Completeness of Length-Weighted Sobolev Metrics on the Space of Curves

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.