- CONVERGENCE TO RICCI SOLITONS 479
the derivative of the conformal factor for g give subsequence convergence,
on any compact set, to a metric g.
If R(go) > 0 and C 00 < oo, then using the Bernstein-Bando-Shi es-
timates, one can show that after a short time T, JJE. 2 JMJ^2 dμ is bounded
uniformly in time (where dμ indicates measure with respect to the evolvingmetric). The evolution equation for J IMl^2 dμ then implies that
1
00
(l 2 2IY'Ml^2 + 3RJMJ^2 dμ) dt < oo.Thus, either M vanishes or R vanishes for the limiting metric g. If M = 0,
then g is a gradient soliton; by Proposition 1.25, it is either fl.at or the cigar
metric. However, C 00 < oo precludes a fl.at limit.
If R(go) > 0 and A > 0, then one may use the Harnack inequality
to show that tRmax is bounded. It follows that the limit g is fl.at in this
ra~. D
One may also study the Ricci fl.ow for a solution of the form (B.1) in
terms of the nonlinear diffusion equation satisfied by the conformal factor
v ~ eu,
(B.2)ov -ot = .6. log v,
where .6. is the standard Laplacian on JR^2. (Note that v = l/(k + lxJ^2 ) for
the cigar, where we write x = (x, y).)
REMARK B.3. As pointed out by Angenent in an appendix to [373], the
equation (B.2) is a limiting case of the porous medium equation
(B.3) ov = .6.vm
at
as the positive exponent m tends to zero. For, substituting t = Tjm in (B.3)
gives
~~ = .6. ( vm ~ 1)
and taking m --t 0 gives ov/or = .6.(logv).
In connection with Ricci fl.ow on JR^2 we also have the following result,
which says that metrics that start near the cigar converge to a cigar (in
the same conformal class), but under weaker assumptions than in the above
theorem.
THEOREM B.4 (Hsu [206]). Suppose that vo E Lf 0 c(lR^2 ) for p > 1, 0 :S:
vo :S: 2/(,6lxl^2 ) for ,6 > 0, and vo - 2/(,6(lxl^2 + ko)) E L^1 (lR^2 ) for ko > 0.
Then there exists a unique positive solution of (B.2), defined for 0 < t < oo,
such that lirnt->O v = vo in L^1 on any compact set and such that
lim e^2 f3tv(e^2 f3t x, t) = -(lxl^2 2 + ki)-^1
t-too ,6