1547671870-The_Ricci_Flow__Chow

46 2. SPECIAL AND LIMIT SOLUTIONS

-1 x (t)

*

FIGURE 4. A typical profile

PROOF. Noting that

"'' _ a + 'l/J; - 1

'f/SS - 'l/J '

one has

'l/J'l/Jt = 'l/J'l/Jss - (n - 1) (1 - 'l/J;) =a - n (l - 'l/J;).

D

COROLLARY 2.29. If g (t) exists for 0 :St < T, then limvr 'l/;(x, t) exists

for each x E [-1, l ].

5.3. The profile of the solution. We call local minima of x ~ 'l/J (x, t)

'necks' and local maxima 'bumps'. We are interested in solutions whose ini-

tial data has at least one neck. The second part of our analysis is to establish

the sense in which the profile of the initial data persists, in particular to show

that the solution will become singular at its smallest neck and nowhere else.

Let g (t) be a solution of the Ricci flow having the form (2.43). We

denote the radius of the smallest neck by

r min ( t) ~ min { 'l/J ( X, t) : 'l/Jx ( X, t) = 0}

for each t such that the solution exists; if the solution has no necks, then

rmin will be undefined. We let x* (t) denote the right-most bump, and call

the region right of x* (t) the 'polar cap'.

The main results of this part of the analysis are as follows.

PROPOSITION 2.30. Let g (t) be a solution to the Ricci flow of the form

(2.43) such that l'l/Js I :S 1 and R > 0 initially. Assume that the solution

keeps at least one neck.

(1) At any time, the derivative 'l/Js has finitely many zeroes. The number

of zeroes is nonincreasing in time, and if 'l/J ever has a degenerate

critical point (to wit, a point such that 'l/Js = 'l/Jss = 0 simultane-

ously) the number of zeroes of 'l/Js drops.