- PERELMAN'S 11;-SOLUTION ON THE n-SPHERE 97
Now recall by Lemma 2.10 (the 'cylinder-to-sphere rule') in Volume One
that a metricg = dr^2 + r.p (r)^2 9can
on (0, oo) x sn-l, where r.p : (0, oo) --+ lR+, extends to a smooth metiic on
JR.n if and only if (19.21)-(19.24) hold.We define 'ljJ : [O, 2L'] --+ [ 0, J2 ( n - 2)], where
L' ~ L + J2 (n - 2)f~ (0) (7r /2 + c:),
by reflecting r.p (r) about the liner= L', i.e.,
• { r.p (r) if r E [O, L'],
'ljJ (r) =:= r.p (2L' - r) if r E [L', 2L'].Then 9L (0) given by (19.17) is a smooth rotationally symmetric metricon sn and (Sn, 9L (0)) contains sn-l ( J2 (n - 2)) x [-L, L] isometrically
(we choose c: > 0 sufficiently small independent of L). Moreover, 9L (0) is
invariant under reflection about the spherical slice { r = L'} c sn.
STEP 2. Curvature is uniformly bounded for the initial metrics.
By (19.26), (19.25), and (19.23) we have that1 - ('l/;')2
Ksph = 'l/; 2is nonnegative and bounded (for example, near r = 0 we have 'l/J' (r)
1+0 (r^2 ), so that Ksph (r) = 0 (1)).^19 We also have that
'l/J"
Krad=-~is also bounded and nonnegative (for example, by (19.19) we have f~' (r) :::; 0
for r?:: 0, so that r.p" (r):::; 0 for r?:: 0). In fact,
(19.27) IRml (gL (0)) ::SC (n),where C (n) < oo depends only on n (in particular, C (n) is independent of
L). Note that the Riemann curvature operator has (n-l)in-^2 ) eigenvalues
equal to Ksph and n-1 eigenvalues equal to Krad· Therefore Rm (gL (0)) ?:: 0.
Let (Sn, 9L (t)), t E [O, TL), be the solution of the Ricci flow with initial
metric 9L (0), where TL < oo is the maximal time of existence. By the
strong maximum principle, since 9L (0) has nonnegative curvature operator
everywhere and positive curvature operator on the caps, we have that 9L (t)
has positive curvature operator for t > 0.
STEP 3. Normalizing the family of solutions.(^19) The boundedness of the curvature also follows from Lemma 2.10 in Volume One on
the smoothness of the metric on sn.