An elliptic proof of the splitting theorems from Lorentzian geometry

TL;DR

This paper introduces an elliptic approach to Lorentzian splitting theorems by replacing the linear d'Alembertian with a non-uniformly elliptic p-d'Alembert operator, simplifying proofs and connecting Lorentzian and Riemannian splitting results.

Contribution

It provides a novel elliptic proof of Lorentzian splitting theorems by using a non-uniformly elliptic operator, bridging Lorentzian and Riemannian geometric frameworks.

Findings

New elliptic proof of Lorentzian splitting theorems

Connection established between Lorentzian and Riemannian splitting results

Simplification of proof techniques in Lorentzian geometry

Abstract

We provide a new proof of the splitting theorems from Lorentzian geometry, in which simplicity is gained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity (non-uniformly) elliptic $p$-d'Alembert operator for this purpose. This allows us to bring the Eschenburg, Galloway, and Newman Lorentzian splitting theorems into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.