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

We show by phase space analysis that there are exactly 17 possible qualitative behaviors for a rotational surface in R3 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 ovaloids 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.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).