Conjecture 1 For generic asymptotically-flat and for generic spatially-compact initial data for Einstein’s vacuum equations, the MGHD is inextendible.

Remark. We are only interested here in initial data specified on compact manifolds, a case to which the conjecture applies. Readers interested in the asymptotically-flat case are referred to, e.g., [90] and references cited therein.

In this form, the statement is due to ChruĊciel; see [17, Section 1.3], based on ideas due to Eardley and Moncrief [32]. It is of course also possible to make the same (or similar) statements in the presence of matter. However, there are some matter models, which exhibit pathologies, and we do not wish to discuss such issues here. The formulation of Conjecture 1 is rather vague; the words “generic” and “inextendible” occur without having been clearly defined. The reason for this is partly that there is no a priori preferred definition of these concepts. Let us discuss them separately.

In the context of a finite-dimensional dynamic system, a generic subset could, e.g., be defined in one of the following ways:

- a set whose complement has measure zero (with respect to the Lebesgue measure, say),
- a set that is open and dense,
- a dense set, i.e., a countable intersection of open sets, which is dense.

Other possibilities are conceivable; see, e.g., [14]. Regardless of the choice of definition, one requirement appears to be quite clear: if a set is generic, then the complement should not be generic. In the case of infinite-dimensional dynamic systems, the case we are interested in here, the measure of the theoretic notion of genericity is not so natural. Consequently, we shall here, unless otherwise stated, take generic to mean open and dense. Nevertheless, such a definition is still not precise; it requires the prior definition of a topology on the set of initial data.

Turning to the meaning of inextendibility, there are several possibilities. First, the inextendibility should refer to a particular differentiability class; a solution could be extendible in one degree of differentiability but inextendible in another (that such a situation can occur is illustrated by [23, 24]). Furthermore, one could say that a solution is extendible if

- it is extendible as a Lorentz manifold, or
- it is extendible as a Lorentz manifold solving Einstein’s equations.

Here, we shall say that a solution is extendible if it is extendible as a Lorentz manifold. In other words,
we shall not require the extension to be a solution to Einstein’s equations. Furthermore, we shall use the
differentiability class C^{2}, since we wish the differentiability class to be strong enough that curvature is still
defined.

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