Chapter 4. Proof of the Compactness Theorem
We think in generalities, but we live in details.- Alfred North Whitehead
There is no royal road to geometry. -Euclid1. Outline of the proof
We now prove the compactness Theorem 3.9. This is a fundamental
result in Riemannian geometry and does not require the Ricci flow. The
compactness theorem is in the spirit of Cheeger [70] and Gromov [169] (see
also Greene and Wu [165], Peters [300], and the book [37]). We follow the
proof for pointed sequences converging in C^00 given by Hamilton [187]; as
Hamilton notes there, things are easier because we can assume bounds on
all covariant derivatives of the curvature.
Theorem 3.9 will be proved in several steps. It is outlined as follows.
STEP A: Construct a sequence of coverings of each manifold M'k which
we can compare to each other. The covers should consist of balls B'f: c Mk
with a number of properties, most notably:
- they are diffeomorphic to Euclidean balls, and for each fixed a they
have the same radii for all sufficiently large k, - they are numbered sequentially in a starting from balls centered at
the origin to balls with centers further and further away from the
origin,
• if we take smaller radii (B'k C B'f:), they are disjoint, and if we take
larger radii (B'f: C B'f: C B'f:), they contain their neighbors,- we can bound the number of balls intersecting a given ball (the
most is I (n, Co), where Co is the curvature bound), and - we can bound the number of these balls that it takes to cover a
large ball in Mk, independent of k for k large (it takes fewer than
A (r) balls to cover B (Ok, r) if k ~ K (r)).
The specifics of this are contained in Lemma 4.18. This process is carried
out in Section 3 of this chapter.
STEP B: Use our nice covering to construct maps F(j : B'f: --+ M.e:
We do this by taking the inverse of the exponential map on the ball B'k C
Mk, identifying the tangent space of Mk at the center of the ball B'f: with
Euclidean space and with the tangent space of M.e at the center of Bf and
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