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 defined by these authors 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 if such an end is asymptotic to a catenoid cousin, then we can associate an axis to it. We also compute the flux of Killing fields through these ends, and we deduce some geometric properties and analogies to minimal surfaces in Euclidean space.
"Flux for Bryant surfaces and applications to embedded ends of finite total curvature." Illinois J. Math. 47 (3) 667 - 698, Fall 2003. https://doi.org/10.1215/ijm/1258138187