For compact-slice cosmological spacetimes, the null-distance completion is bi-Lipschitz to a product taxi space, and monotone families converge uniformly and future-developed, with the limit's causally-null distance equal to the limit tensor's null distance.
Convergence of Lorentzian spaces and curvature bounds for generalized cones
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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.
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math.DG 1years
2026 1verdicts
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Null distance on cosmological spacetimes and monotone convergence
For compact-slice cosmological spacetimes, the null-distance completion is bi-Lipschitz to a product taxi space, and monotone families converge uniformly and future-developed, with the limit's causally-null distance equal to the limit tensor's null distance.