Flux for Bryant surfaces and applications to embedded ends of finite total curvature

We compute the flux of Killing fields through ends of constant mean curvature 1 in hyperbolic space, and we prove a result conjectured by Rossman, Umehara and Yamada : the flux matrix they have defined is equivalent to the flux of Killing fields. We next give a geometric description of embedded ends of finite total curvature. In particular, we show that we can define an axis for these ends that are asymptotic to a catenoid cousin. We also compute the flux of Killing fields through these ends, and we deduce some geometric properties and some analogies with minimal surfaces in Euclidean space.