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

2022/01/11 by Daniele De Gennaro, De Gennaro, Daniele, Anna Kubin +1 · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2201.04174

Abstract

We show that the discrete approximate volume preserving mean curvature flow in the flat torus \mathbbT^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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