- PERELMAN'S 11:-SOLUTION ON THE n-SPHERE 95
(3) the initial scalar curvatures are uniformly bounded from above and
below by positive constants,
( 4) the singularity times are uniformly bounded from above and below
by positive constants,
(5) the diameters tend to infinity.
STEP 1. Construction of the family of initial metrics.
One way of constructing the rotationally symmetric initial metric 9L (0)
on sn is to mollify the warping function of the C^1 metric obtained by gluing
two spherical caps to a round cylinder. In particular, we shall consider a
rotationally symmetric metric of the form(19.17) 9L(O) =dr^2 +1/J(r)^2 9can
on sn, where 9can denotes the standard metric on sn-l (1) and the warping
function 'ljJ is to be defined below. By (20.53) and (20.54) in the notes and
commentary at the end of Chapter 20, we have the standard formulas for
the spherical and radial sectional curvatures:^16
1 - (,,P')2
Ksph = ,,p2 ',,P"
Krad = --;j;' ·To define 'ljJ, we first consider the nondecreasing odd C^1 function f
JR-+ [-1, 1] defined by^17
f(r)"'{
sinr
1
-1ifr E (-7r/2,7r/2),
if r E [7r/2,oo),
if r E (-oo, -Jr /2].Let rJ : JR-+ [O, a] be the standard mollifier defined by
r = { ae^1 /(r
2-^1 ) if lrl < 1,
rJ ( ) · 0 if Ir! 2: 1,where a> 0 is chosen so that JJR rJ (r) dr = 1. Note that rJ is an even function.
Given a small c > 0, define the mollified function f 6 : JR---+ [-1, 1] by
ft: (r) =l;= L f (r + p) ~r/ (~) dp.
Then f 6 is C^00 , nondecreasing, and odd. In particular, f 6 (0) = 0.
(^16) The spherical sectional curvatures are the sectional curvatures of 2-planes contained
in the tangent spaces of the spherical slices { r = canst} whereas the radial sectional cur-
vatures are the sectional curvatures of 2-planes containing the radial vector 8/8r.
(^17) An alternate construction of the warping function is as follows. It should not be
difficult to prove that there exists a 000 function j: [O, oo)--+ [O, 1] such that
f(r) == { sinr ifr E [0,7r/4],
· 1 ifrE[7r/2,oo),0::; j'::; 01 , and -02::; f"::; O; we may take 01 = 1/../2 since f"::; 0. We may use this
function instead of fe in the definition of the warping function.