On the existence of spacetime mean curvature 2-surfaces in Minkowski space

Mario Misas Arcos

In order to accurately define the concept of center of mass within the frame of mathematical relativity, Huisken and Yau (1996) introduce the so-called Constant Mean Curvature foliation. This consist of foliating a an asymptotically Euclidean intial data set with non-vanishing (ADM-) energy by 2-surfaces whose mean curvature vector norm is constant at all points in the surface. The physical system whose center of mass (CoM) we attempt to find would then by represented by the initial data set; and its CoM can be computed by finding the coordinate center of each leaf and then evaluating its limit when the leafs approach infinity. However, although such a foliation was proven to exist for every asymptomatically Euclidean IDS, it often produces a divergent CoM: when taking the limit, it diverges to infinity instead of yielding a point in the physical system. For this reason, Carla Cederbaum and Anna Sakovich (2019) take the concept further by defining the constant spacetime mean curvature foliation, which consists of leafs with constant spacetime mean curvature (a generalization of the concept of mean curvature which takes into account the extrinsic curvature of the physical system) at all points instead, and yields non-divergent CoMs much more consistently. In their work they prove such a foliation to exist for asymptotically Euclidean (AE), nonzero-energy initial data sets; but the question of whether an AE IDS with zero energy can be foliated in this way is still open. The aim of this project is then to study the existence of non-trivial constant spacetime mean curvature surfaces in AE zero-energy IDSs, namely physical systems in Minkwoski space, in order to shed light on the yet unanswered question.