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

TL;DR

This paper extends classical constant mean curvature surface results to more general immersed surfaces in product spaces, providing curvature and height estimates and classifying certain properly embedded surfaces.

Contribution

It introduces new height and curvature estimates for immersed surfaces with mean curvature depending on the angle function in product spaces.

Findings

A priori curvature estimates for graphs

Height estimates for compact graphs

Classification of simply connected, properly embedded surfaces with finite topology

Abstract

The aim of this paper is to extend classic results of the theory of CMC surfaces 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 the existence of a priori curvature estimates for graphs, height estimates for horizontal and vertical compact graphs and a structure-type result, which classifies all the simply connected, properly embedded surfaces with finite topology.