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 $4\pi$.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/3050