Stationary sets of the mean curvature flow with a forcing term

TL;DR

This paper studies the behavior of flat flows for the mean curvature equation with forcing in higher dimensions, showing fattening of tangential balls and characterizing stationary sets as unions of equal-sized, well-separated balls.

Contribution

It generalizes previous planar results to higher dimensions, providing new insights into stationary sets and fattening phenomena under forcing terms.

Findings

Fattening occurs for tangential balls under bounded forcing in higher dimensions.

Stationary sets are characterized as unions of equal-sized, mutually separated balls.

Results extend planar mean curvature flow properties to Euclidean spaces of dimension at least 2.

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.