1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

NOTES AND COMMENTARY 181 (1) (Bieberbach theorem) A closed flat n-manifold is isometric to the quotient of a flat n-torus; see ...

...

Chapter 22. Tools Used in Proof of Pseudolocality We know you've got to blame someone for your own confusion. From "Lunatic Fri ...

184 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY 1.1. Point picking method for points with a-large curvature. In this subsection (M ...

A POINT PICKING METHOD is5 We call the above properties the 'requirements of (x 0 , to)'. Essentially, (22.5) says that we hav ...

186 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY and for every (x, t) E P(xo, to, (cQ)-^112 , -(cQ)-^1 ) we have I Rm l(x, t)::::; ...

1. A POINT PICKING METHOD 187 we are done.) Then we may define (xm+l, tm+i) E Ma to be a point such that (22.10) (22.11) 1 0 < ...

188 22. TOOLS USED IN PROOF OF PSEUDOLOOALITY We note the following elementary variant of Perelman's Claim 1. EXERCISE 22.5. Let ...

1. A POINT PICKING METHOD 189 for all (x, t) such that (22.19) t-~Q-^1 ::; t::; t and dg(f)(x,x)::; ~Q-~, where Q ~I Rm l(x,[). ...

190 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY Suppose that there exists ti > l - ~aQ-^1 which is the first time, going backwa ...

HEAT KERNELS UNDER CHEEGER-GROMOV LIMITS 191 Integrating the above inequality on the interval [ti,~' we obtain (note that t-ti ...

192 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY (2) The limit (M~, 900 (t), Xoo), the exhaustion {UihEN' the solutions {(Mf,9i(t), ...

HEAT KERNELS UNDER CHEEGER-GROMOV LIMITS 193 be Perelman's differential Harnack quantity. Recall that by (21.32), since 9i ( T ...

194 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY for T E ( 0, (.U]. By the pointed Cheeger---Gromov convergence of 9i ( T) to g 00 ...

HEAT KERNELS UNDER CHEEGER-GROMOV LIMITS 195 EXERCISE 22.10. Use method (i), i.e., mean value inequality, to prove (22.33). Pr ...

196 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY by ffr H](, R' ::::; 0, by the mean curvature of distance spheres comparison theor ...

UPPER BOUND FOR THE LOCAL ENTROPY f B v dμ 197 The following problem is not necessary for our discussion but is of inde- pende ...

198 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY where a :S 1. If (22.46a) (22.46b) a= 200n, b 2: 5EoA, where A;::: 41n, then h is ...

UPPER BOUND FOR THE LOCAL ENTROPY fsvdμ 199 Since t::::; c^2 , this implies that for (x, t) EM x [o, c^2 ] where¢/ (s (x, t))- ...

(^200) 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY where a :::; 1. Given A 2: 67n, suppose that there exists (x, [) E M x (0, E^2 ...