New logarithmic sobolev inequalities and an e{open}-regularity theorem for the ricci flow

Hans Joachim Hein, Aaron Naber

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

In this note, we prove an e{open}-regularity theorem for the Ricci flow. Let (Mn,g(t)) with t{small element of}[-T,0] be a Ricci flow, and let Hx0(y,s) be the conjugate heat kernel centered at some point (x0,0) in the final time slice. By substituting Hx0(-,s) into Perelman's W-functional, we obtain a monotone quantity Wx0(s) that we refer to as the pointed entropy. This satisfies Wx0(s)≤0, and Wx0(s) = 0 if and only if (Mn,g(t)) is isometric to the trivial flow on Rn. 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 (Mn,g(-T)) such that Wx0(s)≥-e{open} implies |Rm|≤r-2 on Pe{open} r(x0,0), where r2≡|s| and Pρ(x,t)≡Bρ(x,t)×(t-ρ2,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 Wx(s). To accomplish this, we require a new log-Sobolev inequality. Perelman's work implies that the metric measure spaces (Mn,g(t),dvolg(t)) satisfy a log-Sobolev; we show that this is also true for the heat kernel weighted spaces (Mn,g(t),Hx0(-,t)dvolg(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 languageEnglish (US)
Pages (from-to)1543-1561
Number of pages19
JournalCommunications on Pure and Applied Mathematics
Volume67
Issue number9
DOIs
StatePublished - Sep 2014

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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