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


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


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.