200 32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS
The next chapter i s devoted to the proof that t he inclusion maps of such pieces
induce injections of their fundamental groups.
2. The main result on nonsingular solutions
In this section we present t he statement of Hamilton's classification theorem
for 3-dimensional nonsingular solutions and give a brief outline of Hamilton's proof.
2.1. Definition and examples of nonsingular solutions.
We consider the normalized Ricci flow (NRF) on a closed manifold Mn :(32.1)f) 2
ot9i1 = -2Rij + ;;:;,r 9iJ,where r (t) = flv 1 Rg(t)dμg(t)/ JM dμg(t) denotes the average scalar curvature. This
fl.ow preserves the volume of g (t) and is obtained from the Ricci fl.ow by rescaling
space and time (see §9.1 of Chapter 6 of Volume One).
DEFINITION 32 .l (Nonsingular solution). A solution (Mn,g (t)) to the NRF
on a closed manifold is called nonsingular if it is defined for all time t E [O, oo)
and IRmg(t) I ::; C for some C < oo.
The NRF on a closed manifold is nonsingular when the initial metric is any one
of the following:
(1) Ricci solitons- t hese are the fixed points of the NRF modulo the group
of diffeomorphisms, in particula r , Einstein metrics.
(2) A metric on a 3-manifold with positive Ricci curvature (see H amilton
[135]).
(3) A metric on a surface (see Hamilton [137] and [67]).
(4) A locally homogeneous metric on a 3-manifold (see Jackson and one of
t he authors [154]).
(5) A metric on an n-manifold with 2-positive curvature operator, in dimen-
sion 4 by Hamilton [136] and H. Chen [65], and in dimensions greater
than 4 by Bohm and Wilking [30].^3
(6) A metric with positive complex sectional curvature, proving the 1 /4-
pinched spherical space form conjecture (see Brendle and Schoen [35]).(7) A Kahler metric on a complex n-manifold with c 1 = 0 or c 1 < 0 and
whose Kahler class is proportional to c 1 (see H.-D. Cao [43]).
In each of the above cases, except for case ( 4) and for non-Einstein solitons in
case (1), we h ave convergence to an Einstein metric as t--+ oo.
On the other hand, the Ricci fl.ow on a closed manifold blows up in finite time
when the init ial metric is any of the following:
(8) Any metric on a surface with x > 0 (this follows from (3) above).
(9) A metric on an n-manifold wit h positive scalar curvature (see [135]).
(10) A locally homogeneous metric on a 3-manifold of class (a) SU(2) or (b)
S^2 x JR (see J ackson and one of the authors [154]).(11) A Kahler metric on a complex n-manifold with c 1 > 0.
(^3) See earlier wor k of Huisken [ 152 ], Margerin [217], [218], and Nishikawa [294], [295] for a
metric on an n-manifold with s ufficiently pointwise-pinched positive sectional curvatur es.