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ORBIFOLD COMPACTNESS FOR SPACES OF RIEMANNIAN METRICS AND APPLICATIONS
 

Summary: ORBIFOLD COMPACTNESS FOR SPACES OF RIEMANNIAN METRICS
AND APPLICATIONS
MICHAEL T. ANDERSON
1. Introduction
The Cheeger-Gromov compactness theorem, cf. [C], [G], [CGv] states that the space of Rie-
mannian n-manifolds (M n ; g) satisfying the bounds
jRj  ; vol  v 0 ; diam  D; (1.1)
is precompact in the C 1; topology. Here R denotes the Riemann curvature tensor, vol the volume
and diam the diameter of (M; g). Thus, for any sequence of metrics g i on n-manifolds M i satisfying
(1.1), there is a subsequence, also called g i ; and di eomorphisms  i : M1 ! M i such that the
metrics  
i g i converge in the C 1; topology to a limit metric g 1 on M1 ; for any < 1. In particular,
there are only nitely many di eomorphism types of n-manifolds M which admit Riemannian
metrics satisfying (1.1). In addition, the convergence is in the weak L 2;p topology, and the limit
metric g 1 is L 2;p ; for any p < 1:
While conceptually important, this result is of somewhat limited applicability in itself, since the
bound on the full curvature tensor R is very strong and only realizable in special situations. A
direct generalization of this result to the more natural situation where one imposes bounds on the
Ricci curvature, i.e.
jRicj  ; vol  v 0 ; diam  D; (1.2)

  

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

 

Collections: Mathematics