1547671870-The_Ricci_Flow__Chow

PROVING THE GLOBAL ESTIMATES 229 In order to put (7.3) into the form of a heat equation, we need ~ \lk Rm. Since for any tenso ...

230 7. DERIVATIVE ESTIMATES where f3m is a constant to be chosen momentarily. We will explain the full strategy behind this choi ...

THE COMPACTNESS THEOREM 231 for 0 :S: t :S: a/ K , where Cm~ J /3m (m - 1)! +a (Cm+ f3C:n) depends only on m, n, and max {a, 1 ...

232 7. DERIVATIVE ESTIMATES of the Ricci flow existing for t E (a , w) with the properties that sup [sect [goo] I :::::; K M~x(a ...

CHAPTER 8 Singularities and the limits of their dilations In this chapter, we shall survey the standard classification of maxima ...

234 8. DILATIONS OF SINGULARITIES for all x E Mn. Then since !Rm (t)I = CR, where C > 0 is a constant depending only on g ( 0 ...

SINGULARITY MODELS 235 An Einstein solution with R > 0 is Type I; an Einstein solution with R < 0 is Type III; and a non ...

236 8. DILATIONS OF SINGULARITIES the curvature bound sup IRmoo(·, t)l · ltl < oo. M~x(-oo,O] One says ( M~, 900 ( t)) is an ...

PARABOLIC DILATIONS 237 REMARK 8.6. The distinction between ancient and eternal Type II mod- els should be carefully noted. It ...

238 8. DILATIONS OF SINGULARITIES In order to develop intuition for the dilation criteria introduced below, recall that if the s ...

PARABOLIC DILATIONS 239 Once we have chosen a sequence (xi, ti) of points and times such that ti / T E (0, oo], we consider a ...

240 8. DILATIONS OF SINGULARITIES DEFINITION 8.16. A solution (Mn, g(t)) of the Ricci fl.ow on a time interval [O, T) is K;-nonc ...

DILATIONS OF FINITE-TIME SINGULARITIES 241 PROOF. Recall from Lemma 7.4 that :t 1Rml 2 ::; ~ 1Rml 2 + C 1Rml 3 for some C depe ...

242 8. DILATIONS OF SINGULARITIES which is the curvature bound we need for a subsequence to converge to a Type I singularity mod ...

DILATIONS OF FINITE-TIME SINGULARITIES 243 it follows from (8.12) that (M~, g 00 (t)) exhibits a Type I singularity at w E [c, ...

244 8. DILATIONS OF SINGULARITIES hence if t E [-IRm(xi,ti)I (ti - tE:) ,wi)· Note that Wi _____, w and that i---->oo .lim IR ...

DILATIONS OF FINITE-TIME SINGULARITIES 245 or equivalently, IR mi. ( x, t )I < ( ~ (~ - ) ti) I IRm ( (xi, ) ti)I I. c Ti ...

246 8. DILATIONS OF SINGULARITIES Thus we may replace IRml by R in the discussion above, whence it follows that Roa (x, t) :S 1 ...

DILATIONS OF INFINITE-TIME SINGULARITIES 247 Now ai --+ oo, but we do not know that Wi also tends to infinity. This motivates ...

248 8. DILATIONS OF SINGULARITIES by noting a consequence of the formula (8.24) d d Lt( 'Yt) = - 2 1 1 ~g (S, S) ds -1 (\7 sS, V ...