The Penrose singularity theorem, MOTS stability, and horizon topology in weighted spacetimes

We consider versions of the Penrose singularity theorem and the Hawking horizon topology theorem in weighted spacetimes that contain weighted versions of trapped surfaces, for arbitrary spacetime dimension and synthetic dimension. We find that suitable generalizations of the unweighted theorems hold under weighted energy conditions. Our results also provide further evidence in favour of a weighted scalar curvature that differs from the trace of the weighted Ricci curvature. When the synthetic dimension is a positive integer, these weighted curvatures have a natural interpretation in terms of warped product metrics.