Prescribed mean curvature equation on torus

TL;DR

This paper extends the prescribed mean curvature problem to higher-dimensional tori, proving the existence of graphs with specified mean curvature vectors under certain conditions.

Contribution

It introduces a new existence result for prescribed mean curvature graphs on n-dimensional tori, generalizing previous one-dimensional cases.

Findings

Existence of prescribed mean curvature graphs on $ $-dimensional tori.

Conditions involving Sobolev norms and monotonicity are sufficient.

Generalization from 1D to higher dimensions achieved.

Abstract

Prescribed mean curvature problems on the torus has been considered in one dimension. In this paper, we prove the existence of a graph on the $n$-dimensional torus $\mathbb {T}^n$, the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.