New logarithmic sobolev inequalities and an ɛ-regularity theorem for the ricci flow

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 H_x0(y,s) be the conjugate heat kernel centered at some point (x₀,0) 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 (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 W_x0(s)≥-e{open} implies |Rm|≤r^-2 on P_{e{open} r}(x₀,0), where r²≡|s| and P_ρ(x,t)≡B_ρ(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 W_x(s). To accomplish this, we require a new log-Sobolev inequality. Perelman's work implies that the metric measure spaces (Mⁿ,g(t),dvol_g(t)) satisfy a log-Sobolev; we show that this is also true for the heat kernel weighted spaces (Mⁿ,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.