Proposition 3 is open in the C^{2} × C^{1}-topology on initial data and is open in the
C^{2} × C^{1}-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:

Proposition 4 Given , there is an open neighborhood of the initial data for in the
C^{1} × C^{0} topology such that for each corresponding solution , for all
.

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

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