1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

SINGULARITY MODELS AND K;-SOLUTIONS 81 and n such that if (Mn, g) is a complete Riemannian manifold, p E M, and r E (0, oo) ar ...

82 19. GEOMETRIC PROPERTIES OF K-SOLUTIONS (ii) There is a constant C < oo such that the scalar curvature R9 (x, t) ~ C for a ...

SINGULARITY MODELS AND 11:-SOLUTIONS 83 As a higher-dimensional analogue of the 2-dimensional cigar soliton so- lution we have ...

84 19. GEOMETRIC PROPERTIES OF ii:-SOLUTIONS Now · -w(ro)^1 - 2 < - C^2 r- 0 1 and llaf w' (ro + w (ro) r) drl ::::; C llaf ( ...

THE t;;-NONCOLLAPSED CONDITION 85 HINT: Estimate the higher derivatives of w (r). 2. The fl;-noncollapsed condition In this se ...

86 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS 2.1.2. Examples. Now we consider some examples. EXAMPLE 19.16 (Cylinders have AVR = ...

THE K;-NONCOLLAPSED CONDITION 87 by (19.9) there exists a constant C < oo such that R (x) d (x, 0)^2 :::; C for all x EM. T ...

88 19. GEOMETRIC PROPERTIES OF K:-SOLUTIONS There are two classes of solutions to Ricci flow which have infinite as- ymptotic sc ...

THE 1"-NONCOLLAPSED CONDITION satisfies the trace Harnack estimate 8R · (19.11) 8t + 2 (\/ R, X) + 2 Re (X, X) 2: 0 89 for all ...

90 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS this solution. One may simply apply Hamilton's matrix Harnack estimate on each inter ...

THE 1£-NONCOLLAPSED CONDITION 91 CLAIM 19.32 (Entropy and A:-noncollapsing for ancient solutions). Let (Mn,g(t)), t E (-oo,O], ...

92 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS PROOF. Let H (x,p 0 , t) = (41T't)-nl^2 e-f(x,t) be the heat kernel (we drop the sub ...

PERELMAN'S iv-SOLUTION ON THE n-SPHERE From (19.15) and (19.16) we obtain Wlin (f (t), t):::; ~ + C (n) + ln VolB (po, Jt) -n ...

94 19. GEOMETRIC PROPERTIES OF wSOLUTIONS THEOREM 19.36 (Perelman's K-solution). For any n 2: 3, there exists a rotationally sym ...

PERELMAN'S 11:-SOLUTION ON THE n-SPHERE 95 (3) the initial scalar curvatures are uniformly bounded from above and below by pos ...

96 19. GEOMETRIC PROPERTIES OF 11:-SOLUTIONS Moreover, for all r E [(rr/2) + c-,oo) we have fc:(r) r E (-oo, - (7r/2) - c-] we h ...

PERELMAN'S 11;-SOLUTION ON THE n-SPHERE 97 Now recall by Lemma 2.10 (the 'cylinder-to-sphere rule') in Volume One that a metri ...

98 19. GEOMETRIC PROPERTIES OF A;-SOLUTIONS By Theorem 6.3 in Volume One and Theorem 11.2 in Part II (i.e., Hamilton's convergen ...

PERELMAN'S K-SOLUTION ON THE n-SPHERE 99 On the other hand, since the initial scalar curvature satisfies R(gL (0)) :2: c > ...

100 19. GEOMETRIC PROPERTIES OF A;-SOLUTIONS Proof. We divide the proof of this claim into a few steps. (i) We first show that t ...