Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

Emre Baspinar (1) , Giovanna Citti (2)

1 MATHNEURO - Mathématiques pour les Neurosciences
2 Dipartimento di Matematica [Bologna]

We provide a uniqueness result for a class of viscosity solutions to sub-Riemannian mean curvature flows. In a sub-Riemannian setting, uniqueness cannot be deduced by the comparison principle, which is known only for graphs and for radially symmetry surfaces. Here we use a definition of continuous viscosity solutions of sub-Riemannian mean curvature flows motivated from a regularized Riemannian approximation of the flow. With this definition, we prove that any continuous viscosity solution of the equation is a limit of a sequence of solutions of Riemannian flow and obtain as a consequence uniqueness and the comparison principle. The results are provided in the settings of both 3-dimensional rototranslation group ${SE}(2)$ and Carnot groups of step 2, which are particularly important due to their relation to the surface completion problem of a model of the visual cortex.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]; Théorie des groupes [math.GR]

Emre Baspinar, Giovanna Citti. Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures. SIAM Journal on Mathematical Analysis, 2019, 51 (3), pp.2633-2659. ⟨10.1137/17M1150797⟩. ⟨hal-02319482⟩