Markus Wolff: Uniqueness of STCMC surfaces in asymptotically Schwarzschildean lightcones

The spacetime mean curvature of a surface (of co-dimension 2) in an ambient spacetime is defined as the (Lorentzian) length of the co-dimension 2 mean curvature vector. Here, we consider surfaces of constant spacetime mean curvature (STCMC) along a given asymptotically Schwarzschildean lightcone. In joint work in preparation with Klaus Kröncke, we show that the lightcone under consideration admits an asymptotic foliation of STCMC surfaces. In the initial data set case asymtotic foliations by STCMC surfaces have previously been constructed by Cederbaum-Sakovich.

In this talk, I will present the corresponding uniqueness statement by showing that STCMC surfaces are unique within a suitably defined a-priori class. As a consequence the constructed foliation is unique in the a-priori class in the sense that any STCMC surface in the a-priori class sufficiently far out in the asymptotic region is a leaf of the foliation.