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

TL;DR

This paper classifies rotational elliptic Weingarten surfaces in three-dimensional space, analyzes their singularities, and applies the results to halfspace theorems and sphere classifications, providing new insights into their geometric behavior.

Contribution

It provides a complete phase space classification of rotational elliptic Weingarten surfaces and studies their singularities, leading to new global geometric results.

Findings

Exactly 17 qualitative behaviors for rotational elliptic Weingarten surfaces.

A sharp halfspace theorem for elliptic Weingarten equations of finite order.

Classification of peaked elliptic Weingarten spheres with limited singularities.

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.