Chapter 3. The Compactness Theorem for Ricci Flow
Although this may seem a paradox, all exact science is dominated by the idea
of approximation. -Bertrand RussellThe compactness of solutions to geometric and analytic equations, when
it is true, is fundamental in the study of geometric analysis. In this chapter
we state and prove Hamilton's compactness theorem for solutions of the Ricci
flow assuming Cheeger and Gromov's compactness theorem for Riemannian
manifolds with bounded geometry (proved in Chapter 4). In Section 3 of
this chapter we also give various versions of the compactness theorem for
solutions of the Ricci flow.
Throughout this chapter, quantities depending on the metric 9k (or
9k (t)) will have a subscript k; for instance, '\lk and Rmk denote the Rie-
mannian connection and Riemannian curvature tensor of 9k. Quantities
without a subscript depend on the background metric g. Often we suppress
the t dependence in our notation where it is understood that the metrics
depend on time while being defined on a space-time set. Given a sequence
of quantities indexed by {k}, when we talk about a subsequence, most of
the time we shall still use the indices { k} although we should use the indices
{jk}.
1. Introduction and statements of the compactness theorems
Given a sequence of solutions (Mk, 9k (t)) to the Ricci flow, Hamilton's
Cheeger-Gromov-type compactness theorem states that in the presence of
injectivity radii and curvature bounds we can take a 000 limit of a subse-
quence. The role of the compactness theorem in Ricci flow is primarily to
understand singularity formation. This is most effective when the compact-
ness theorem is combined with monotonicity formulas and other geometric
and analytic techniques, in part because these formulas and techniques en-
able us to gain more information about the limit and sometimes enable us
to classify singularity models. This has been particularly successful in low
dimensions. In latter parts of this volume we shall see some examples of
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