11.4 Verification of genericity, openness

If it is possible to prove that ð’Ē is open and dense, it is justified to call it a generic set. A first step in this direction is given by [79Jump To The Next Citation Point, Proposition 1.15, p. 988]:

Proposition 3 ð’Ēl,m is open in the C2 × C1-topology on initial data and ð’Ēl,m,c is open in the C2 × C1-topology on the subset of initial data satisfying Equation (49View Equation).

It is of interest to note that the topology can be weakened somewhat if the only information of interest concerning the asymptotics is that the asymptotic velocity is different from 1. In fact, [79, Proposition 1.16, p. 988] states:

Proposition 4 Given z ∈ ð’Ēl,m, there is an open neighborhood of the initial data for z in the C1 × C0 topology such that for each corresponding solution ˆz, v∞ [ˆz](𝜃) ∈ (0,1) ∪ (1, 2) for all 𝜃 ∈ S1.

Recall that an asymptotic velocity different from 1 implies curvature blow up.


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