The entropy formula for linear heat equation

Journal of Geometric Analysis

Volumn 14, Issue 1, 2004, Pages 87-100

  Ni, Lei ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Entropy formula; heat equation; logarithmic Sobolev

Indexed keywords

EID: 84867951130     PISSN: 10506926     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02921867     Document Type: Article

References (26)

1. D. Bakry D. Concordet M. Ledoux 1995 Optimal heat kernel bounds under logarithmic Sobolev inequalities ESAIM Probab. Statist. 1 391 407 10.1051/ps:1997115 1486642
2. W. Beckner 1999 Geometric asymptotics and the logarithmic Sobolev inequality Forum Math. 11 105 137 10.1515/form.11.1.105 1673903 0917.58049
3. W. Beckner M. Pearson 1998 On sharp Sobolev and the logarithmic Sobolev inequality Bull. London Math. Soc. 30 80 84 10.1112/S0024609397003901 1479040
4. Cao, H.D. and Ni, L. Matrix Li-Yau-Hamilton estimates for the heat equation on Kaehler manifolds, submitted, arXiv: math.DG/0211283.
5. Cao, H.D.; Chen, B-L.; Zhu, X-P. Ricci flow on compact Kahler manifolds of positive bisectional curvature, preprint.
6. S.-Y. Cheng P. Li S.-T. Yau 1981 On the upper estimate of heat kernel of a complete Riemannian manifold Am. J. Math. 103 1021 1063 10.2307/2374257 630777 0484.53035
7. B. Chow 1998 Interpolating between Li-Yau's and Hamilton's Harnack inequalities on a surface J. Partial Differential Equations 11 2 137 140 1626999 0943.58017
8. Chow, B.; Chu, S-C.; Lu, P.; Ni, L. Notes on Perelman's papers on Ricci flow.
9. E. Davies 1987 Explicit constants for Gaussian upper bounds on heat kernels Am. J. Math. 109 319 334 10.2307/2374577 0659.35009
10. K. Ecker 2002 Logarithmic Sobolev inequalities on submanifolds of Euclidean spaces J. Reine Angew. Mat. 552 105 118 1758578
11. L. Gross 1975 Logarithmic Sobolev inequalities Am. J. Math. 97 4 1061 1083 10.2307/2373688
12. Hamilton, R. The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262,Contemp. Math.;71, Am. Math. Soc.; Providence, RI, (1988).
13. R. Hamilton 1993 A matrix Harnack estimate for the heat equation Comm. Anal. Geom. 1 113 126 1230276 0799.53048
14. Kleiner, B. and Lott, J. Notes on Perelman's paper.
15. M. Ledoux 1999 On manifolds with non-negative Ricci curvature and Sobolev inequalities Comm. Anal. Geom. 7 347 353 1685586 0953.53025
16. P. Li 1980 On the Sobolev constant andp-spectrum of a compact Riemannian manifold Ann. Scient. Éc. Norm. Sup. 13 451 469 0466.53023
17. P. Li 1986 Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature Ann. of Math. 124 1 1 21 10.2307/1971385 847950
18. P. Li S.-T. Yau 1986 On the parabolic kernel of the Schrödinger operator Acta Math. 156 139 168 10.1007/BF02399203 834612
19. L. Ni 2002 Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact Kahler manifolds Indiana Univ. Math. J. 51 3 679 704 10.1512/iumj.2002.51.2175 1911050 1035.53032
20. Perelman, G. The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/ 0211159.
21. Perelman, G. Informal talks and discussions.
22. O.S. Rothaus 1985 Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities J. Funct. Anal. 64 296 313 10.1016/0022- 1236(85)90079-5 812396 0578.46028
23. A.J. Stam 1959 Some inequalities satisfied by the quantities of informations of Fisher and Shannon Inform, and Control 2 101 112 10.1016/S0019-9958(59)90348-1 109101 0085.34701
24. Sesum, N.; Tian, G.; Wang, X. Notes on Perelman's paper on entropy formula for Ricci flow and its geometric applications.
25. R.S. Varadhan 1967 On the behavior of the fundamental solution of the heat equation with variable coefficients Comm. Pure Applied Math. 20 431 455 208191 0155.16503 10.1002/cpa.3160200404
26. F.B. Weissler 1978 Logarithmic Sobolev inequalities for the heat-diffusion semi-group Trans. AMS. 237 255 269 10.2307/1997621 479373 0376.47019