Calculus of Variations and Geometric Measure Theory

D. De Gennaro - A. Kubin

Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

created by kubin1 on 13 Jan 2022
modified by degennaro on 12 Jan 2023

[BibTeX]

Accepted Paper

Inserted: 13 jan 2022
Last Updated: 12 jan 2023

Journal: Calculus of Variations and Partial Differential Equations
Year: 2022

ArXiv: 2201.04174 PDF

Abstract:

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.