A Schoen type theorem for elliptic special Weingarten surfaces of minimal type

In this talk we show that a complete (connected), embedded outside a compact set, elliptic special Weingarten surface of minimal type in R3 of finite total curvature and two ends is rotationally symmetric; in particular, it must be one of the rotational special catenoids described by R. Sa Earp and E. Toubiana. This answers in the positive a question posed in 1993 by R. Sa Earp.