I have been studying some aspects of Ricci flow, namely existence, uniqueness, finite time extinction, the preservation of curvature bounds via the maximum principle, and the modifications of Ricci flow that lead to long time solutions with limiting geometries of interest like constant curvature metrics.
As a physicist, I naturally wanted to see what the analogue of Ricci flow is for Lorentzian manifolds. A couple of problems arise: (1) globally hyperbolic spacetimes are not closed, (2) non compact manifolds without further conditions may render the flow ill-defined for existence of solutions, and (3) the Laplacian in this case is not elliptic given the metric is not positive.
Does anyone have any good references where progress is made on some of these problems in the Lorentzian setting? I would also appreciate any insight from the community here, as I am a bit out of my depth.