Classification of Willmore surfaces with vanishing Gaussian curvature

TL;DR

This paper classifies certain special surfaces called Willmore surfaces with zero Gaussian curvature, explores their cone structures, and provides a Bernstein-type theorem as an application.

Contribution

It offers a complete classification of simply-connected, complete Willmore surfaces with zero Gaussian curvature and analyzes their cone structures, including a new Bernstein-type theorem.

Findings

Classification of simply-connected, complete Willmore surfaces with zero Gaussian curvature.

Classification of Willmore cones.

A Bernstein-type theorem for these surfaces.

Abstract

We classify simply-connected, complete Willmore surfaces with vanishing Gaussian curvature. We also study the Willmore cones and give a classification. As an application, we give a Bernstein-type theorem.