  We study completeness properties of reparametrization invariant Sobolev
metrics of order $n\ge 2$ on the space of manifold valued open and closed
immersed curves. In particular, for several important cases of metrics, we show
that Sobolev immersions are metrically and geodesically complete (thus the
geodesic equation is globally well-posed). These results were previously known
only for closed curves with values in Euclidean space. For the class of
constant coefficient Sobolev metrics on open curves, we show that they are
metrically incomplete, and that this incompleteness only arises from curves
that vanish completely (unlike "local" failures that occur in lower order
metrics).
