128 3. THE COMPACTNESS THEOREM FOR RICCI FLOW
this:
I Compactness Theorem I I No local collapsing I ~ I Monotonicity I
I Singularity analysis I
In general, there are three scenarios in which we shall apply the compact-
ness theorem for the Ricci flow. The compactness result may be applied to
study solutions (Mn, g (t)) to the Ricci flow defined on time intervals (a, w),
where w ::S oo is maximal, i.e., the singularity time. To understand the limit-
ing behavior of the solution g (t) as t approaches w, we shall take a sequence
of times tk ---+ w and consider dilations of the solution g (t) about the times
tk and a sequence of points Ok E M by defining
(3.1) gk (t) = Kkg (tk + K-,;^1 t),
where Kk = IRm (Ok, tk)i is the norm of Rm (g (tk)) at the point Ok· We
are interested in determining when there exists a subsequence of pointed
solutions to the Ricci fl.ow (M, gk (t), Ok) which limits to a complete solution
(M~, g 00 (t), 000 ). This limit solution reflects some aspects of what the
singularity looks like near (Ok, tk)· Similarly, when a = -oo for solution
(M, g (t)), which arises when we already have a (first) limit solution of a
finite time singularity, we may consider sequences tk ---+ -oo and take a
second limit, now backward in time. Yet other limits that we shall consider
arise from dimension reduction on a limit solution. Here tk remains fixed
whereas Ok tends to spatial infinity. Many of the topics in this volume are
related to the study of the geometry (and topology) of the limits of these
solutions when they exist.
1.1. Definition of convergence. Now we review the definition of C^00 -
convergence on compact sets in a smooth manifold Mn. By convergence on
a compact set in GP we mean the following.
DEFINITION 3.1 (GP-convergence). Let K c M be a compact set and
let fokhEN, goo, and g be Riemannian metrics on M. For p E {O} UN we
say that gk converges in GP to g 00 uniformly on K if for every s > 0
there exists ko = ko ( E) such that for k 2 ko,
sup sup IVa (gk - goo) lg< E,
o::;a::;pxEKwhere the covariant derivative \7 is with respect to g.
Note that since we are on a compact set, the choice of metric g on Kdoes not affect the convergence. For instance, we may choose g = g 00 •
In regards to C^00 -convergence on manifolds, with the noncompact case
in mind, we have the following. We say that a sequence of open sets {UkhEN
in a manifold Mn is an exhaustion of M by open sets if for any compact
set KC M there exists ko EN such that Uk~ K for all k 2 ko.