Abstract
We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.
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Bakry, D., Concordet, D., and Ledoux, M. Optimal heat kernel bounds under logarithmic Sobolev inequalities,ESAIM Probab. Statist.,1, 391–407, (1995/97).
Beckner, W. Geometric asymptotics and the logarithmic Sobolev inequality,Forum Math.,11, 105–137, (1999).
Beckner, W. and Pearson, M. On sharp Sobolev and the logarithmic Sobolev inequality,Bull. London Math. Soc.,30, 80–84, (1998).
Cao, H.D. and Ni, L. Matrix Li-Yau-Hamilton estimates for the heat equation on Kaehler manifolds, submitted, arXiv: math.DG/0211283.
Cao, H.D., Chen, B-L., and Zhu, X-P. Ricci flow on compact Kahler manifolds of positive bisectional curvature, preprint.
Cheng, S.-Y., Li, P., and Yau, S.-T. On the upper estimate of heat kernel of a complete Riemannian manifold,Am. J. Math.,103, 1021–1063, (1981).
Chow, B. Interpolating between Li-Yau’s and Hamilton’s Harnack inequalities on a surface,J. Partial Differential Equations,11(2), 137–140, (1998).
Chow, B., Chu, S-C., Lu, P., and Ni, L. Notes on Perelman’s papers on Ricci flow.
Davies, E. Explicit constants for Gaussian upper bounds on heat kernels,Am. J. Math.,109, 319–334, (1987).
Ecker, K. Logarithmic Sobolev inequalities on submanifolds of Euclidean spaces,J. Reine Angew. Mat.,552, 105–118, (2002).
Gross, L. Logarithmic Sobolev inequalities,Am. J. Math.,97(4), 1061–1083, (1975).
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).
Hamilton, R. A matrix Harnack estimate for the heat equation,Comm. Anal. Geom.,1, 113–126, (1993).
Kleiner, B. and Lott, J. Notes on Perelman’s paper.
Ledoux, M. On manifolds with non-negative Ricci curvature and Sobolev inequalities,Comm. Anal. Geom.,7, 347–353, (1999).
Li, P. On the Sobolev constant andp-spectrum of a compact Riemannian manifold,Ann. Scient. Éc. Norm. Sup.,13, 451–469, (1980).
Li, P. Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature,Ann. of Math.,124(1), 1–21, (1986).
Li, P. and Yau, S.-T. On the parabolic kernel of the Schrödinger operator,Acta Math.,156, 139–168, (1986).
Ni, L. Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact Kahler manifolds,Indiana Univ. Math. J.,51(3), 679–704, (2002).
Perelman, G. The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/ 0211159.
Perelman, G. Informal talks and discussions.
Rothaus, O.S. Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities,J. Funct. Anal.,64, 296–313, (1985).
Stam, A.J. Some inequalities satisfied by the quantities of informations of Fisher and Shannon,Inform, and Control,2, 101–112, (1959).
Sesum, N., Tian, G., and Wang, X. Notes on Perelman’s paper on entropy formula for Ricci flow and its geometric applications.
Varadhan, R.S. On the behavior of the fundamental solution of the heat equation with variable coefficients,Comm. Pure Applied Math.,20, 431–455, (1967).
Weissler, F.B. Logarithmic Sobolev inequalities for the heat-diffusion semi-group,Trans. AMS.,237, 255–269, (1978).
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Ni, L. The entropy formula for linear heat equation. J Geom Anal 14, 87–100 (2004). https://doi.org/10.1007/BF02921867
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DOI: https://doi.org/10.1007/BF02921867