1547671870-The_Ricci_Flow__Chow

DILATIONS OF INFINITE-TIME SINGULARITIES 249 If (8.25) holds for some c > 0, there exists a time Tc: < oo such that for ...

250 8. DILATIONS OF SINGULARITIES fort > -a. Hence the pointed limit solution (M~, 900 (t), x 00 ), if it exists, is defined ...

NOTES AND COMMENTARY 251 defined as before. Each solution (g 00 )i exists on the dilated time interval (- oo, (w - ti) IRmoo (xi ...

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CHAPTER 9 Type I singularities Suppose that (M^3 , g (t)) is a solution of the Ricci fl.ow which becomes singular at some time T ...

254 9. TYPE I SINGULARITIES possible pattern, (0, 0, 0, ), may be ruled out by choosing a dilation sequence so that (8.9) holds, ...

2. POSITIVE CURVATURE IS PRESERVED 255 ( c) a cigar product (IR^3 , g ( t)), where g ( t) is the self-similar solu- tion corresp ...

256 by TYPE I SINGULARITIES p : JP r--+ min JP (V, V) , IVl=l then v (JP) = p (JP) and p(sJP+ (1-s)Q) 2 sp(JP) + (1-s)p(Q). Th ...

POSITIVE SECTIONAL CURVATURE DOMINATES 257 study of (6.32), we first consider the subcase that μ = v < 0. We then have the ...

258 9. TYPE I SINGULARITIES Namely, large negative sectional curvatures can occur only in the presence of much larger positive s ...

POSITIVE SECTIONAL CURVATURE DOMINATES 259 u FIGURE l. Boundary of the region Kin the (u, v)-plane. Now consider the map <P ...

260 9. TYPE I SINGULARITIES PROOF. By (6.32), we calculate v^2 dt d D (M) = -v dt d (tr M) + (tr M) dt d v - v dv dt d d = -v dt ...

POSITIVE SECTIONAL CURVATURE DOMINATES 261 COROLLARY 9.7. In dimension n = 3, every Type I limit of a Type I sin- gularity or ...

262 9. TYPE I SINGULARITIES in the sequel to this volume.) In terms of the solution g (t), this shows that R (x, t) 2: Iv (x, t) ...

NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 263 If for every c > 0, there exist some T < T and fJ > 0 such that ther ...

264 9. TYPE I SINGULARITIES If v < - A, then lvl :::; R/3 by Theorem 9.4; so 1 :::; N:::; 2, and we have IRml :::; lvl + lμI ...

NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 265 Collecting terms as in Lemma 6.34, we obtain !!..._ F = !:::..F^2 ( l - E) (' ...

266 9. TYPE I SINGULARITIES and IR~l2 = ~ [ (>, μ)2 + (,\ v)2 + (μ _ v)2] :::; ~ (>-2 + μ2 + v2) :::; 4 ,\2. So if Iv\ : ...

NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 267 On the other hand, if condition (9.7) holds at (x, t), Lemma 9.12 implies the ...

268 9. TYPE I SINGULARITIES and hence 8F < b..F+ 2(1- c:) (\!F \!R)- Cea F. ot - R+ p ' T-t So the maximum principle implies ...