A BEEMIAN SAMPLER:
1966-2002 13
That is how matters stood until1985, when a copy of P. Williams' Ph.D. thesis
[92], "Completeness and its stability on manifolds with connection," was received
unexpectedly in
the mail. First, this result revealed that there was a significant gap
in the previous arguments for the
cr
-stability of geodesic completeness in Lor(M),
and that in fact neither geodesic completeness nor geodesic incompleteness was
cr
-stable, although a stronger topology could be placed on Lor(M) which made
geodesic completeness stable.
From a certain perspective, a good
portion of Beem and his co-authors' research
during the next decade can be viewed as trying to understand the more complicated
geometry of the space of geodesics, once it was realized that Proposition 6.4 on page
175 of
[21]
failed to be valid.
In Williams
[92],
explicit studies were made of the system of null geodesics on
the 2-torus
T
2
which has background flat metric g
=
dx dy. For the first example,
Williams studied the sequence of metrics
(5.2)
(
sinx)
gn
=
dx dy
+ ----;:---
dy
2
and observed that x
=
0
represents an incomplete null geodesic on
(T2
,
gn) for all
n.
Hence, null geodesic completeness fails to be
cr
-stable. For the second example,
Williams considered
(5.3) gn
=
dxdy
+
(1-
cosx
+
1/n)dy2
and observed that while
dx dy
+
(1-
cosx)dy2
contains an incomplete null geodesic, the metrics gn are all null geodesically com-
plete. Hence, null geodesic incompleteness fails to be
cr
-stable.
In a series of papers, of which we will only discuss results from Beem and
Parker
[30], [31],
the concept of pseudoconvex geodesic system was formulated.
We will let Parker give the PDE motivation for this work in his essay in this
conference proceedings, and content ourselves with giving some indications of how
this theory applies to the questions of stability of completeness and incompleteness
for geodesic systems. The theory was first formulated for metric connections and
later broadened to linear connections. Here is how the concept is defined for the
nonspacelike geodesics of a space-time (
M, g).
DEFINITION 5.1. The space-time (M,g) is causally pseudoconvex iff for each
compact subset
K
of
M,
there is a compact subset
K'
of
M
such that if"'(:
[a, b] --"
M is a nonspacelike geodesic with
"'!(a)
E
K and "'f(b)
E
K, then
"'!([a,
b]) is contained
inK'.
As well as being a convexity statement akin to taking the convex hull of a set,
this condition can be understood as a kind of internal completeness condition. It
rules out incompleteness arising, for example, by taking a causal diamond
M
=
J+(p)
n
I-(q)
and deleting a single point.
A second condition Beem and Parker imposed was that neither end of any of
the geodesics in the geodesic system should be totally imprisoned in a compact set.
DEFINITION 5.2. The space-time (M, g) is causally disprisoning if, for each
inextendible nonspacelike geodesic "'( :
(a, b) --" M
and any
t
0
E
(a, b),
both sets
{"Y(t) :
a
t::;
t0
}
and
{"Y(t) : t0
::;
t
b}
fail to have compact closure.
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