A Fary-Milnor theorem for CAT(0) spaces

We discuss Jordan curves of finite total curvature in CAT(0)
spaces. Our main result says that if such a curve $\gamma$ has total curvature $\leq4\pi$ then the following dichotomy occurs.
Either $\gamma$ spans an embedded disc, or the total curvature of
$\gamma$ equals $4\pi$ and $\gamma$ bounds a star shaped subset,
intrinsically isometric to a Euclidean cone of cone angle equal to $4\pi$.