The Logarithmic Sobolev and Sobolev Inequalities Along the Ricci Flow

Abstract

Based on Perelman’s entropy monotonicity, uniform logarithmic Sobolev inequalities along the Ricci flow are derived. Then uniform Sobolev inequalities along the Ricci flow are derived via harmonic analysis of the integral transform of the relevant heat operator. These inequalities are fundamental analytic properties of the Ricci flow. They are also extended to the volume-normalized Ricci flow and the Kähler–Ricci flow.

Keywords

Sobolev inequality / Logarithmic Sobolev inequality / Ricci flow / Heat operator

References

[1] Carron, G.: Inégalités isopérimétriques de Faber-Krahn et conséquences. Actes de la Tables Ronde de Géométrie Différentielle (Luminy, 1992), pp. 205–232, Sémin. Congr., 1, Soc. Math. France, paris, (1996)

[2] Davies EB. Heat Kernel and Spectral Theory. 1989 Cambridge: Cambridge University Press

[3] Gallot S. Isoperimetric inequalities based on integral norms of Ricci curvature. Astérisque. 1988, 157–158 191-216

[4] Gallot S. Inégalités isopérimétriques et analytiques sur les variétés riemanniennes. Astérisque. 1988, 163–164 31-91

[5] Hamilton RS, Isenberg J. Quasi-convergence of Ricci flow for a class of metrics. Commun. Anal. Geom.. 1993, 1 543-559

[6] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159v1 (2002). Accessed Nov 2002

[7] Perelman, G.: Ricc flow with surgery on three-manifolds. arXiv:math/0303109v1 (2003). Accessed March 2003

[8] Sadosky C. Interpolation of Operators and Singular Integrals: An Introduction to Harmonic Analysis. 1979 New York and Basel: Marcel Dekker Inc.

[9] Seeley RT. Complex powers of an elliptic operator. Proc. Symp. Pure Math.. 1967, 10 288-307

[10] Sesum, N., Tian, G.: Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman) and some applications, preprint

[11] Ye, R.: Curvature estimates for the Ricci flow I. In: Calculus of Variations and Partial Differential Equations, vol. 31, pp. 417–437. arXiv:math/0509142 (2008)

[12] Ye, R.: Some results on the Sobolev inequality. Lecture Notes

[13] Ye, R.: Notes on the logarithmic Sobolev inequality and its application to the Ricci flow, Lecture Notes

[14] Ye, R.: The logarithmic Sobolev inequality along the Ricci flow. arXiv:0707.2424

[15] Ye, R.: The logarithmic Sobolev inequality along the Ricci flow in dimension 2. arXiv:0708.2003

[16] Ye, R.: The logarithmic Sobolev inequality along the Ricci flow: the case $\lambda _0(g_0)=0$. arXiv:0708.2005 (2005)

[17] Ye, R.: Sobolev inequalities, Riesz transforms and the Ricci flow. arXiv:0709.0512

[18] Ye, R.: Sobolev inequalities and $\kappa $-noncollapsing estimates for the Ricci flow with surgeries, preprint

[19] Zhang, Q. S.: A uniform Sobolev inequality under Ricci flow. arXiv:0706.1594v1

[20] Zhang QS. A uniform Sobolev inequality under Ricci flow. Int. Math. Res. Not.. 2007, 17 1-17

[21] Zhang, Q.S.: Erratum to: A uniform Sobolev inequality under Ricci flow. Int. Math. Res. Not. 19 (2007)