1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

THE COMPACTNESS OF /;;-SOLUTIONS 141 The limit also satisfies R9= (y 00 , 0) = 1 and R9= (x 00 , 0) = 0. However this contradi ...

142 20. COMPACTNESS OF THE SPACE OF 11:-SOLUTIONS Claim. There exists a constant C1 < oo independent of k such that (20.31) R ...

THE COMPACTNESS OF 11;-SOLUTIONS r43 ask--+ oo and for w E B-gk(o) (wk, sk),r^8 R-gk (w,O) ~ 2R-gk (wk,O). Then applying the c ...

144 20. COMPACTNESS OF THE SPACE OF 11:-SOLUTIONS where a < 1 is a small positive constant to be chosen later. On the other h ...

THE COMPACTNESS OF /';;-SOLUTIONS 145 for (y,t) E B 9 k(- 2 r~) (zk,rk) x [-r~,O]. Using IRc-gkl:::; R9k, we obtain that fort ...

146 20. COMPACTNESS OF THE SPACE OF /\;-SOLUTIONS for any y E Bgk(O) (xk, r). This estimate and the 11;-noncollapsing condition ...

THE COMPACTNESS OF />;-SOLUTIONS 147 By Lemma 18.15 there exists Wk E B 9 k(o) (xk, r*) and Sk E (0, r* + 1) such that (a) ...

(^148) 20. COMPACTNESS .OF THE SPACE OF 11:-SOLUTIONS . First we prove a claim by contradiction. (This claim corresponds to (20. ...

DERIVATIVE ESTIMATES AND SOME CONJECTURES 149 with (20.47) The limit solution ( M~, 900 ( t)) is complete with nonnegative cur ...

150 20. COMPACTNESS OF THE SPACE OF 1;;-SOLUTIONS PROOF. If the theorem is not true, then for some K > 0 and n 2:: 2 there ex ...

DERIVATIVE ESTIMATES AND SOME CONJECTURES 151 for x E M and t :::; 0. More generally, for any R and m there exists 'Tl ( R, m) ...

152 20. COMPACTNESS OF THE SPACE OF 11;-SOLUTIONS PROBLEM 20.22. Can one show that for any complete noncompact 11,- solution wit ...

DERIVATIVE ESTIMATES AND SOME CONJECTURES 153 1 R(O'(u))::::; R-1 (0'(0)) + (1-28)u· From this we may obtain COROLLARY 20.24. ...

154 20. COMPACTNESS OF THE SPACE OF K-SOLUTIONS Very recently this conjecture has been solved. Daskalopoulos, Hamilton, and Sesu ...

NOTES AND COMMENTARY 155 §3. Originally, Theorem 20.9 was proved for noncompact ,.,;-solutions in dimension 3 (see §11.7 of [1 ...

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Chapter 21. Perelman's Pseudolocality Theorem You speak to me in riddles and you speak to me in rhymes. From "Possession" by Sa ...

158 21. PERELMAN'S PSEUDOLOCALITY THEOREM In contrast, Hamilton's matrix Harnack estimate for the Ricci flow is apparently diffi ...

STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY 159 n-dimensional volume of n. We have the following sharp Euclidean isoperi- m ...

160 21. PERELMAN'S PSEUDOLOCALITY THEOREM for any x E Bg(t) (x 1 , corJro) and t E (0, (crJro)^2 ]. In particular, applying this ...