# Rotational hypersurfaces of prescribed mean curvature

## TL;DR

This paper classifies rotational hypersurfaces in Euclidean space with prescribed mean curvature using phase space analysis, revealing a Delaunay-type classification for even prescribed functions and demonstrating richer behaviors than constant mean curvature cases.

## Contribution

It introduces a phase space analysis approach to classify rotational hypersurfaces with prescribed mean curvature, extending Delaunay-type results and highlighting complex behaviors.

## Key findings

- Delaunay-type classification for even prescribed functions
- Existence of diverse hypersurface behaviors beyond constant mean curvature
- Rich variety of rotational hypersurfaces with non-constant prescribed mean curvature

## Abstract

We use a phase space analysis to give some classification results for rotational hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in $\mathbb{S}^n$, we show that a Delaunay-type classification holds for this class of hypersurfaces. We also exhibit examples showing that the behavior of rotational hypersurfaces of prescribed (non-constant) mean curvature is much richer than in the constant mean curvature case.