Singularity theorems and the Lorentzian splitting theorem for the Bakry-Emery-Ricci tensor

TL;DR

This paper extends classical singularity and splitting theorems in Lorentzian geometry to settings with Bakry-Emery-Ricci curvature, broadening their applicability under weaker curvature conditions.

Contribution

It proves that the Hawking-Penrose singularity theorems and Lorentzian splitting theorem hold under nonnegative Bakry-Emery-Ricci curvature, including cases with infinite m and bounded functions.

Findings

Theorems hold for finite m with nonnegative Ric_f^m.

Theorems hold for infinite m if f is bounded above.

Results generalize classical theorems to weighted curvature settings.

Abstract

We consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature $Ric_f^m$ in timelike directions. We prove that they still hold when $m$ is finite, and when $m$ is infinite, they hold under the additional assumption that $f$ is bounded from above.