# Half-space theorems for properly immersed surfaces in $\mathbb{R}^3$ with prescribed mean curvature

## TL;DR

This paper establishes half-space theorems for properly immersed surfaces in three-dimensional space with prescribed mean curvature depending on the Gauss map, extending classical results for minimal and constant mean curvature surfaces.

## Contribution

It introduces new half-space theorems for surfaces with prescribed mean curvature functions related to the Gauss map, generalizing known results for minimal and constant mean curvature surfaces.

## Key findings

- Derived conditions for half-space theorems in prescribed mean curvature surfaces.
- Analyzed the asymptotic behavior of a family of embedded annuli analogous to minimal catenoids.

## Abstract

Motivated by the large ammount of results obtained for minimal and positive constant mean curvature surfaces in several ambient spaces, the aim of this paper is to obtain half-space theorems for properly immersed surfaces in $\mathbb{R}^3$ whose mean curvature is given as a prescribed function of its Gauss map. In order to achieve this purpose, we will study the behavior at infinity of a 1-parameter family of properly embedded annuli that are analogous to the usual minimal catenoids.