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

Abstract

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 $8\ddot{I}€$.

30 pages, 8 figures

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