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 + ---- :--- dy2 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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