Proposition 3 is open in the C2 × C1-topology on initial data and is open in the C2 × C1-topology on the subset of initial data satisfying Equation (49).
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:
Recall that an asymptotic velocity different from 1 implies curvature blow up.
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