arXiv subjects

Alec Metsch

Publications and source records attributed to Alec Metsch.

6 recordsLinked to original sources

arXiv2026

Lipschitz continuity of the cut time for globally hyperbolic spacetimes

We prove that, for a fixed point in a globally hyperbolic spacetime, the focalization time is locally Lipschitz continuous on the open subset of the future causal cone where it is finite. We also show that the cut time is locally Lipschitz continuous in a neighborhood of any timelike tangent vector whose associated geodesic is defined at least up to (and including) its cut time. Furthermore, we derive quantitative estimates for the Lipschitz constant near the null cone and provide a criterion ensuring that the Lipschitz property extends to null directions. As a consequence, we show that the cut locus of a point has Hausdorff codimension at least $1$. These results extend classical Lipschitz continuity results for complete Riemannian manifolds due to Itoh-Tanaka and Li-Nirenberg. Our approach follows the method of Itoh-Tanaka, suitably adapted to the Lorentzian setting.

math.DG↗

arXiv2026

A Lorentzian Lasry-Lions regularization theorem

The main goal of this paper is to establish a general Lorentzian Lasry-Lions regularization theorem: let $u$ be a function defined on a globally hyperbolic spacetime. Assume that its forward Lax--Oleinik evolution $Tu$ is locally semiconcave in a neighbourhood of $(t_0,y_0)$ and has future-directed timelike superdifferentials there. Then, for $t$ close to $t_0$ and sufficiently small $s>0$, the function $\hat T_s\circ T_tu$ is of class $C_{\mathrm{loc}}^{1,1}$ in a neighbourhood of $y_0$. We give sufficient conditions ensuring the assumptions of the theorem and present an application to optimal transport: under quite general assumptions, for any two intermediate measures along a displacement interpolation, there exists a $C^{1,1}_{loc}$-regular maximizing pair in the dual formulation.

math.OC↗

arXiv2025

Semiconvexity of (weak) Kantorovich potentials in the Lorentzian optimal transport problem

We study semiconvexity properties of (weak) Kantorovich potentials for the Lorentzian optimal transport problem with the standard cost function $c$. We show that, in general, this regularity - known in the Riemannian context - does not extend to the Lorentzian setting. Nevertheless, we provide a general regularity result for $c$-convex functions and show that, under suitable general assumptions on the measures, this yields semiconvexity of the (weak) potentials at least on an open set of full measure. This, in turn, allows us to conclude the existence and uniqueness of an optimal transport map.

math.OC↗

arXiv2025

On the locus of multiple maximizing geodesics on a globally hyperbolic spacetime

Extending the recent work of Cannarsa, Cheng and Fathi, we investigate topological properties of the locus ${\cal NU}(M,g)$ of multiple maximizing geodesics on a globally hyperbolic spacetime $(M,g)$, i.e.\ the set of causally related pairs $(x,y)$ for which there exists more than one maximizing geodesic (up to reparametrization) from $x$ to $y$. We will prove that this set is locally contractible. We will also define the notion of a Lorentzian Aubry set ${\cal A}$ and prove that the inclusions ${\cal NU}(M,g)\hookrightarrow \operatorname{Cut}_M\hookrightarrow J^+\backslash {\cal A}$ are homotopy equivalences.

math.OC↗

arXiv2025

$C_{loc}^{1,1}$ optimal pairs in the dual optimal transport problem for a Lorentzian cost along displacement interpolations

We consider the optimal transportation problem on a globally hyperbolic spacetime with a cost function $c$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is given by the squared Riemannian distance. Building upon methods of weak KAM theory, we will establish the existence of $C_{loc}^{1,1}$ optimal pairs for the dual optimal transport problem for probability measures along displacement interpolations.

math.OC↗

arXiv2024

Optimal transport and regularity of weak Kantorovich potentials on a globally hyperbolic spacetime

We consider the optimal transportation problem on a globally hyperbolic spacetime for some cost function $c_2$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is the Riemannian distance squared. Building on insights from previous studies on the Riemannian and Lorentzian case, our main goal is to investigate the regularity of $\pi$-solutions (weak versions of Kantorovich potentials), from which we can conclude, in a classical way, the existence, uniqueness and structure of an optimal transport map between given Borel probability measures $\mu$ and $\nu$, under suitable assumptions.

math.OC↗