arXiv subjects

Matteo Zanardini

Publications and source records attributed to Matteo Zanardini.

5 recordsLinked to original sources

arXiv2026

Ideal points, directed completion and the case of the maximally extended Schwarzschild spacetime

We study the directed completion of Lorentzian pre-length spaces, and we show that, under some natural assumptions which cover the case of smooth globally hyperbolic spacetimes, it coincides with the future causal completion of Geroch--Kronheimer--Penrose. Moreover, we provide applications of our findings by characterizing the directed completion of the Kruskal--Szekeres spacetime in terms of its radial null geodesics.

math.DG↗

arXiv2026

Submetries in non-positive signature and applications

We consider a generalization of the notion of submetry tailored to the setting of spacetimes. We show that, under suitable completeness assumptions, Lorentzian submetries between smooth spacetimes correspond to locally C^{1,1} Semi-Riemannian submersions. Moreover, we establish some applications regarding timelike curvature bounds, isometric group actions, acausal foliations and Lorentz-Wasserstein geometry.

math.DG↗

arXiv2026

Lorentz meets Ptolemy

We consider a Lorentzian analogue of the Ptolemy inequality and we prove that in the setting of globally hyperbolic spacetimes it is equivalent to a global timelike sectional curvature bound from above by zero. We investigate the link between the Ptolemy inequality and the hyperbolic inversion and establish some applications and rigidity properties.

math.DG↗

arXiv2026

PDE aspects of the dynamical optimal transport in the Lorentzian setting

One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable `continuity inequality', to which we shall refer to as `causal continuity inequality'. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou--Brenier formula.

math.AP↗

arXiv2024

Microstructures and anti-phase boundaries in long-range lattice systems

We study the effect of long-range interactions in non-convex one-dimensional lattice systems in the simplified yet meaningful assumption that the relevant long-range interactions are between $M$-neighbours for some $M\ge 2$ and are convex. If short-range interactions are non-convex we then have a competition between short-range oscillations and long-range ordering. In the case of a double-well nearest-neighbour potential, thanks to a recent result by Braides, Causin, Solci and Truskinovsky, we are able to show that such a competition generates $M$-periodic minimizers whose arrangements are driven by an interfacial energy. Given $M$, the shape of such minimizers is universal, and independent of the details of the energies, but the number and shapes of such minimizers increases as $M$ diverges.

math.AP↗