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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 surfacesextends widely, among others, the well-known theory of surfaces with constantmean curvature. In this paper we give necessary and sufficient conditions forthe existence of prescribed mean curvature spheres, and we describe completesurfaces of revolution proving that they behave as the Delaunay surfaces of CMCtype.

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