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

TL;DR

This paper investigates surfaces in product spaces with prescribed mean curvature depending on their angle, extending classical CMC surface theory, and characterizes when such surfaces exist and their geometric behavior.

Contribution

It provides necessary and sufficient conditions for the existence of prescribed mean curvature spheres and describes the structure of complete revolution surfaces.

Findings

Existence criteria for prescribed mean curvature spheres.

Complete revolution surfaces behave like Delaunay surfaces.

Extension of CMC surface theory to variable mean curvature cases.

Abstract

The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(\kappa)\times\mathbb{R}$, 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.