arXiv subjects

Matteo Calisti

Publications and source records attributed to Matteo Calisti.

4 recordsLinked to original sources

arXiv2025

Generalized cones admitting a curvature-dimension condition

We study (generalized) cones over metric spaces, both in Riemannian and Lorentzian signature. In particular, we establish synthetic lower Ricci curvature bounds \`a la Lott-Villani-Sturm and Ohta in the metric measure case, and \`a la Cavalletti-Mondino in Lorentzian signature. Here, a generalized cone is a warped product of a one-dimensional base space, which will be positive or negative definite, over a fiber that is a metric space. We prove that Riemannian or Lorentzian generalized cones over $\mathsf{CD}$-spaces satisfy the (timelike) measure contraction property $\mathsf{(T)MCP}$ - a weaker version of a (timelike) curvature-dimension condition $\mathsf{(T)CD}$. Conversely, if the generalized cone is a $\mathsf{(T)CD}$-space, then the fiber is a $\mathsf{CD}$-space with the appropriate bounds on Ricci curvature and dimension. In proving these results we develop a novel and powerful two-dimensional localization technique, which we expect to be interesting in its own right and useful in other circumstances. We conclude by giving several applications including synthetic singularity and splitting theorems for generalized cones. The final application is that we propose a new definition for lower curvature bounds for metric and metric measure spaces via lower curvature bounds for generalized cones over the given space.

math.DG↗

arXiv2025

Hawking's singularity theorem for Lipschitz Lorentzian metrics

We prove Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity. The proof rests on (1) new estimates for the Ricci curvature of regularising smooth metrics that are based upon a quite general Friedrichs-type lemma and (2) the replacement of the usual focusing techniques for timelike geodesics -- which in the absence of a classical ODE-theory for the initial value problem are no longer available -- by a worldvolume estimate based on a segment-type inequality that allows one to control the volume of the set of points in a spacelike surface that possess long maximisers.

math.DG↗

arXiv2024

A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes

We introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the maximal weak subslope of an arbitrary causal function, which plays the role of the (Lorentzian) modulus of its differential. It is shown to satisfy certain chain and Leibniz rules, certify a locality property, and be compatible with its smooth analog. In this setup, we propose a quadraticity condition termed infinitesimal Minkowskianity, which singles out genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian in a weak form under the timelike measure contraction property. As a particular case, this extends Eschenburg's classical estimate past the timelike cut locus.

math.DG↗

arXiv2022

Timelike Ricci bounds for low regularity spacetimes by optimal transport

We prove that a globally hyperbolic smooth spacetime endowed with a $\smash{\mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $\smash{\mathrm{C}^{1,1}}$, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.

gr-qc↗