  In this article we prove completeness results for Sobolev metrics with
nonconstant coefficients on the space of immersed curves and on the space of
unparametrized curves. We provide necessary as well as sufficient conditions
for the coefficients of the Riemannian metric for the metric to be metrically
complete and we construct examples of incomplete metrics. This work is an
extension of previous work on completeness of Sobolev metrics with constant
coefficients.
