Volume comparison theorems in Finsler spacetimes

TL;DR

This paper extends volume comparison theorems to Lorentz-Finsler spacetimes with curvature bounds, using Riccati equations to establish Bishop--Gromov and G"unther comparisons for standard sets.

Contribution

It introduces volume comparison results in Lorentz-Finsler geometry under curvature bounds, applying Riccati techniques to Lorentzian volumes.

Findings

Established Bishop--Gromov volume comparison for Lorentzian volumes.

Proved G"unther volume comparison under upper flag curvature bounds.

Extended classical comparison theorems to Lorentz-Finsler spacetimes.

Abstract

In a $(1+n)$-dimensional Lorentz--Finsler manifold with $N$-Bakry--\'Emery Ricci curvature bounded below for $N\in(n,\infty]$, using the Riccati equation techniques, we established the Bishop--Gromov volume comparison for the so-called standard sets for comparisons in Lorentzian volumes (SCLVs).We also established the G\"unther volume comparison for SCLVs when the flag curvature was bounded above.