Surfaces with prescribed mean curvature in $\mathbb{H}^2\times\mathbb{R}$

TL;DR

This paper investigates rotational surfaces in hyperbolic space cross real line with prescribed mean curvature, generalizing constant mean curvature surfaces and translating solitons, using phase plane analysis for classification and construction.

Contribution

It introduces a framework for studying surfaces with prescribed mean curvature in , including new classifications and explicit examples, extending known special cases.

Findings

Constructed entire rotational graphs and catenoid-type surfaces.

Classified solutions when the prescribed function is linear.

Extended understanding of surfaces with prescribed mean curvature in .

Abstract

In this paper we study rotational surfaces in the space $\mathbb{H}^2\times\mathbb{R}$ whose mean curvature is given as a prescribed function of their angle function. These surfaces generalize, among others, the ones of constant mean curvature and the translating solitons of the mean curvature flow. Using a phase plane analysis we construct entire rotational graphs, catenoid-type surfaces, and exhibit a classification result when the prescribed function is linear.