- BASIC RICCI FLOW 461
(See pp. 187-189 in Section 4 of Chapter 6 in Volume One.)
The PDE (Vl-6.27) governing the behavior of Rm corresponds to the
ODE(Vl-6.28)In dimension 3, if Mo is diagonal, then M (t) remains diagonal, and its
eigenvalues satisfyd).. 2
dt =).. + μv,(Vl-6.32) dμ dt = μ^2 + >..v,
dv 2
dt = v + >..μ.From now on we shall assume ).. 2: μ 2: v, a condition which is preserved
under the ODE.
Theorem 4.8 on p. 101 of Volume One applied to the Riemann curvature
operator Rm yields the following.
THEOREM A.25 (Maximum principle for Rm: ODE to .. PDE). Let g ( t)be a solution to the Ricci flow on a closed manifold Mn and let K (t) be a
closed subset of E ~ A^2 V ®s A^2 V for all t E [O, T) satisfying the following
properties:
(1) the space-time track UtE[O,T) (K (t) x {t}) is a closed subset of Ex
[O,T);
(2) K (t) is invariant under parallel translation by ~ (t) for all t E
[O, T);
(3) Kx (t) ~ K (t)n7r-l (x) is a closed convex subset of Ex for all x EM
and t E [O, T); and
(4) every solution M of the ODE (Vl-6.28) with Rm (to) E Kx (to) de-
fined in each fiber Ex remains in Kx (t) for all t 2: to and to E [O, T).
If (i* Rm) (0) E JC (0), then (i* Rm) (t) E JC (t) for all t E [O, T).2.4. 3-manifolds with positive Ricci curvature. The following fa-
mous theorem of Hamilton started the Ricci flow (RF). (See Theorem 6.3
on p. 173 of Volume One.)
THEOREM A.26 (RF on closed 3-manifolds with Re> 0). Let (M^3 , go) be
a closed Riemannian 3-manifold of positive Ricci curvature. Then a unique
solution g (t) of the normalized Ricci flow with g (0) = go exists for all
positive time; and as t --t oo, the metrics g(t) converge exponentially fast
in every Ck-norm, k E N, to a metric g= of constant positive sectional
curvature.
The above result generalizes to orbifolds (see [191]).