1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. PROPERTIES OF SINGULARITY MODELS 4 1 oo such that if we set gi(t) ~ K ig(ti+Ki-^1 t), then (Mn,gi(t),(xi ,O)) con- verges in ...

42 28. SPECIAL ANCIENT SOLUTIONS estimate for Xo = 'Y (2kc-^1 l^2 ) , where 0::::: k::::: l zc:1;2 - ~J ~ f'..^1 That is, e Volg ...

PROPERTIES OF SINGULARITY MODELS 43 OPTIMISTIC CONJECTURE 28 .18. The asymptotic cone of a noncompact sin- gularity model at a ...

44 28. SPECIAL ANCIENT SOLUTIONS B 9 , 1 (o) (xi;, j^112 ) x [-j, O] is }-close to the corresponding subset of some ,,;-solution ...

PROPERTIES OF SINGULARITY MODELS 45 THEOREM 28.24 (Existence of asymptotic shrinker). Let (Mn, g ( T)), T E [O, oo), be a r;,- ...

46 28. SPECIAL ANCIENT SOLUTIONS 2.4. Classification of asymptotic cones of 3-dimensional K-solutions. Recall that 3-dimensional ...

PROPERTIES OF SINGULARITY MODELS 47 STEP 2. Existence of r::-necks. Fix a time t, which without loss of generality we may assu ...

48 28. SPECIAL ANCIENT SOLUTIONS of g (0) h ave a positive lower bound on /(i· By this and the assumption that limi--+oo Ei = 0, ...

3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 49 On the other h and, s ince the geodesic segments 'Y1 I (O,t ., 1 ,e J an ...

50 28. SPECIAL ANCIENT SOLUTIONS Consider the following notion of the area of the boundary of a set. DEFINITION 28.33. The geome ...

2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 51 Choosing co E (0, oo) so that1-l^2 (f-^1 [0, c 0 )nB(r)) = 1-l^2 (f-^1 [c ...

52 28. SPECIAL ANCIENT SOLUTIONS is continuous, nondecreasing, and surjective. To see the continuity of Gk we argue as follows. ...

2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 53 The cigar soliton has finite width equal to 2n. F irst, by taking F to be ...

54 28. SPECIAL ANCIENT SOLUTIONS M x ( -oo, 0) -+ JR satisfies 6.f = R. Then, for any smooth bounded domain n c M and any time t ...

2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 55 3.4. Vanishing of the first boundary term in the limit. First, we show th ...

56 28. SPECIA L ANCIENT SOLUTIONS Since R > 0, distances are decreasing under the Ricci fl.ow. Hence, if p E M - Ba(C" +c^1 / ...

3. 2 -DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 57 of a comparison principle for metrics more technical, we instead use co ...

58 28. SPECIAL ANCIENT SOLUTIONS 2 s By (28.41), we have for the cigar soliton that u(t) = e 4 'e+e 2 s. In particular, e•/+i :: ...

2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 59 We conclude that we have the lower bound u(s, B, t) ~ e-~W-(O,t) on [1, o ...

60 28. SPECIAL ANCIENT SOLUTIONS We shall use the following Trudinger- Moser-type inequality of Brezis and Merle [36]. LEMMA 28. ...