1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY 161 result, one requires a curvature bound on a whole parabolic cylinder in ord ...

162 21. PERELMAN'S PSEUDOLOCALITY THEOREM be interesting to consider the following class of metrics: for o > 0. In view of Pe ...

STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY 163 applies to a time interval much smaller than the time elapsed before the si ...

(^164) 21. PERELMAN'S PSEUDOLOCALITY THEOREM Now we can define Topping's incomplete solution of the Ricci flow. Let 7f: JR---+ 5 ...

STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY 165 REMARK 21.8. By taking products of Topping's example (M^2 , g'M_ (t)), t E ...

166 21. PERELMAN'S PSEUDOLOCALITY THEOREM where r can be made as small as we like compared to ro. Now using the control over the ...

SETTING UP THE PROOF BY CONTRADICTION AND POINT PICKING 167 THEOREM 21.9 (ro = 1 case of pseudolocality). For every a > 0 a ...

168 21. PERELMAN'S PSEUDOLOCALITY THEOREM (Mi,gi(t),xoi), t E [O,cr], where Ci E (O,coi], for i EN, of the Riccifiow with bounde ...

SETTING UP THE PROOF BY CONTRADICTION AND POINT PICKING 169 REMARK 21.13. Call Ei the 'size' of the parabolic cylinder, which ...

i7o 21. PERELMAN'S PSEUDOLOCALITY THEOREM Let^8 (xii, lii) E Mia denote the bad points given by Counterstatement A and Lemma 21. ...

LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS 171 3. Local entropies are nontrivial near bad points With the aim of obtaining ...

(^172) 21. PERELMAN'S PSEUDOLOCALITY THEOREM 3.1.2. A uniform negative upper bound for the local entropies. We shall prove the f ...

LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS 173 3.2. Proof that the local entropies are nontrivial near (xi, ti)· PROOF OF ...

174 21. PERELMAN'S PSEUDOLOCALITY THEOREM to obtain a subsequence converging, in the C^00 pointed Cheeger-Gromov sense, to a smo ...

LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS 175 By applying the strong maximum principle (since v 00 :::; 0, we may ap- ply ...

176 21. PERELMAN'S PSEUDOLOCALITY THEOREM 900 (t) is fiat.^12 However, 900 (t) being fiat contradicts (21.43), which says that J ...

LOCAL ENTROPIES ARE NONTRIVIAL NEAR BAD POINTS 177 Note that, a priori, M 00 may be either compact or noncompact. We shall rul ...

178 21. PERELMAN'S PSEUDOLOOALITY THEOREM Fix a time in the interval (t', 0). Since, by (21.64), f 00 is a proper strictly con- ...

NOTES AND COMMENTARY 179 with JM Hi dμgi(o) E (0, 1] and with supp Hi C Bgi(o) (xoi, 20AiEoi) c Bgi(o) (xoi, 1). Here the quan ...

180 21. PERELMAN'S PSEUDOLOCALITY THEOREM LEMMA 21.19 (Curvature gap for Type III solutions). If an eternal so- lution (Mn,g(t)) ...