An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space

TL;DR

This paper develops a method to construct smooth functions in Euclidean space whose level sets are hyperspheres with prescribed mean curvature, filling the entire space with such hyperspheres.

Contribution

It introduces an inverse approach to generate functions with level sets as hyperspheres of specified mean curvature, expanding the understanding of curvature-driven geometric structures.

Findings

Complete filling of Euclidean space with hyperspheres of prescribed mean curvature

Construction of smooth functions with level sets as hyperspheres

Extension of classical curvature problems to a broader class of functions

Abstract

We construct families of smooth functions $H\colon\mathbb{R}^{n+1}\to\mathbb{R}$ such that the Euclidean $(n+1)$-space is completely filled by not necessarily round hyperspheres of mean curvature $H$ at every point.