xx CONTENTS OF PART I OF VOLUME TWO
issue in applying the Riemannian compactness theorem is showing that the
limiting metric/solution is Kahler; fortunately this is easily handled.
Chapter 4. In this chapter we first give an outline of the proof of the
Cheeger-Gromov compactness theorem for pointed Riemannian manifolds.
The proof of the compactness theorem for pointed manifolds is rather tech-
nical and involves a few steps. The main step is to define, after passing to a
subsequence, approximate isometries wk from balls B (Ok, k) in Mk to balls
B ( Ok+l, k + 1) in Mk+i · The manifold M 00 is defined as the direct limit of
the directed system wk. Convergence to M 00 and the completeness of this
limit follows from wk being approximate isometries.
The ideas in the proof of the main step are as follows. In each of the
manifolds Mk in an appropriate subsequence, starting with the origins Ok =
x£, one constructs a net (sequence) of points { xk} :l~) which will be the
centers of balls Bk of the appropriate radii (technically one considers balls of
different radii for what follows). By passing to a subsequence and appealing
to the Arzela-Ascoli theorem repeatedly, we may assume these Riemannian
balls have a limit as k --+ oo for each a. Furthermore, we may also assume
that the balls cover larger and larger balls centered at Ok and that the
intersections of balls Bk and Bf are independent of kin the limit. Choosing
frames at the centers of these balls yields local coordinate charts Hf: (this
depends on a decay estimate for the injectivity radius and our choice of the
radii of the balls) and we can define overlap maps J~(J = ( Hf)-l o Hf:.
By passing to a subsequence, we may assume the J~(J converge as k--+ oo
for each a and (3. The local coordinate charts also define maps between
manifolds by FfJ, = Hf o (Hf)-^1. Now we can define approximate isometries
Fke : B (Ok, k) --+ Me by taking a partition of unity and averaging the maps
FfJ,. Technically, this is accomplished using the so-called center of mass and
nonlinear averages technique. This brings us to the remaining step, which
is to show that these maps are indeed approximate isometries.
Chapter 5. In Volume One we saw the integral monotonicity formula
for Hamilton's entropy for solutions to the Ricci flow on surfaces with pos-
itive curvature. There we also saw various curvature pinching estimates,
the gradient of the scalar curvature estimate, and higher derivative of cur-
vature estimates. Other monotonicity-type formulas, for the evolution of
the lengths and areas of stable minimal geodesics and surfaces, yielded in-
jectivity radius estimates in various special cases in low dimensions. In a
generalized sense, all of these estimates may be thought of as monotonicity
formulas.
In this chapter we address Perelman's energy formula. One of the main
ideas here is the introduction of an auxiliary function, which serves several
purposes. It fixes the volume form, it satisfies a backward heat-type equa-
tion, it is used to understand the action of the diffeomorphism group, and it
relates to gradient Ricci solitons. We discuss the first variation formula for
the energy functional and the modified Ricci flow as the gradient flow for