470 A. BASIC RICCI FLOW THEORY3.6. Necklike points in Type I solutions. (See Section 4 in Chap-
ter 9 of Volume One.) We say that (x, t) is a Type I c-essential point
if(Vl-p. 262a)c
IRrn(x,t)l 2: T-t > 0.We say that (x, t) is a 8-necklike point if there exists a unit 2-forrn e at
(x, t) such that(Vl-p. 262b) IRrn-R (e ® B)I :::; 8 IRrnl.The following result can be used to show that necks must form. in Type I
solutions where the underlying manifold is not diffeornorphic to a spherical
space form.. (See Theorem. 9.9 on p. 262 of Volume One.)
THEOREM A.51 (Necklike points in 3d Type I singular solutions). Let(M^3 , g (t)) be a closed solution of the Ricci flow on a maximal time interval
0 :::; t < T < oo. If the normalized flow does not converge to a metric of
constant positive sectional curvature, then there exists a constant c > 0 such
that for all T E [O, T) and 8 > 0, there are x E M and t E [T, T) such that
(x, t) is a Type I c-essential point and a 8-necklike point.The analogous result for ancient solutions is the following. (See Theo-
rem. 9.19 on p. 272 of Volume One.)THEOREM A.52 (Necklike points in 3d Type I ancient solutions). Let(M^3 ,g (t)) be a complete ancient solution of the Ricci flow with positive
sectional curvature. Suppose thatsup ltl^7 R (x, t) < oo
Mx(-oo,O]for some "( > 0. Then either ( M, g ( t)) is isometric to a spherical space
form or else there exists a constant c > 0 such that for all T E (-oo, OJ and
8 > 0, there are x EM and t E (-oo, T) such that (x, t) is an ancient Type
I c-essential point and a 8-necklike point.
4. More Ricci fl.ow theory and ancient solutions
In this section we summarize some additional basic aspects of Ricci flow.
We warn the reader that 'basic' here (and in the previous sections) means
neither 'should be obvious' nor 'should be known by every graduate student'
nor 'should be easy to prove'.
4.1. Strong maximum principle. For solutions with nonnegative
curvature operator, the strong maxim.urn principle implies a certain type
of rigidity in the case where the curvature operator is not strictly positive
(see [179] or Theorem. 6.60 in [111]).