1547671870-The_Ricci_Flow__Chow

5. THE NECKPINCH 59

For n = 2 we get

W(s)^2

--n · R -> n - 1 - 2B = 1 - 2B.

To estimate a, we write

a= WWss - Ws^2 + 1 = "2(W^1 2 )ss - 2Ws^2 + 1,

which implies that
B2s2
a = B - 2 A+ Bs2 + 1,
hence that 1 - B < a < 1 + B. D

Now for A and B chosen as above, we define

, _. _ { W (s) if Isl ::; SA,B

'l/JA' B(s)- mm{W(s)coss}- cos S 1 .fS A,B < I I < S _ 7r 12 ,

where SA,B is the unique positive solution of cos s = v A+ Bs^2. The

function ,(/; A,B is piecewise smooth an satisfies I ts,(/; A,B I ::; 1 for all s E

[-7r/2irr/2] \ {SA,B}· Moreover, it is easy to check that the metric ds^2 +

;J;~,B9can has positive scalar curvature.

We now smooth out the corner that ,(/; A,B has at S A,B. First we construct

a new function ~ A,B which coincides with ,(/; A,B outside a small interval

IE; ~ (SA,B - E, SA,B + E) and has ~;/;A,B constant in IE. This constant

may be chosen so that ,;/; A,B is C^1. Since ,(/; A,B is increasing for 0 ::; s < 5 A,B

and decreasing thereafter, we have ts;/; A,B < 0 in IE. Moreover, we also

have I ts;/; A,B I < 1 in IE. Hence the metric ds^2 + ;/J~,B9can will have R > 0

everywhere.

Now the function ;/;~,B is C^1 ' and its second derivative ~;/;A,B has

simple jump discontinuities at SA,B ± E. So we may smooth it in arbitrarily

small neighborhoods of the two points SA,B ± E in such a way that the

smoothed function 'ljJ A,B is c=, satisfies R > 0 everywhere it is defined, and

coincides with ,(/; A,B outside of hE.

Finally, we observe that is possible to perform the construction above in

such a way that the initial neck is small enough that the solution will en-

counter a neckpinch singularity before its diameter shrinks to zero. Indeed,

it is not hard to check that one may choose A > 0 sufficiently small so that

for all A E ( 0, A), the smoothed function 'ljJ A,B will coincide with ,(/; A,B in

the interval Isl ::; a. Moreover, its derivatives ~'l/JA,B will be bounded for

all Isl~ a uniformly in A E (o,A).

We have proved the following.

LEMMA 2.44. There exists a family of initial metrics

9A,B = ds^2 + 'l/JA,B(s) 9can