Elliptic Weingarten surfaces: Singularities, rotational examples and the halfspace theorem. (July 2023)

• Record Type: Journal Article
• Authors: Fernández, Isabel; Mira, Pablo
• Is Part Of: Nonlinear analysis. Volume 232(2023)
• Publication Date: 2023-07
• Subjects: 53A10 -- 53C42 -- 35J15 -- 35J60; Weingarten surfaces -- Fully nonlinear elliptic equations -- Phase space analysis -- Halfspace theorem -- Isolated singularities -- Rotational surfaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2023.113244
• Languages: English
• ISSNs: 0362-546X

Abstract

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 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.