New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow
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In this note, we prove an ɛ‐regularity theorem for the Ricci flow. Let (Mn,g(t)) with t ∊ [−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 ɛ > 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) ≥ −ɛ implies |Rm| ≤ r−2 on Pɛ r(x0,0), where r2 ≡ |s| and Pρ(x,t) ≡ Bρ(x,t) × (t−ρ2,t] is our notation for parabolic balls.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/4fd333757dda9bf5614aaefcb67092f3018dbc04