Classification of Willmore Surfaces with Vanishing Gaussian Curvature

We classify simply-connected, orientable, complete Willmore surfaces with vanishing Gaussian curvature. also study the cones in $${\mathbb {R}}^{3}$$ and give a classification. As an application, we show that for embedding $$f:{\mathbb {R}}^{2} \rightarrow {\mathbb , if its corresponding curvature is nonnegative image of Gauss map lies $$\big \{\theta \in {S}}^{2}: d_{{\mathbb {S}}^{2}}(\theta ,\theta _{0}) \le \alpha <\frac{\pi }{2} \big \}$$ $$f({\mathbb {R}}^{2})$$ plane.