1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

REDUCED DISTANCE OF TYPE A SOLUTIONS 141 PROOF. Let (q, l) E JC x [a 00 + c:, w 00 - c:]. By Lemma 7.34 in Part I, the time de ...

142 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS satisfying the following properties: (i) ep,T is locally Lipschitz. (ii) The ...

l. REDUCED DISTANCE OF TYPE A SOLUTIONS 143 independent of i. Using the Banach- Alaoglu theorem, there exists a subsequence such ...

144 30. TYPE I SINGULARIT IES AND ANCIENT SOLUTIONS which holds in the weak se nse (see Lemma 7.129(ii) in P art I); i.e., we h ...

2. REDUCED VOLUME AT THE SINGULAR T IME FOR TYP E I SOLUTIONS 145 2. Reduced volume at the singular time for Type I solutions It ...

146 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS Before we prove Theorem 30.13 at the end of this section, we first discuss a ...

REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS 147 Therefore, applying this and the Type A assumption to (30.50), we ...

148 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS STEP 3. (30.57a) (30.57b) Define The ODI f or u (s) in (30.48). By taking z = ...

REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS 149 (2) By (7 .94) in Part I, I~: I (x, t) ::; ~ JV'RJ2 + ~ JRI + 2 ( ...

150 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS in Part III with K = w"'!_t and radius = K -^1 /^2 , we have (30.66) ( () ) ( ...

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS 151 Integrating (30.69) over [f, w] yields dg(f) h'1(f), 1'2(f)) :S4 ...

152 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS differentiable int, ft~ (t) ;::: 0 and ~ (t) :S: 1. Fix any compact subset K ...

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS 153 We wish to use (30.76) to prove that d~? (t) ;:::: 0. Claim 2. T ...

154 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS n n £ Since u 00 ~ (47r)-2 (w 00 - t)-^2 e-= satisfies D~u 00 = 0 by (30.80), ...

3. TYPE I SOLUTIONS HAVE SHRINKER SINGULARITY MODELS 155 PROPOSITION 30.19 (Limits of Type I solutions with fixed basepoints are ...

156 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS (-47rt)-n/^2 e-e;(i.f.>;(x),t)dμi.f.>;g,(t) (x) are bounded above on M ...

3. TYPE I SOLUTIONS HAVE SHRINKER SINGULARITY MODELS 157 Generalizing the above discussion, we have DEFINITION 30.21 (Set of Typ ...

158 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS The following says that bounded scalar curvature, finite-time singular soluti ...

SOME RESULTS ON TYPE I ANCIENT SOLUTIONS 159 same reasoning used to obtain (30.96) (using Perelman's pseudolocality theorem), ...

160 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS We shall use the following elementary result. LEMMA 30.29 (Characterization o ...