Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature

TL;DR

This paper constructs and analyzes a foliation of asymptotically Schwarzschildean lightcones using surfaces of constant spacetime mean curvature, demonstrating convergence and uniqueness properties via geometric flow methods.

Contribution

It introduces a new method for constructing and proving the uniqueness of STCMC foliations in asymptotically Schwarzschildean spacetimes using null mean curvature flow.

Findings

Exponential convergence of initial data to STCMC surfaces.

Existence and uniqueness of the asymptotic foliation.

Application of geometric flow techniques in a Lorentzian setting.

Abstract

We construct asymptotic foliations of asymtotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (STCMC). Our construction is motivated by the approach of Huisken-Yau for the Riemannian setting in employing a geometric flow. We prove that initial data within a sufficient a-priori class converges exponentially to an STCMC surface under area preserving null mean curvature flow. Further, we show that the resulting STCMC surfaces form an asymptotic foliation that is unique within the a-priori class.