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

created by kubin1 on 13 Jan 2022

[BibTeX]

preprint

Inserted: 13 jan 2022

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.

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