1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

REDUCED VOLUME FOR RICCI FLOW analysis (see Corollary 15 on p. 110 of [313]), we have ip(q,T,h) = ~ (cp1(q,T,h)-cp1(q,T,0)) = ...

398 8. APPLICATIONS OF THE REDUCED DISTANCE for some constant C4 < oo. From (8.34), (8.35), and (8.31), it is easy to see tha ...

the reduced volume 3. A weakened no local collapsing theorem via the monotonicity of A weakened no local collapsing theorem via ...

400 8. APPLICATIONS OF THE REDUCED DISTANCE we have l·l~(T) s eCoT l·l~(To) = e^00 T l·l~(o) acting on vector fields for T E [To ...

NO LOCAL COLLAPSING VIA REDUCED VOLUME MONOTONICITY 401 3.2. The weakened no local collapsing theorem. In Lemma 8.9 we saw how ...

402 8. APPLICATIONS OF THE REDUCED DISTANCE (2) there exist r 1 > 0 and v1 > 0 such that Vol_g(o) B_g(o) (x, r1) ;::: v1 f ...

NO LOCAL COLLAPSING VIA REDUCED VOLUME MONOTONICITY 403 Shi's local derivative estimate implies that there is a constant c 2 = ...

404 8. APPLICATIONS OF THE REDUCED DISTANCE along ')'V, and hence we can use (7.48) to get^4 Iv (7)12 g*(T) < -:--e6(n-l)ec-1 ...

NO LOCAL·COLLAPSING VIA REDUCED VOLUME MONOTONICITY 405 PROOF OF PROPOSITION 8.28. Let c1 be chosen as in Lemma 8.29 and let / ...

406 8. APPLICATIONS OF THE REDUCED DISTANCE Therefore 1 2 n-2 n-2 Noting that rn-^2 e-2r :::;; (n - 2)2 e--2-, we get V2 - ( Er ...

BACKWARD LIMIT OF ANCIENT K-SOLUTION IS A SHRINKER 407 Given Po E M, we have the reduced length f (q, r) ~ f9 (q, r), £-Jacobi ...

408 8. APPLICATIONS OF THE REDUCED DISTANCE PROOF. Given that we know how curvature changes under scaling, it is easy to check t ...

BACKWARD LIMIT OF ANCIENT A;-SOLUTION IS A SHRINKER 409 PROOF. First we shall prove that there exists o > 0 such that for a ...

410 8. APPLICATIONS OF THE REDUCED DISTANCE For any c > 0, Lemma 8.35, after parabolic rescaling g ( T) by Ti, yields the cur ...

BACKWARD LIMIT OF ANCIENT i.;-SOLUTION IS A SHRINKER 411 [A-^1 , A] for some subsequence. Since c: is arbitrary, (i) then foll ...

412 8. APPLICATIONS OF THE REDUCED DISTANCE there exists a neighborhood B 900 (e) ( q, ~ inj 900 (e) (q)) and a smooth function ...

BACKWARD LIMIT OF ANCIENT ~-SOLUTION IS A SHRINKER 413 4.4. The limit of the reduced volume. Note that the limit g 00 (e) is d ...

414 8. APPLICATIONS OF THE REDUCED DISTANCE (ii) Note that V(oo) < 1 follows from Corollary 8.17(iii). To see V(oo) > 0, w ...

BACKWARD LIMIT OF ANCIENT 11;-SOLUTION IS A SHRINKER 415 where 'lj;( ()) 2: 0 is an arbitrarily smooth function with compact s ...

416 8. APPLICATIONS OF THE REDUCED DISTANCE 4.6. The limit is nonflat. We argue by contradiction. If goo(B) is fiat for some Bo, ...