Invariant hypersurfaces with linear prescribed mean curvature

TL;DR

This paper investigates hypersurfaces in Euclidean space with mean curvature linearly dependent on the Gauss map, providing explicit parametrizations and classifying rotationally invariant cases, linking to manifolds with density.

Contribution

It introduces a new class of invariant hypersurfaces with linear prescribed mean curvature and offers explicit parametrizations and classifications for special cases.

Findings

Explicit parametrizations of constant curvature hypersurfaces

Classification of rotationally invariant hypersurfaces

Connection to manifolds with density and constant weighted mean curvature

Abstract

Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. In this paper we obtain explicit parametrizations of constant curvature hypersurfaces, and also give a classification of rotationally invariant hypersurfaces.