A Delaunay-type classification result for prescribed mean curvature surfaces in $\mathbb{M}^2(\kappa)\times\mathbb R$

Antonio Bueno

Vol. 313 (2021), No. 1, 45–74

DOI: 10.2140/pjm.2021.313.45

Abstract

The purpose of this paper is to study immersed surfaces in the product spaces ${\mathbb{𝕄}}^2\left(\kappa \right)\times ℝ$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In this paper we give necessary and sufficient conditions for the existence of prescribed mean curvature spheres, and we describe complete surfaces of revolution proving that they behave as the Delaunay surfaces of CMC type.

Keywords

prescribed mean curvature, product space, rotational surface, existence of spheres, Delaunay-type classification

Mathematical Subject Classification 2010

Primary: 34C05, 34C40, 53A10, 53C42

Milestones

Received: 17 June 2019

Revised: 4 January 2021

Accepted: 16 April 2021

Published: 17 September 2021