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New logarithmic Sobolev inequalities and an ε-regularity theorem for the Ricci flow

In this note we prove a new ε-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into Perelman's W-functional produces a monotone function W_{x}(s) of s \in [-T,0], the pointed entropy, with W_{x}(s) <= 0, and W_{x}(s) = 0 iff (M,g(t)) is isometric to the trivial flow on R^n. Our main theorem asserts the following: There exists an ε>0, depending only on T and on lower scalar curvature and μ-entropy bounds for (M,g(-T)), such that W_{x_0}(s) > -εimplies |Rm|< r^{-2} on P_{εr}(x,0), where r^2 = |s| and P_r(x,t) \equiv B_r(x,t)\times (t-r^2,t] is the parabolic ball. 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. It is well known by Perelman that the metric measure spaces (M,g(t),dv_{g(t)}) satisfy a log-Sobolev; however we prove that this is also true for the conjugate heat kernel weighted spaces (M,g(t),H_{x}(-,t)\,dv_{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 an average Gaussian upper bound on the conjugate heat kernel that only depends on a two-sided scalar curvature bound.