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

We construct asymptotic foliations of asymtotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (STCMC). For a surface in an ambient spacetime, the spacetime mean curvature is defined as the (Lorentzian) length of the co-dimension 2 mean curvature vector. Asymtotic foliations of asymptotically flat spacelike hypersurfaces by STCMC surfaces have previously been constructed by Cederbaum-Sakovich.

Our construction is motivated by the approach of Huisken-Yau for the Riemannian setting in employing a geometric flow. We show that initial data within a sufficient a-priori class converges exponentially to an STCMC surface under area preserving null mean curvature flow. We further show that the resulting STCMC surfaces form an asymptotic foliation. This is joint work with Klaus Kröncke.