1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2 -DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 61 We conclude that since 2 ~ < 2, 1 ( o lw(x)I ) d ( ) 4n 2 exp μIE x & ...

62 28. SPECIAL ANCIENT SOLUTIONS where C 1 = e^40 I 71"^3 sup1R2 R. By (28.63) with E = 2 1 we have that ew - E L^2 (B11"). Now ...

ANCIENT SOLUTIONS WITH POSITIVE CURVATURE 63 LEMMA 28.54 (Pointwise subconvergence to a flat cylinder). The sequence of soluti ...

64 28. SPECIAL ANCIENT SOLUTIONS PROOF. STEP l. Diameter bound. Let g (T) = g (-T). By the Type l as- sumption, we h ave Rcg(r): ...

ANCIENT SOLUTIONS WITH POSITIVE CURVATURE 65 By Lemma 6.35(2) in Part I (note that,\ (g (t)) > 0 since R (g (t)) > 0), w ...

66 28. SPECIAL ANCIENT SOLUTIONS By what amounts to the maximum principle, the function Gmax (t) ~max G (x, t) xEM satisfies the ...

NOTES AND COMMENTARY 67 finite-width case), and one of the authors [83] (noncompact infinite-width case). The last result is i ...

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Chapter 29. Compact 2-Dimensional Ancient Solutions Picture paragraphs unloaded , wise words being quoted. From "Hail Mary" by ...

70 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS In §2 we review the various standard coordinates in which we express the Ricci fl ...

THE RICCI FLOW EQUATION ON 52 AND SOME INTUITION 71 circle of length 27rr and let 51 = 51 (1). We recall the following standar ...

72 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS We shall find it useful to consider solutions (including the King- Rosenau solu- ...

THE KING-ROSENAU SOLUTION IN THE VARIOUS COORDINATES 73 2.2. Intuition based on earlier results. To gain intuition, we note th ...

74 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Let 9KR ~ (vKR)-^1 gcyl ~ (vKR)-^1 9euc· The ansatz (29.8) is equivalent to (29.1 ...

THE K ING-ROSENAU SOLUTION IN THE VARIO US COORDINATES 75 3.2. A quantity that is constant on the King-Rosenau solution. Now w ...

76 29. COMPACT^2 -DIMENSIONAL ANCIENT SOLUTIONS Now we consider some other Cheeger- Gromov limits. Let ti --* -oo and define vfR ...

A PRIORI ESTIMATES FOR THE PRESSURE FUNCTION 77 We start with a second-order quantity in v. Let <p : 52 ---+JR be any C^3 f ...

78 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS LEMMA 29 .8. (29.33) J[!Vvl~2 [[w 2 .p :::; C(p) for p ?_ 1, [[IVvl;2 J[c 1 .a :: ...

THE ALMOST EVERYWHERE VANISHING OF R 00 PROOF. We compute that \7^3 (v 2 2 ) =\7^2 (v\7v) =v\7^3 v+2\7v®\7^2 v+\7^2 v®\7v, so ...

sa 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Therefore mg(-l) (E 1 ;i) = 0 for each i E N. Since Ea mg(-l)(Ea) = 0. We conclud ...