We construct asymptotic foliations of asymptotically 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.

We consider null mean curvature flow along the standard lightcone in the de Sitter spacetime. This flow was first studied by Roesch– Scheuer along null hypersurfaces for the detection of MOTS, and independently by the author in the specific case of the standard Minkowski lightcone. Similar to the Minkowski case, null mean curvature flow along the de Sitter lightcone can be related to 2d-Ricci flow for surfaces of genus 0 by an appropriate rescaling. Building on this rescaling procedure, we analyse singularity formation, asymptotic behavior and ancient solutions to the flow.

Motivated by solving the constraint equations in the evolutionary form suggested by R & aacute;cz in 2016, we propose a family of asymptotically flat initial data sets which are "asymptotically spherically symmetric" at infinity. Within this family, we obtain Penrose-like energy estimates and establish the existence of solutions for the constraint equations in the spherical symmetric and totally umbilic cases.

It is a well-known fact that the Schwarzschild spacetime admits a maximal spacetime extension in null coordinates which extends the exterior Schwarzschild region past the Killing horizon, called the Kruskal–Szekeres extension. This method of extending the Schwarzschild spacetime was later generalized by Brill–Hayward to a class of spacetimes of “profile h” across non-degenerate Killing horizons. Circumventing analytical subtleties in their approach, we reconfirm this fact by reformulating the problem as an ODE, and showing that the ODE admits a solution if and only if the naturally arising Killing horizon is non-degenerate. Notably, this approach lends itself to discussing regularity across the horizon for non-smooth metrics. We will discuss applications to the study of photon surfaces, extending results by Cederbaum–Galloway and Cederbaum–Jahns–Vičánek-Martínez beyond the Killing horizon. In particular, our analysis asserts that photon surfaces approaching the Killing horizon must necessarily cross it.