

[13]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, , ().   Google Scholar

[14]

R. Schoen and S. T. Yau, Lectures on Differential Geometry,, International Press, (1994).   Google Scholar

[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.  doi: 10.1112/S0024609306018947.  Google Scholar

[16]

J. Wang, Global heat kernel estimates,, Pacific J. Math., 178(2) (1997), 377.  doi: 10.2140/pjm.1997.178.377.  Google Scholar

[17]

F. W. Warner, Extension of the Rauch comparison theorem to submanifolds,, Trans. Amer. Math. Soc., 122 (1966), 341.   Google Scholar

[18]

Q. S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman,, Int. Math. Res. Notice, (2006).  doi: 10.1155/IMRN/2006/92314.  Google Scholar

show all references

References:
[1]

A. Chau, L. F. Tam and C. Yu, Pseudolocality for the Ricci flow and applications,, Canad. J. Math., 63 (2011), 55.  doi: 10.4153/CJM-2010-076-2.  Google Scholar

[2]

I. Chavel, Riemannian geometry: A modern introduction,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511616822.  Google Scholar

[3]

R. Chen, Neumann eigenvalue estimate on a compact Riemannian manifold,, Proc. AMS, 108 (1990), 961.  doi: 10.1090/S0002-9939-1990-0993745-X.  Google Scholar

[4]

B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow,, Graduate Studies in Mathematics, (2006).   Google Scholar

[5]

R. S. Hamilton, The formation of singularities in the Ricci flow,, in Surveys in differential geometry, (1995), 7.   Google Scholar

[6]

S. Y. Hsu, Uniqueness of solutions of Ricci flow on complete noncompact manifolds,, , ().   Google Scholar

[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.  doi: 10.1016/j.jfa.2008.05.014.  Google Scholar

[8]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type,, Transl. Math. Mono. Vol 23, (1968).   Google Scholar

[9]

P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator,, Acta Math., 156 (1986), 153.  doi: 10.1007/BF02399203.  Google Scholar

[10]

L. Ni, The entropy formula for linear heat equation,, J. Geometric Analysis, 14 (2004), 87.  doi: 10.1007/BF02921867.  Google Scholar

[11]

L. Ni, Addenda to "The entropy formula for linear heat equation'',, J. Geometric Analysis, 14 (2004), 369.  doi: 10.1007/BF02922078.  Google Scholar

[12]

L. Ni, A note on Perelman's LYH-type inequality,, Comm. Anal. and Geom., 14 (2006), 883.  doi: 10.4310/CAG.2006.v14.n5.a3.  Google Scholar

[13]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, , ().   Google Scholar

[14]

R. Schoen and S. T. Yau, Lectures on Differential Geometry,, International Press, (1994).   Google Scholar

[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.  doi: 10.1112/S0024609306018947.  Google Scholar

[16]

J. Wang, Global heat kernel estimates,, Pacific J. Math., 178(2) (1997), 377.  doi: 10.2140/pjm.1997.178.377.  Google Scholar

[17]

F. W. Warner, Extension of the Rauch comparison theorem to submanifolds,, Trans. Amer. Math. Soc., 122 (1966), 341.   Google Scholar

[18]

Q. S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman,, Int. Math. Res. Notice, (2006).  doi: 10.1155/IMRN/2006/92314.  Google Scholar

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