Alexandrov's Patchwork and the Bonnet-Myers Theorem for Lorentzian length spaces

Abstract

We present key initial results in the study of global timelike curvature bounds within the Lorentzian pre-length space framework. Most notably, we construct a Lorentzian analogue to Alexandrov's Patchwork, thus proving that suitably nice Lorentzian pre-length spaces with local upper timelike curvature bound also satisfy a corresponding global upper bound. Additionally, for spaces with global lower bound on their timelike curvature, we provide a Bonnet-Myers style result, constraining their finite diameter. Throughout, we make the natural comparisons to the metric case, concluding with a discussion of potential applications and ongoing work.

The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

“…The timelike curvature bounds were first introduced in [KS18], and slightly modified in [BNR15]. To describe timelike curvature bounds, we will compare certain distances to distances in comparison spaces: the Lorentzian model spaces L 2 (K) of constant sectional curvature K.…”

“…3.4.2] implies a sectional curvature version. In the setting of synthetic sectional curvature bounds (see [KS18]), there also is a Bonnet-Myers theorem [BNR15,Thm. 4.11,Rem.…”