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

TL;DR

This paper extends the classification of minimal and Weingarten surfaces in three-dimensional space, establishing new curvature-topology relations, classifying certain surfaces, and confirming symmetry properties for specific finite total curvature surfaces.

Contribution

It generalizes classical minimal surface theory to elliptic special Weingarten surfaces, providing new classification results and symmetry properties, and answering a longstanding question from 1993.

Findings

Classified planes as the only elliptic special Weingarten surfaces with total curvature less than 4π.

Proved that certain elliptic special Weingarten surfaces with two ends are rotationally symmetric.

Established that special catenoids are the only non-flat surfaces with total curvature less than 8π.

Abstract

We extend the theory of complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature to the wider class of elliptic special Weingarten surfaces of finite total curvature; in particular, we extend the seminal works of L. Jorge and W. Meeks and R. Schoen. Specifically, we extend the Jorge-Meeks formula relating the total curvature and the topology of the surface and we use it to classify planes as the only elliptic special Weingarten surfaces whose total curvature is less than $4 \pi$. Moreover, we show that a complete (connected), embedded outside a compact set, elliptic special Weingarten surface of minimal type in $\mathbb{R}^3$ 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. We also prove that the special catenoids are the only connected non-flat special Weingarten surfaces whose total curvature is less than $8 \pi$.