Convergence of Lorentzian spaces and curvature bounds for generalized cones

TL;DR

This paper introduces a new convergence notion for Lorentzian spaces, proves its stability for curvature bounds, and applies it to generalized cones, establishing sharp curvature bounds and pre-compactness results.

Contribution

It defines $ ext{l}$-convergence for Lorentzian pre-length spaces, demonstrating stability of curvature bounds and applying it to generalized cones with convergence criteria.

Findings

$ ext{l}$-convergence extends previous notions and preserves curvature bounds.

Generalized cones with converging bases and functions satisfy sharp curvature bounds.

Pre-compactness holds for smooth cones with uniform Ricci curvature bounds.

Abstract

The goal of this article is twofold. We introduce a notion of convergence for Lorentzian pre-length spaces, $\ell$-convergence, that extends previous convergence notions in this context. We show that timelike curvature and timelike curvature-dimension bounds are stable under (measured) $\ell$-convergence. Then, we show that $\ell$-convergence is well adapted for generalized Lorentzian cones: a sequence of generalized cones $-I_i\times_{f_i}X_i$ converges in $\ell$ sense if the base $I_i$ and the fiber $X_i$ converge in GH sense and the functions $f_i$ converge uniformly. We use this to show sharp timelike curvature and timelike curvature-dimension bounds for such cones. Finally, we obtain a pre-compactness theorem for $\ell$-convergence in the class of smooth generalized cones that have a uniform lower bound on the full Ricci (or Riemann) curvature tensor.