42 1. RICCI SOLITONS
0
with equality only if Re = 0. Thus, if Rmin < 0, then R is constant at this
0time. By applying the same argument at every point, we get Re = 0 and
the metric is Einstein.
Otherwise, assume Rmin = O; by (1.67), we know that b.R ::S 0 at p.
Let n be the open set where b.R < 0 at this time. If n is nonempty,
then R can only attain its infimum on n (which is zero) at a point on
an. Applying the Hopf maximum principle shows that the outward normal
derivative of R must be negative there; but this is impossible because \7 R =
0 there. Thus, n is empty and b.R 2:: 0 everywhere. Since R achieves its
maximum somewhere, by the maximum principle R is constant (in fact,
identically zero), and the metric is Einstein by applying the same argument
~~~. D
REMARK 1.67. Note that a breather with positive scalar curvature on
a closed manifold is a shrinking breather. In Chapter 6 we will describe
Perelman's result that shrinking breathers on closed manifolds are Ricci
solitons. The proof of this uses his entropy formula. In Chapter 5 another
proof that steady or expanding breathers are Einstein will be given.It was shown in Proposition 5.10 of Volume One that the only solitons for
the normalized flow on compact surfaces were constant curvature metrics.
We will now prove the generalization of this to compact 3-manifolds, by first
obtaining a curvature pinching estimate for the sectional curvatures. As a
corollary to Theorem A.31, we have the following (see Hamilton [186] and
one of the authors [218]).
COROLLARY 1.68 (Hamilton-Ivey estimate). Assume the normalization
infxEM3 v (x, 0) 2:: -1 on the initial metric, where v (x, t) denotes the small-
est eigenvalue of the curvature operator. There exists a continuous positive
nondecreasing function 'ljJ : JR -t JR with 'ljJ ( u) / u decreasing for u > 0 and
'ljJ (u) ju -t 0 as u -too, such that for any solution (M^3 ,g (t)) of the Ricci
flow on a closed 3-manifold, we have v 2:: -'lj; (R). That is,
Rm 2:: -'lj; (R) id,where id: A^2 -t A^2 is the identity.
PROOF. By Theorem A.31, wherever v < 0, we have R 2:: lvl(log lvl - 3).
In particular, if v ::S -e^6 , then R 2:: ~ lvl log lvl. The function f (u) = u
logu is increasing for u 2:: 1/e and hence has an inverse f-^1 : [-1/e, oo) -t
[1/e, oo). If v ::S -e^6 , then R 2:: 3e^6 and lvl ::S f-^1 (2R). If we let
'ljJ (u) = { f-^1 (2u) ~f u 2:: 3e^6 ,
e^6 if u < 3e^6 ,
then v 2:: -'lj; (R) at all points in space and time. It is easy to see that
the function 'ljJ has all of the properties claimed in the statement of the
corollary. D