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

Daniele de Gennaro (1) , Anna Kubin (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Polito - Politecnico di Torino = Polytechnic of Turin

We show that the discrete approximate volume preserving mean curvature flow in the flat torus 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.

Mots clés: Geometric evolution equations; Minimizing movement; 53C24; 49M25; 49Q20

Domaines: Mathématiques [math]

Daniele de Gennaro, Anna Kubin. Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus. Calculus of Variations and Partial Differential Equations, 2023, pp.103. ⟨10.1007/s00526-023-02439-0⟩. ⟨hal-03697158⟩