1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

THAT Q = 0 IMPLIES THE SOLUTION IS THE KING-ROSENAU SOLUTION 121 Second, by substituting (29.182) into the LHS of (29.170), we ...

122 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Observe that by taking the difference of (29.186d) and (29.186e), we obtain the ...

THAT Q = 0 IMPLIES THE SOLUTION IS THE KING-ROSENAU SOLUTION 123 By (29.186a) and (29.188), we find that A (B - E) is constant ...

124 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Since 1 = c;, we have that ~g\ is independent oft. On the other hand, by pulling ...

15. The quantity Q must be identically zero and (29.199) 8 at l~eucvil 2 = V~euc l~eucvil 2 2v l~eucval 2 4~eucV l~eucvil 2 ...

126 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS PROOF. Since Q is a subsolution of the heat equation by (29.140), the maxi- mum ...

THE QUANTITY Q MUST BE IDENTICALLY ZERO 127 by (29.7), we have a2w as2 > O; i.e., s H W(s- , t) is convex. On the other han ...

128 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS for each s. We claim, for i sufficiently large, that (29.210) aw as (s , ti)::; ...

THE EQUIVALENCE OF Q AND Q 129 The annulus ([-1, 1] x 51 , gi(O)) is isometric to (Bo(es,+^1 ) - B 0 (e^8 '-^1 ), g(ti )). So ...

130 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS The proof of this lemma shall occupy the rest of this section. The totally symme ...

THE EQUIVALENCE OF Q AND Q 131 Using (29.18), we compute that the components of the second covariant deriv- ative of v are \i' ...

132 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Therefore, in terms of iJ we may rewrite a and (3 as (using also that cos 'If; = ...

CHAPTER 30 Type I Singularities and Ancient Solutions It' s a tad bit late. From "Regulate" by Warren G featuring Nate Dogg Th ...

134 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS there exist M < oo and r E [1, ~) such that (30.l) on M x (a, w). M JRmJ ( ...

l. REDUCED DISTANCE OF TYPE A SOLUTIONS 135 1.2. Motivation and set-up for the reduced distance based at the singular time. Let ...

136 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS leads to the well-definedness and monotonicity of the reduced volume at the s ...

l. REDUCED DISTANCE OF TYPE A SOLUTIONS 137 on Mix [a 00 ,wi)· Given any (p,t) E M 00 x [to,w 00 ) and any nonzero tangent vecto ...

138 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS we obtain (30.17) 1 w=-~ d2 (q,p=) L i (q, t) ~ 4K (t) VWi - t goo(to) _ 2 ...

REDUCED DISTANCE OF TYPE A SOLUTIONS 139 whose graph joins (;(q), l) to (pi,wi).^2 By Lemma 7.32 in Part I , we have that the ...

140 3 0. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS where ( D = 2 exp 2nM (Woo-t)-- (r- 1) ). vw -a d2 _ 9oo _ (t o~) (q,poo_ ) ...