The logarithmic Sobolev inequality along the Ricci flow

Abstract

We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any restriction on time. One application of it is a uniform kappa-noncollapsing estimate which holds true for all time. We also obtain similar results for bounded time without assuming the eigenvalue condition. The results extend to the Ricci flow with surgeries.

One appendix is added. A theorem on the Sobolev inequality along the Ricci flow with surgeries of Perelman is added. Two nonlocal Sobolev inequalities are also added. These results were obtained at the time of the posting of the first version of the paper. They were originally planned as parts of two upcoming papers of the author

References in corpus (1)

• A uniform Sobolev inequality under Ricci flow

Cited by in corpus (30)

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• The Calabi flow on Kähler surface with bounded Sobolev constant
• Uniform Sobolev inequalities for manifolds evolving by Ricci flow
• Regularity of Kähler-Ricci flows on Fano manifolds
• Compactness results for the Kähler-Ricci flow
• Entropy Functionals, Sobolev Inequalities and kappa-Noncollapsing Estimates along the Ricci Flow
• The Log Entropy Functional Along the Ricci Flow
• The logarithmic Sobolev inequality along the Ricci flow in dimension 2
• Degeneration of Kähler-Ricci solitons
• Stability of Kähler-Ricci flow
• An upper bound of the heat kernel along the harmonic-Ricci flow
• First variation of the Log Entropy functional along the Ricci flow
• Gaussian upper bounds for the heat kernel on evolving manifolds
• Uniform Sobolev inequality along the Sasaki-Ricci flow
• Isoperimetric inequality along the twisted Kähler-Ricci flow
• Evolving hypersurfaces by their mean curvature in the background manifold evolving by Ricci flow
• Ricci flow with bounded curvature integrals
• The (logarithmic) Sobolev inequalities along geometric flow and applications
• Isoperimetric inequality under Kähler Ricci flow
• Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries
• The Mabuchi metric and the Kähler-Ricci flow
• A compactness result for Fano manifolds and Kähler Ricci flows
• Volume bounds of the Ricci flow on closed manifolds
• Time-dependent Sobolev inequality along the Ricci flow
• Multiplier ideal sheaves and geometric problems
• A remark on odd dimensional normalized Ricci flow
• Stability of the conical Kähler-Ricci flows on Fano manifolds