1547671870-The_Ricci_Flow__Chow

5. THE NECKPINCH 43

PROOF. Applying the maximum principle to (2.54), one concludes that
at any maximum of v which exceeds 1, one has

Vt :S ~ n -^1 ( 1 - V V 2 ) < 0.

Similarly, at any minimum of v with v < -1, one has Vt > 0. The result
follows once we observe that (2.48) implies sup Iv(·, O)I ~ 1. 0

Denoting the second derivative of 'ljJ by w ~ 'l/Jm one calculates that

v w^2 v^2 n - 1

(2.55a) Wt= Wss + (n - 2) -;j;Ws - 2--:;;; - (4n - 5) 'l/J 2 w + ~w

v^2 (l- v^2 )

(2.55b) - 2(n-1) 'ljJ 3

Together, the evolution equations for v and w help us obtain a good

estimate for a.

LEMMA 2.22. Under the Ricci flow, the quantity a evolves by

at= ass+ (n - 4) ~as - 4 (n - 1) ~~a.

PROOF. Noting that

a = 'ljJw - v^2 +1,

one computes that

and

ass= 'l/JsWs + 'l/JWss - VsW - VWs = 'l/JWss - W^2.

Then recalling equations (2.47), (2.54), and (2.55), one derives the equation

at = W'l/Jt + 'l/JWt - 2VVt

= w { w - (n - 1)

1

~ v

2

}

• 'ljJ { Wss + (n - 2)7 - 2~

2

• (4n - 5)~ }

+(n - 1)¥ - 2(n - l)v

2

(~3v

2

)

-2v{ws+(n- 2) v; + (n-l)v(l; 2 v

2

)}

v^2 w v^2 (1 -v^2 )

= 'l/Jwss - w^2 + (n - 4) VWs - (5n - 8) T -4(n - 1) 'l/J 2

v v^2

=ass+ (n- 4)-;j;as - 4(n - 1) 'ljJ 2 a.

Applying the maximum principle proves that a is uniformly bounded.

0