The inverse F-curvature flow in ARW spaces

TL;DR

This paper studies the inverse F-curvature flow in Lorentzian ARW spaces, proving long-term existence and convergence, and explores its role in modeling cosmological transitions from big crunch to big bang.

Contribution

It introduces the inverse F-curvature flow in ARW spaces and demonstrates its existence, convergence, and application to cosmological spacetime transitions.

Findings

Existence of the inverse F-curvature flow for all times.

Convergence of the rescaled flow to a smooth function.

Application to modeling big crunch to big bang transition.

Abstract

We consider the so-called inverse $F$-curvature flow (IFCF) $\dot x = -F^{-1}\nu$ in ARW spaces, i.e. in Lorentzian manifolds with a special future singularity. Here, $F$ denotes a curvature function of class $(K^*)$, which is homogenous of degree one, e.g. the $n$-th root of the Gaussian curvature, and $\nu$ the past directed normal. We prove existence of the IFCF for all times and convergence of the rescaled scalar solution in $C^{\infty}(S_0)$ to a smooth function. Using the rescaled IFCF we maintain a transition from big crunch to big bang into a mirrored spacetime.