| Issue |
ESAIM: M2AN
Volume 56, Number 6, November-December 2022
|
|
|---|---|---|
| Page(s) | 1939 - 1954 | |
| DOI | https://doi.org/10.1051/m2an/2022059 | |
| Published online | 14 September 2022 | |
Computing the cut locus of a Riemannian manifold via optimal transport
1
Inria, Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000 Lille, France
2
Cemosis, IRMA UMR 7501, CNRS, Université de Strasbourg, Strasbourg, France
3
Department of Mathematics “Tullio Levi-Civita”, University of Padua, Via Trieste 63, Padova, Italy
* Corresponding author: enrico.facca@inria.fr
Received:
14
June
2021
Accepted:
6
July
2022
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.
Mathematics Subject Classification: 35J70 / 49K20 / 49M25 / 49Q20 / 58J05 / 65K10 / 65N30
Key words: Cut locus / Riemannian geometry / Optimal Transport problem / Monge–Kantorovich equations / geodesic distance
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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