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.

中文翻译：

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通过最优传输计算黎曼流形的切割轨迹

在本文中，我们将紧凑黎曼流形上点的切割轨迹的新特征描述为 Monge-Kantorovich 方程的最优输运密度解的零集，这是最优输运问题的 PDE 公式，成本等于测地距离。将此结果与基于所谓的动态 Monge-Kantorovich 方法的最佳传输数值求解器相结合，我们提出了一种新颖的框架，用于流形中点的切割轨迹的数值近似。我们展示了所提出的方法在嵌入 $R^{3}$ 的二维表面上的几个示例上的适用性，并讨论了优点和局限性。