Computing the Cut Locus of a Riemannian Manifold via Optimal Transport

TL;DR

This paper introduces a novel PDE-based characterization of the cut locus on Riemannian manifolds using optimal transport theory, and proposes a numerical method for its approximation with practical examples.

Contribution

The paper provides a new PDE formulation linking the cut locus to optimal transport density and develops a numerical approach for computing it on 2D surfaces.

Findings

The cut locus corresponds to the zero set of the optimal transport density.

The proposed numerical method successfully approximates the cut locus on embedded surfaces.

The approach demonstrates advantages and limitations through practical examples.

Abstract

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 $R^{3}$ and discuss advantages and limitations.