## Abstract

In this note, we prove an e{open}-regularity theorem for the Ricci flow. Let (M^{n},g(t)) with t{small element of}[-T,0] be a Ricci flow, and let H_{x0}(y,s) be the conjugate heat kernel centered at some point (x_{0},0) in the final time slice. By substituting H_{x0}(-,s) into Perelman's W-functional, we obtain a monotone quantity W_{x0}(s) that we refer to as the pointed entropy. This satisfies W_{x0}(s)≤0, and W_{x0}(s) = 0 if and only if (M^{n},g(t)) is isometric to the trivial flow on R^{n}. Then our main theorem asserts the following: There exists e{open}>0, depending only on T and on lower scalar curvature and μ-entropy bounds for the initial slice (M^{n},g(-T)) such that W_{x0}(s)≥-e{open} implies |Rm|≤r^{-2} on P_{e{open} r}(x_{0},0), where r^{2}≡|s| and P_{ρ}(x,t)≡B_{ρ}(x,t)×(t-ρ^{2},t] is our notation for parabolic balls. The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s-average of W_{x}(s). To accomplish this, we require a new log-Sobolev inequality. Perelman's work implies that the metric measure spaces (M^{n},g(t),dvol_{g(t)}) satisfy a log-Sobolev; we show that this is also true for the heat kernel weighted spaces (M^{n},g(t),H_{x0}(-,t)dvol_{g(t)}). Our log-Sobolev constants for these weighted spaces are in fact universal and sharp. The weighted log-Sobolev has other consequences as well, including certain average Gaussian upper bounds on the conjugate heat kernel.

Original language | English (US) |
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Pages (from-to) | 1543-1561 |

Number of pages | 19 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 67 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2014 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics