1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

PERELMAN'S K:-SOLUTION ON THE n-SPHERE 101 where Re - denotes the Ricci tensor of ?fL· We then compute -d d Rmax ,,.._ :S 2 ma ...

102 19. GEOMETRIC PROPERTIES OF i;;-SOLUTIONS (iii) The diameter of 9L (to (L)) does not exceed -8 (n -1) to (L). By integrating ...

PERELMAN'S i;;-SOLUTION ON THE n-SPHERE 103 STEP 6. Taking a sequential limit to get Perelman's K-solution. The family of solu ...

104 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS Note that the King-Rosenau solution is qualitatively different from Perel- man's ,, ...

2-AND 3-DIMENSIONAL w-SOLUTIONS 105 Let 9i (t) ~ R (xi, 0) g ( R (xi, 0)-^1 t) be the rescaled solutions. Since ~~ ;:::: 0 and ...

106 19. GEOMETRIC PROPERTIES OF K-SOLUTIONS Hence, by Proposition 18.12, fixing p E M, there exist sequences {xi}~ 1 and {ri}~ 1 ...

EXISTENCE OF AN ASYMPTOTIC SHRINKER 107 5.1. Two estimates for the reduced distance function. In this subsection we discuss tw ...

108 19. GEOMETRIC PROPERTIES OF 1;;-SOLUTIONS PROOF. We follow the aforementioned references. Let '"Ya be a minimal £-geodesic w ...

EXISTENCE OF AN ASYMPTOTIC SHRINKER 109 From this and (19.48) we conclude that the second pair of terms on the RHS of (19.47) ...

110 19. GEOMETRIC PROPERTIES OF 1;;-SOLUTIONS for q E B 9 ( 7 ) (la (r), r (r)). Then the RHS of (19.53) is Now let r (r) be of ...

EXISTENCE OF AN ASYMPTOTIC SHRINKER in Thus 2(n-1) (rt 7 ) +K(7)r(7)) :::::: J6(n+1) 7i/47-3/4 ( ve (q2, 7) + ~7-i/47i/4). By ...

112 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS be a point such that£ (qn 7) :::; n/2. Then for any E > 0 and A > 1, there ex ...

EXISTENCE OF AN ASYMPTOTIC SHRINKER 113 Given an r > 0, if for any to E [r^2 , T) and any qo EM the solution g (t) is not s ...

114 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS (2) There exists a subsequence, still denoted by (qi, Ti), such that the sequence ( ...

EXISTENCE OF AN ASYMPTOTIC SHRINKER 115 STEP 3. Limit of the reduced volumes. The reduced volume (with basepoint po) of the so ...

ll6 19. GEOMETRIC PROPERTIES OF iv-SOLUTIONS On the other hand, by (19.66) and the volume comparison theorem (again Rm 9 ri ( e) ...

THE fl;-GAP THEOREM FOR 3-DIMENSIONAL fl;-SOLUTIONS 117 Hence it follows from the strong maximum principle that the round cyli ...

118 19. GEOMETRIC PROPERTIES OF 1"-SOLUTIONS such that the sequence { (M^3 , g 7 i ( B), qi)} converges to a nonfl.at 3-dimensio ...

THE ~-GAP THEOREM FOR 3-DIMENSIONAL ~-SOLUTIONS 119 since o E (0, 1).^32 From (19.70) and Vol (Bsz(p) (xoo, JU) x BIR (iJ 00 , ...

120 19. GEOMETRIC PROPERTIES OF 11;-SOLUTIONS Since the RHS tends to zero as 11; 1 tends to zero, this inequality gives a unifor ...