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The aim of this paper is to extend classic results of the theory of CMCsurfaces in the product spaces to the class of immersed surfaces in$\mathbb{M}^2(\kappa)\times\mathbb{R}$ whose mean curvature is given as a $C^1$function depending on their angle function. We cover topics such as theexistence of a priori curvature estimates for graphs, height estimates forhorizontal and vertical compact graphs and a structure-type result, whichclassifies all the simply connected, properly embedded surfaces with finitetopology.

Prescribed mean curvature surfaces in the product spaces  $\mathbb{M}^2(\kappa)\times\mathbb{R}$; Height estimates and classification  results for properly embedded surfaces