Title: A New Entropy Formula and Gradient Estimates for the Linear Heat Equation on Static Manifold
Author(s): Abimbola Abolarinwa
Pages: 1-17
Cite as:
Abimbola Abolarinwa, A New Entropy Formula and Gradient Estimates for the Linear Heat Equation on Static Manifold, Int. J. Anal. Appl., 6 (1) (2014), 1-17.


In this paper we prove a new monotonicity formula for the heat equation via a generalized family of entropy functionals. This family of entropy formulas generalizes both Perelman’s entropy for evolving metric and Ni’s entropy on static manifold. We show that this entropy satisfies a pointwise differential inequality for heat kernel. The consequences of which are various gradient and Harnack estimates for all positive solutions to the heat equation on compact manifold.

Full Text: PDF



  1. A. Abolarinwa, Differential Harnack and logarithmic Sobolev inequalities along RicciHarmonic map flow, To appear Google Scholar

  2. A. Abolarinwa, Analysis of eigenvalues and conjugate heat kernel under the Ricci flow, PhD Thesis, University of Sussex, (2014). Google Scholar

  3. D. Bakry, D. Concordet and M. Ledoux, Optimal heat kernel bounds under logarithmic Sobolev inequalities, ESAIM Probab. Statist., 1,(1995), 391-407. Google Scholar

  4. D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality, Revist. Mat. Iberoamericana 22, (2006), 683-702. Google Scholar

  5. J. Cheeger and S-T. Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34(4)(1981), 465-480. Google Scholar

  6. B. Chow, S. Chu, D. Glickenstein, C. Guenther, J. Idenberd. T. Ivey, D. Knopf, P. Lu, F. Luo and L. Ni, The Ricci Flow: Techniques and Applications. Part II, Analytic Aspect, AMS, Providence, RI, (2008). Google Scholar

  7. E. B. Davies, Heat Kernel and Spectral theory. Cambridge University Press (1989). Google Scholar

  8. N. Garofalo and E. Lanconelli, Asymptotic behaviour of fundamental solutions and potential theory of parabolic operators with variable coefficients, Math. Ann. 283(2)(1989), 211-239. Google Scholar

  9. L. Gross Logarithmic Sobolev inequalities, America J. Math 97(1)(1975), 1061-1083. Google Scholar

  10. L. Gross Logarithmic Sobolev inequalities and contractivity properties of semigroups, Dirichlet Form, Lecture Notes in Mathematics Volume 1563, (1993), 54-88. Google Scholar

  11. R. Hamilton, A matrix Harnack estimate for the heat equation, Commun. Anal. Geom., 1, (1993), 113-126. Google Scholar

  12. G. Huang, Z. Huang, H. Li, Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds, Ann. Glob. Anal. Geom., 23(3) (1993), 209-232. Google Scholar

  13. S. Kuang, Qi S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255(4)( 2008), 1008-1023. Google Scholar

  14. J. Li, X. Xu, Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Advances in Math., 226 (2011), 4456-4491. Google Scholar

  15. P. Li, S-T. Yau, On the parabolic kernel of the Schr¨odinger operator, Acta Math. 156 (1986), 153-201 Google Scholar

  16. R. M¨uller, Differential Harnack Inequalities and the Ricci Flow. European Mathematics Society, (2006). Google Scholar

  17. L. Ni, The Entropy Formula for Linear Heat Equation, Journal of Geom. Analysis 14(2)(2004), 86-96 Google Scholar

  18. L. Ni, Addenda to ”The Entropy Formula for Linear Heat Equation”, Journal of Geom. Analysis 14(2)(2004), 229-334. Google Scholar

  19. L. Ni, A note on Perelman’s Li-Yau-Hamilton inequality, Comm. Anal. Geom 14(2006), 883-905. Google Scholar

  20. G. Perelman, The entropy formula for the Ricci flow and its geometric application, arXiv:math.DG/0211159v1 (2002). Google Scholar

  21. P. Souplet, Qi S. Zhang, Sharp gradient estimate and Yaus Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc. 38(2006), 1045-1053. Google Scholar

  22. F. B. Weissler, Logarithmic Sobolev Inequalities for the Heat-Diffusion Semigroup, Trans Am Math. Soc., 237(1978), 255-269. Google Scholar


Copyright © 2021 IJAA, unless otherwise stated.