Curvature-dimension bounds for Lorentzian splitting theorems

TL;DR

This paper extends Lorentzian splitting and singularity theorems to include all synthetic dimensions using curvature-dimension bounds, revealing new rigidity phenomena and broadening the scope of Lorentzian Bakry-Émery theory.

Contribution

It generalizes key Lorentzian geometric theorems to all synthetic dimensions, including negative and critical values, with detailed rigidity analysis.

Findings

Extended Hawking-Penrose singularity theorem to all synthetic dimensions.

Proved Lorentzian splitting theorem with reduced rigidity at critical dimension.

Unified curvature-dimension bounds for null and timelike cases across all synthetic dimensions.

Abstract

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-\'Emery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when $N=1$. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry-\'Emery-Ricci tensor for all $N\in (-\infty, 2]\cup (n,\infty)$ and for the $N=\infty$ case as well, with reduced rigidity if $N=2$. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-\'Emery theory now cover all synthetic dimensions for which such theorems are possible. The splitting theorems are found always to exhibit reduced rigidity at the critical synthetic dimension.