14.2 General T3-case

In the general T3-case, the result is an immediate consequence of Proposition 3, Theorem 3 and the fact that solutions are future causally-geodesically complete. Let us quote the exact statement [83, Corollary 1, p. 1190–1191]:

Theorem 8 Consider the set of smooth, periodic initial data 𝒮i,p,c to Equations (16View Equation)–(17View Equation) satisfying Equation (50View Equation). There is a subset 𝒢i,c of 𝒮i,p,c with the following properties

For the sake of completeness, let us also recall the definition of C2-inextendibility.

Definition 9 Let (M, g) be a connected Lorentz manifold, which is at least C2. Assume there is a connected C2 Lorentz manifold ( ˆM ,ˆg) of the same dimension as M and an isometric embedding i : M → Mˆ such that i(M ) ⁄= Mˆ. Then M is said to be C2-extendible. If (M, g) is not C2-extendible, it is said to be C2-inextendible.

It might be possible to obtain this result using different methods. In fact, let 𝒢 be the set of initial data such that the corresponding solutions have an asymptotic velocity, which is different from one on a dense subset of the singularity. Endowing the initial data with the C∞-topology, it is possible to show that 𝒢 is a dense Gδ set [77]. Due to its definition, it is clear that solutions corresponding to initial data in 𝒢 have the property that the curvature blows up on a dense subset of the singularity. It would be natural to expect the corresponding solutions to be inextendible, but providing a proof is nontrivial. Important steps in the direction of proving this statement were taken in [22]. However, to the best of our knowledge, there is, as yet, no result to this effect.

  Go to previous page Go up Go to next page