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Summary: REGULARITY FOR LORENTZ METRICS UNDER CURVATURE BOUNDS
MICHAEL T. ANDERSON
Abstract. Let (M, g) be an (n+1)-dimensional space time, with bounded curvature, with respect
to a bounded framing. If (M, g) is vacuum, or satis es a mild condition on the stress-energy tensor,
then we show that (M, g) locally admits coordinate systems in which the Lorentz metric g is well-
controlled in the (space-time) Sobolev space L 2;p , for any p < 1.
1. Introduction
A well-known issue in the geometry of space-times is to understand the regularity of metrics
with given bounds on the curvature tensor. This issue arises frequently in discussions and analysis
of the behavior at the boundary and de nitions of singularities for space-times, c.f. [11], [6], [7],
[14] for example.
More speci cally, it has been an open problem for some time, cf. [4]-[6], [14] for instance, whether
a space-time (M, g) which has curvature bounded in L 1 in a suitable sense has coordinate charts
in which the metric g = g is C 1;
\L 2;p ; for any
< 1, p < 1. Here C k;
is the Holder space of
functions whose k th derivatives are Holder continuous of order
, while L k;p is the Sobolev space of
functions with k weak derivatives in L p :
The purpose of this paper is to provide an aÆrmative solution to this problem, at least for
vacuum space-times or space-times satisfying a mild condition on the stress-energy tensor.
The solution of the corresponding problem in Riemannian geometry has been known for some
time, and it is useful to state the exact result in this context before considering the Lorentzian
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