Computing the cut locus of a Riemannian manifold via optimal transport

Enrico Facca (1) , Luca Berti (2) , Francesco Fassò (3) , Mario Putti (3)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 IRMA - Institut de Recherche Mathématique Avancée
3 Dipartimento di Matematica [Padova]

In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge–Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver, based on the so-called dynamical Monge–Kantorovich approach, we propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. We show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in ℝ3, and discuss advantages and limitations.

Mots clés: Cut locus; Riemannian geometry; Optimal Transport problem; Monge–Kantorovich equations; geodesic distance; Cut locus

Domaines: Mathématiques [math]

Enrico Facca, Luca Berti, Francesco Fassò, Mario Putti. Computing the cut locus of a Riemannian manifold via optimal transport. ESAIM: Mathematical Modelling and Numerical Analysis, 2022, 56 (6), pp.1939-1954. ⟨10.1051/m2an/2022059⟩. ⟨hal-03778833v2⟩