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arXiv · 2606.21997

Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations

Gregory J. Galloway·Robert J. McCann·Argam Ohanyan
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Abstract

The families of Lorentzian distance functions to and from points along a complete timelike line in a spacetime are known to be locally equi-Lipschitz continuous in a neighborhood of the line. This is an essential component in the proof of the classical Lorentzian splitting theorems. We show that this property fails in general for families of Lorentzian distances to and from the level sets of a given Cauchy temporal function. Moreover, we formulate conjectures based on the existence of Cauchy temporal functions in cosmological and timelike geodesically complete spacetimes such that the Lorentz distances to its level sets are equi-Lipschitz, which are equivalent to Bartnik's splitting conjecture.

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Gregory J. Galloway, Robert J. McCann, Argam Ohanyan. 2026-09-09. Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations. https://arxiv.org/abs/2606.21997

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