A chord-arc curve is a locally rectifiable curve satisfying the property that the length of the shortest arc on the curve joining any two points is comparable to the distance between these two points. In this talk, we will introduce the open problem of the connectivity of the manifold of chord arc curves, already mentioned by G. David in his thesis in 1981, and present some recent results that allow us to transfer the connectivity problem to a problem involving the spectrum of a Beurling- type operator on a particular weighted space.