Stationary sets of the mean curvature flow with a forcing term

Abstract

We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space ${\mathbb{R}}^n$ of dimension at least 2. Our main results states that tangential balls in ${\mathbb{R}}^n$ under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in ${\mathbb{R}}^n$ for a constant forcing term as finite unions of equisized balls with mutually positive distance.

Minor corrections and two new reference added