 
Summary: CHEEGERGROMOV THEORY AND APPLICATIONS TO GENERAL
RELATIVITY
MICHAEL T. ANDERSON
Contents
1. Background: Examples and Denitions. 1
2. Convergence/Compactness. 4
3. Collapse/Formation of Cusps. 9
4. Applications to Static and Stationary SpaceTimes. 12
5. Lorentzian Analogues and Open Problems. 16
6. Future Asymptotics and Geometrization of 3Manifolds. 20
References 23
This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a
given manifold M , and some recent applications of this theory to general relativity. The basic
point of view of convergence/degeneration described here originates in the work of Gromov, cf.
[31][33], with important prior work of Cheeger [16], leading to the joint work of [18].
This CheegerGromov theory assumes L 1 bounds on the full curvature tensor. For reasons
discussed below, we focus mainly on the generalizations of this theory to spaces with L 1 ; (or L p )
bounds on the Ricci curvature. Although versions of the results described hold in any dimension,
for the most part we restrict the discussion to 3 and 4 dimensions, where stronger results hold and
the applications to general relativity are most direct. The rst three sections survey the theory in
