New Logarithmic Sobolev Inequalities and an epsilon-Regularity Theorem for the Ricci Flow

作者：Hein Hans Joachim^*; Naber Aaron

来源：Communications on Pure and Applied Mathematics, 2014, 67(9): 1543-1561.

DOI：10.1002/cpa.21474

In this note, we prove an epsilon-regularity theorem for the Ricci flow. Let (M-n, g(t)) with t is an 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 epsilon > 0, depending only on T and on lower scalar curvature and mu-entropy bounds for the initial slice (M-n, g(-T)) such that W-x0(s) >= -epsilon implies vertical bar Rm vertical bar <= r(-2) on P-epsilon r(x(0), 0), where r(2) equivalent to vertical bar s vertical bar and P-rho(x, t) equivalent to B-rho(x, t) x (t - rho(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.

• 出版日期2014-9
• 单位MIT