1547671870-The_Ricci_Flow__Chow

44 2. SPECIAL AND LIMIT SOLUTIONS

COROLLARY 2.23. sup Ja (-, t) i :::::; a~ sup la(-, O)J.

We can also show that the curvature is controlled by the radius.

COROLLARY 2.24. If g (t) is a solution of the Ricci flow of the form

(2.43), then there exists C depending only on n and g (0) such that

c

JRmJ ::S '1/J 2.

PROOF. Since vis bounded by Lemma 2.21, so is 'lj;^2 Ki = 1 - v^2. Then

since a is bounded, it follows that 'lj;^2 Ko = 'lj;^2 Ki - a is bounded as well.

Since

IRmJ^2 = 2nKJ + n (n - 1) Kf,

the result follows. 0

Now to simplify the notation, we define

1 -'lj;2

K=. K. - o=-'l/Jss 'l/J and L ~Ki= '1/J 2 s,

recalling that Ko and Ki are the sectional curvatures defined in (2.49) and
(2.50), respectively. Noting that the Laplacian of a radially symmetric func-

tion f is given by

.6f - a

2

f 'l/Js 8f

• 8s 2 + n 'l/J as,

we compute the evolution of the sectional curvatures. One finds that K
evolves by

Kt = ·.6K + 2 (n - 1) KL - 2K^2 - 2 (n - 1) ~~ (K + L).

The evolution equation for L is derived as follows.

LEMMA 2.25. Under the Ricci flow, the quantity L evolves by

Lt= .6L + 2 ~Ls+ 2 [K^2 + (n - l)L^2 ]

= .6L-4~~ (K +L) +2 [K^2 + (n-l)L^2 ].

PROOF. Using equations (2.47) and (2.54), one computes that

Lt= -2'1/J'l/Js ) L

2 ('1/Js t-2-:;j;'l/Jt

= - 2 '1/J~~ss - 2 ( n - 1) ~~ ( K + L) + 2 ( ~~ - L) K + 2 ( n - 1) L^2.

Then observing that

Ls= - 2 ~ (K + L),