New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow

In this note, we prove an ɛ‐regularity theorem for the Ricci flow. Let ( M n , g ( t )) with t ∊ [− 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 ɛ > 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 ) ≥ −ɛ implies |Rm| ≤ r −2 on P ɛ 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 ), d vol 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 ) d vol 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.