Elliptic special Weingarten surfaces of minimal type in $\mathbb{R}^3$ of finite total curvature

We study complete connected embedded elliptic special Weingarten surfaces of minimal type ($f$-\emph{surfaces}, for short) in $\mathbb{R}^3$ with finite total curvature, under the standing assumption that $f$ is a non-negative and uniformly elliptic function.

First, using the recent asymptotic theory of Barbieri, Gálvez, Lian, and Zhang for embedded ends, we derive a Jorge--Meeks type formula for this class of surfaces.

Next, we adapt the Alexandrov reflection method to the non-cylindrically bounded setting and prove a Schoen-type theorem: a complete connected embedded $f$-surface of finite total curvature with two embedded ends must be rotationally symmetric.

In particular, it is one of the special catenoids constructed by Sa Earp and Toubiana.

This gives a positive answer to a question raised by Sa Earp in 1993.

As a consequence, we show that planes and special catenoids are the only complete connected embedded $f$-surfaces of finite total curvature whose absolute total curvature is less than $8\pi$.