1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

NOTES AND COMMENTARY 121 the effect on the £-length of a path / : [O, f] -+ M by a variation v ( T) of g ( T). When we compute ...

(^122) 19. GEOMETRIC PROPERTIES OF t;;-SOLUTIONS As a special case we may take Vij = 2Rij, so that V = 2R, which implies 82Rc.C( ...

CHAPTER 20 Chapter 20. Compactness of the Space of /\;-Solutions It never would come to me working on a mystery. From "Runnin' ...

124 20. COMPACTNESS OF THE SPACE OF KrSOLUTIONS In §3, we prove that for any n 2: 3 the collection of n-dimensional K,- solution ...

ASCR AND AVR OF ,,;-SOLUTIONS 125 PROOF. We prove the corollary by contradiction. Suppose that there exists a complete noncomp ...

126 20. COMPACTNESS OF THE SPACE OF tv-SOLUTIONS The assumption that 9(t) is K-noncollapsed at all scales^3 implies that injec- ...

ASCR AND AVR OF 11;-SOLUTIONS 127 Case B. 0 < ASCR (to) < oo for some to ::::; 0. In this case a semi-global blow-down l ...

128 20. COMPACTNESS OF THE SPACE OF Ii-SOLUTIONS Recall that (Ni (b,B) ,gi (0)) C (M,gi (0)) and note that d 9 i(o) (p, xi) = V ...

ALMOST ,,;-SOLUTIONS V' ..!L 88 j = 0 at a point, we have at that point, By' Y V'a (-/\-.^0 0) = ( V'a-0) /\-.+-/\^0 0 ( V'a-. ...

130 20. COMPACTNESS OF THE SPACE OF ,,;-SOLUTIONS 2.1. Volume collapsing on large space-time scales. The following is Corollary ...

ALMOST ~-SOLUTIONS 131 In particular the dilated solutions gk(t) = Qk · 9k(Q"T;^1 t) satisfy Vol fJk(o)BfJk(o) (xk, Ak) (20.8) ...

132 20. COMPACTNESS OF THE SPACE OF /\;-SOLUTIONS for all (x,t) EM x (to,O] satisfying dg(tj(x,xo)::; iro. (b) (N oncollapsed at ...

ALMOST K;-SOLUTIONS 133 Note that^9 (20.16) Bgk(t*k) (Y*k' 1 1 0 Ak 12 R; 1 l 2 (Yk' tk)) c Bgk(t*k) ( xok, ~~~). Since Re 2:: ...

134 20. COMPACTNESS OF THE SPACE OF /;;-SOLUTIONS such that for sufficiently large k 1/2 Volgk(t.k)Bgk(t.k)(Yk,A(c)R~ (Yk't*k) ...

ALMOST /<;-SOLUTIONS i35 STEP 2. We shall prove the following. Claim 2. For To (w) defined by (20.20) below, we have ti '.S ...

136 20. COMPACTNESS OF THE SPACE OF t;;-SOLUTIONS It follows from the contradiction assumption lt1I :S To (w) that we have 2 (n ...

THE COMPACTNESS OF /\:-SOLUTIONS 137 converges in the 000 Cheeger-Gromov topology to a limit solution (M~, 900 (t), X 00 ), t ...

(^138) 20. COMPACTNESS OF THE SPACE OF IV-SOLUTIONS and space-time points (xk, 0) with R 9 k (xk, 0) = 1, there exists a converg ...

THE COMPACTNESS OF A;-SOLUTIONS 139 REMARK 20.13 (Heuristic). A priori, it is conceivable that it is pos- sible for a sequence ...

140 20. COMPACTNESS OF THE SPACE OF 11;-SOLUTIONS Hence, by Theorem 20.1, which says ASCR (gk (0)) = oo if Mk is noncom- pact, w ...