Elliptic Weingarten surfaces: singularities, rotational examples and the halfspace theorem

Abstract

We show by phase space analysis that there are exactly 17 possible qualitative behaviors for a rotational surface in ${\mathbb{R}}^3$ that satisfies an arbitrary elliptic Weingarten equation $W({\kappa }_1,{\kappa }_2)=0$, and study the singularities of such examples. As global applications of this classification, we prove a sharp halfspace theorem for general elliptic Weingarten equations of finite order, and a classification of peaked elliptic Weingarten spheres with at most two singularities. In the case that $W$ is not elliptic, we give a negative answer to a question by Yau regarding the uniqueness of rotational ellipsoids.

28 pages, 13 figures