- INTRODUCTION; STATEMENTS OF THE COMPACTNESS THEOREMS 129
DEFINITION 3.2 ( C^00 -convergence uniformly on compact sets). Suppose{UkhEN is an exhaustion of a smooth manifold Mn by open sets and 9k
are Riemannian metrics on Uk. We say that (Uk, 9k) converges in C^00 to
(M, 900) uniformly on compact sets in M if for any compact set K C M
and any p > 0 there exists ko = ko (K,p) such that {9kh>k - 0 converges in
CP to 900 uniformly on K.In order to look at convergence of manifolds which come from dilations
about a singularity, we must ensure that the form of convergence can handle
diameters going to infinity. When this happens, a basepoint, or origin, is
carried along with the manifold and the Riemannian metric to distinguish
what parts of the manifolds in the sequence we are keeping in focus. This
allows us to compare spaces that either have diameters going to infinity or
are noncompact.
DEFINITION 3.3 (Pointed manifolds and solutions). A pointed Rie-mannian manifold is a 3-tuple (Mn, 9, 0), where (M, 9) is a Riemannian
manifold and 0 E M is a choice of point (called the origin, or basepoint).
If the metric 9 is complete, the 3-tuple is called a complete pointed Rie-
mannian manifold. We say that (Mn, 9 (t), 0), t E (a, w), is a pointed
solution to the Ricci fl.ow if (M,9(t)) is a solution to the Ricci flow.
REMARK 3.4. In [187] Hamilton considered marked Riemannian man-
ifolds (and marked solutions to the Ricci flow), where one is also given a
frame F = {ea} ~=l at 0 orthonormal with respect to the metric 9 ( 0) with
0 E (a, w). Since for most applications, the choice of frame is not essential,
we restrict ourselves to considering pointed Riemannian manifolds in this
chapter.
Convergence of pointed Riemannian manifolds is defined in a way which
takes into account the action of basepoint-preserving diffeomorphisms on
the space of metrics.
DEFINITION 3. 5 ( C^00 -convergence of manifolds after diffeomorphisms).A sequence {(Mk, 9k, Ok)}kEN of complete pointed Riemannian manifolds
converges to a complete pointed Riemannian manifold (M~, 900 , Ooo) if
there exist
(1) an exhaustion {UkhEN of Moo by open sets with Ooo E Uk and
(2) a sequence of diffeomorphisms <Pk : Uk ----+ Vk ~ <Pk (Uk) c Mk with<Pk ( Ooo) =Ok
such that (Uk, <Pk [ 9k lvk]) converges in C^00 to (M 00 , 900) uniformly on
compact sets in Moo.
We shall also call the above convergence Cheeger-Gromov conver-
gence in 000 • The corresponding definition for sequences of pointed solu-
tions of the Ricci flow is given by the following.