1547671870-The_Ricci_Flow__Chow

THE NECKPINCH 49 where U is the (2,0) tensor given by U =(Ki - Ko) [(n - 1) ds^2 + ~^2 gcan]. To verify (2.56) at any given po ...

50 2. SPECIAL AND LIMIT SOLUTIONS PROOF. By Lemma 2.28, we may let Co be an upper bound for l(-iP^2 )tl· Choose t 1 E (0, T) so ...

THE NECKPINCH 51 We have l?fsl 2: 8 in the region Q2 = [x2, 1) x [t1, T). If we take 77 < 482 < 4, then we have L(u) > ...

52 2. SPECIAL AND LIMIT SOLUTIONS curvature of 9k is uniformly bounded by I Rm I :S 1, with equality attained at Pk at t = 0. On ...

THE NECKPINCH 53 log K1 breaks scale invariance. Our bound on F will thus show that Ko/ K 1 becomes small whenever K1 is large ...

54 2. SPECIAL AND LIMIT SOLUTIONS where (2.58) K P = ( n - 1) log L - 2 L (log L - 1) and (2.59) K2 Q = n - 1 - L 2 (log L - 1) ...

THE NECKPINCH 55 To complete the proof we must deal with possibility that Lmin (0) < e^2. In this case, we consider a resca ...

56 2. SPECIAL AND LIMIT SOLUTIONS Taking C larger if necessary, we may assume by Lemma 5.4 that If; log L :S C as well. This imp ...

THE NECKPINCH 57 Now we integrate once again to get 1 '1/J du v?J(J" >. Tmin. /logrmin - 1 V logu Substitute u = rminV· T ...

58 2. SPECIAL AND LIMIT SOLUTIONS In this way, one obtains part (3) of Proposition 2.36 as an immediate corol- lary of Lemma 2.4 ...

THE NECKPINCH 59 For n = 2 we get W(s)^2 --n · R -> n - 1 - 2B = 1 - 2B. To estimate a, we write a= WWss - Ws^2 + 1 = "2(W^ ...

60 2. SPECIAL AND LIMIT SOLUTIONS such that each metric in the family has positive scalar curvature, satisfies I ls 1JiA,B I :S ...

THE NECKPINCH 61 Since Lemma 2.31 implies that 7./Js 2 0 when 0 ::; s ::; s(t), one may then estimate at any x E [xD(t), x(t)] ...

62 2. SPECIAL AND LIMIT SOLUTIONS estimates at (s, f) that V - t - Vt -< c (-f}S f)t - p I - --n 'l/J - 2 ) V - (n - 1) (1 - ...

THE DEGENERATE NECKPINCH 63 FIGURE 5. A lopsided dumbell shrinking to a round point As o: increases from 0 to 1, we may thus i ...

64 2. SPECIAL AND LIMIT SOLUTIONS FIGURE 6. A cusp forming On the other hand, the singularity is classified as Type Ila if sup I ...

THE DEGENERATE NECKPINCH In Theorem 6.45, we shall prove that T < oo only if lim K (t) = oo. t / T 65 Because the metric g ...

66 2. SPECIAL AND LIMIT SOLUTIONS Notes and commentary Although no examples of slowly forming (Type Ila) singularities on com- p ...

CHAPTER 3 Short time existence A foundational step in the study of any system of evolutionary par- tial differential equations i ...

68 3. SHORT TIME EXISTENCE Hence D LEMMA 3.2. The variation of the Levi-Civita connection r is given by (3.3) atrij a k = Ike ) ...