1547671870-The_Ricci_Flow__Chow

42 2. SPECIAL AND LIMIT SOLUTIONS

and its scalar curvature is

(2.52) R = 2nKo + n (n - 1) K1.

5 .2. Bounds on curvature and other derivatives. The first part
of our analysis of a neckpinch singularity is to obtain bounds for the first
and second derivatives of '!/J. To do so, it will be useful to consider the scale-
invariant measure of the difference between the two sectional curvatures
defined by

(2.53) a ~ '!/J^2 (K1 - Ko).

We collect the key results of this part of the analysis in our first proposition.

PROPOSITION 2.20. Let g (t) be a solution of the Ricci flow having the

form (2.43) and satisfying the bounds l'!/Jsl :S 1 and R > 0 initially.

(1) For as long as the solution exists, IVis I :S 1.

(2) For as long as the solution exists, one has the bound

a a

--'!/J2 <Ki- - Ko< - -'!/J2'

where a ~ sup la(-, 0) I·

( 3) There exists C = C ( n, g ( 0)) such that for as long as the solution

exists,

c

IRml :S '!/J 2.

( 4) The quantity ( K 1) min is nondecreasing.

(5) For as long as the solution exists, one has R > 0 and '!/Jt < 0.

(6) 'ljJ^2 is a uniformly Lipschitz-continuous function of time; in fact,

one has

l('!/J2)tl :S2(a+n).

(7) If g (t) exists for 0 :St < T, then the limit

'ljJ ( x, T) ~ lim 'ljJ ( x, t)

t/T

exists for each x E [-1, l ].

We establish the claims above in the remainder of this subsection. De-

noting the first derivative by v ~ '!/J 8 , one first computes that

(2.54) Vt= V 8 s + n-2 ~VV 8 + ~ n - l (1 -V^2 ) V.

This equation has a happy consequence.

LEMMA 2.21. Assume g (t) is a solution to the Ricci flow having the

form (2.43) and satisfying (2.48) for 0 :St< T. Then

1 :S sup Iv(-, t) I :S sup Iv(·, O)I.