Stationary sets of the mean curvature flow with a forcing term

Abstract

We consider the flat flow solution to the mean curvature equation with forcing in ℝ n {\mathbb{R}^{n}} . Our main result states that tangential balls in ℝ n {\mathbb{R}^{n}} under a flat flow with a bounded forcing term will experience fattening, which generalizes the result in [N. Fusco, V. Julin and M. Morini, Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane, preprint 2020, https://arxiv.org/abs/2004.07734] from the planar case to higher dimensions. Then, as in the planar case, we characterize stationary sets in ℝ n {\mathbb{R}^{n}} for a constant forcing term as finite unions of equisize balls with mutually positive distance.

Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus

“…In order to obtain the higher-order regularity inequalities, we apply some curvature estimates obtained recently in [25].…”