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CHEEGER-GROMOV THEORY AND APPLICATIONS TO GENERAL MICHAEL T. ANDERSON
 

Summary: CHEEGER-GROMOV THEORY AND APPLICATIONS TO GENERAL
RELATIVITY
MICHAEL T. ANDERSON
Contents
1. Background: Examples and De nitions. 1
2. Convergence/Compactness. 4
3. Collapse/Formation of Cusps. 9
4. Applications to Static and Stationary Space-Times. 12
5. Lorentzian Analogues and Open Problems. 16
6. Future Asymptotics and Geometrization of 3-Manifolds. 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 Cheeger-Gromov 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

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics