Some results for the Perelman LYH-type inequality

Shu-Yu Hsu

doi:10.3934/dcds.2014.34.3535

Article Contents

2014, Volume 34, Issue 9: 3535-3554. Doi: 10.3934/dcds.2014.34.3535

This issue Previous Article Slowly oscillating wavefronts of the KPP-Fisher delayed equation Next Article An averaging theorem for nonlinear Schrödinger equations with small nonlinearities

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

Received: November 30, 2012

Revised: October 31, 2013

Published: August 2014

Abstract Related Papers Cited by

Abstract

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,
Ricci flow,
conjugate heat equation,
Dirichlet fundamental solutions,
Perelman's Li-Yau-Hamilton type inequality.

Mathematics Subject Classification: Primary: 58J35, 58C99; Secondary: 35K05.

Citation:

Related Papers

Cited by

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.