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 R 3 that satisfies an arbitrary elliptic Weingarten equation W (κ 1 , κ 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 2 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.