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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 |
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,, , ().
|
[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,, , ().
|
[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. |
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), 55-85.
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,, , ().
|
[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,, , ().
|
[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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