### 10.6 Value of the asymptotic velocity as a criterion for the existence of expansions

Not only is the asymptotic velocity a geometric quantity (from the wave-map perspective), not only can
it be used as an indicator for curvature blow up, it can also be used as a criterion to determine whether
asymptotic expansions exist or not. There are many results of this form, see, e.g., [12, 76, 83]. However, we
shall only describe some of them, beginning with [79, Proposition 1.5, p. 984] (note that this result was
essentially obtained in a previous paper [74]):
It is worth noting that the above proposition proves that if , then is smooth in the
neighborhood of . In other words, knowledge concerning at one point can sometimes yield
conclusions in the neighborhood of that point; see [79, Remark 1.6, p. 985].

Equation (48) essentially has the same content as Equation (38). In order to see this, define the object
inside the norm on the left-hand side of Equation (48) to be . Then

Using the above expansions and equations, expressions for the higher-order time derivatives can be
derived.