On the Spacetime Mean Curvature of Surfaces in General Relativity

Abstract

The spacetime mean curvature of a 2-surface in a 4-dimensional spacetime is defined as the (Lorentzian) length of the codimension-2 mean curvature vector of the surface in the ambient spacetime. In particular, we are interested in surfaces of constant spacetime mean curvature (STCMC surfaces), i.e., surfaces such that the length of the mean curvature vector is constant along the surface. In recently published work Cederbaum-Sakovich constructed an asymptotic foliation of STCMC surfaces in asymptotically Euclidean initial data sets to define a new notion of center-of-mass in the context of General Relativity. Similar to the work of Cederbaum-Sakovich, we consider the spacetime mean curvature of surfaces that are restricted to a hypersurface in the ambient spacetime. More explicitly, we consider the spacetime mean curvature restricted to either an asymptotically Euclidean, maximal initial data set, a totally umbilic, asymptotically hyperboloidal initial data set, or a null hypersurface in the ambient spacetime. In the asymptotically Euclidean setting we show the existence of weak solutions to inverse spacetime mean curvature flow in maximal initial data sets. This builds on previous work by Huisken-Ilmanen and Moore, and is joint work with Gerhard Huisken. In the asymptotically hyperboloidal setting, we prove a characterization of STCMC surfaces in a class of spherically symmetric spacetimes, building on an Alexandrov Theorem by Brendle. In the null hypersurface setting we consider what is called null mean curvature flow here in the special case of the round Minkowski lightcone. This parabolic flow was first studied by Roesch and Scheuer on null hypersurface to detect marginally outer trapped surfaces. As no such surfaces exists on the Minkowski lightcone, we show that the flow develops singularities in finite time. Using either the rescaled flow or the respective elliptic equation further leads to an estimate for surfaces on the Minkowski lightcone similar to work by De Lellis-Müller in Euclidean space.