No 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.

We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in R n+1 is close to a constant in L n -sense, then the set is close to a union of disjoint balls with respect to the Hausdorff distance. This result is more general than the previous quantifications of the Alexandrov theorem and using it we are able to show that in R 2 and R 3 a weak solution of the volume preserving mean curvature flow starting from a set of finite perimeter asymptotically convergences to a disjoint union of equisize balls, up to possible translations. Here by weak solution we mean a flat flow, obtained via the minimizing movements scheme.

In this paper we establish a new stability result for smooth volume preserving mean curvature flows in flat torus T n in dimensions n = 3, 4. The result says roughly that if an initial set is near to a strictly stable set in T n in H 3 -sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in W 2,5 -sense.