- CONVERGENCE TO RICCI SOLITONS 481
However' the time-derivative a I at under the Ricci fl.ow and the radial-
derivative a/ ar do not commute; in fact,
[:t'! ] = 2v2!-
We now outline the proof of Theorem B.5; again, details may be found in
[220].
First, long-time existence is proved by obtaining an estimate, depending
only C and Z, for lww"'I· Then, convergence to a soliton is proved by
examining the evolution of the quantityQ ~ R + ((v1 + v2)w/w^1 )
2
.
Comparing with (1.58) shows that Q coincides with R+ l\7 fl^2 when g is the
Bryant soliton. Thus, Q is constant for the soliton. In general, it evolves by
the equation(B.9) (a ) (ww')2 (aQ)
2
at - Ll Q = - (1 + (w')^2 + wv2)^2 ar(wv2 ww' ) aQ-^2 w' + 1 + ( w')^2 + wv2 (vi+ v^2 ) ar '
where
L'.l = (~)2 -2w' ~
ar war
is the Laplacian with respect to g for functions depending on r and t only.
Given (B.5) and (B.6), the paraboloid condition (B.8) is eq~ivalent to
Q having a positive lower bound. In fact, (B.6) implies that
(B.10) (ww')^2 ~ (1 + Z)^2 /Q.
As part of the proof of long-time existence, one shows that conditions (B.5)
and (B.6) persist in time. These, together with (B.10), imply that
(~ at -L'.l) 2': -A Q ( aQ) ar 2 - B I aQ ar I
for some positive constants A, B. Applying the maximum principle to the
corresponding inequality for </> = 1/Q shows that the lower bound on Q
persists in time.
Finally, existence of a limiting metric is proven using the compactness
theorem, and showing that it is a nontrivial soliton comes by appealing to
Theorem A.64 and the positive lower bound for Q.
REMARK B.6. The generalization of Theorem B.5 to rotationally sym-
metric Ricci fl.ow in higher dimensions should be straightforward. It may
even be possible to generalize at least the long-time existence to the non-
rotationally-symmetric case by finding a generalization for the quantity Q
(open problem). It would also be interesting to find conditions under which
the fl.ow converges to the product of the cigar metric with the real line.