In this project we study geometrically defined asymptotic foliations of initial data sets in General Relativity. They allow the
construction of asymptotic coordinate systems which are well adapted to the study of physical invariants such as mass and the center of mass. The type of foliations under consideration includes surfaces of constant mean curvature, constant expansion, and constant spacetime mean curvature.
The main goals of this project are:
- Compare the different foliations and their related coordinate systems. Of particular interest is the influence of the value of physical invariants on the shape and position of the surfaces.
- Of particular interest is to find coordinate systems that do not depend on the choice of initial data sets for a given space-time. This could in particular lead to coordinate independent versions of the Regge-Teitelboim asymptotic parity conditions.
- Do all of the above this with as general asymptotic conditions as possible.
In 2015, Mantoulidis and Schoen constructed 3-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be ``far away'' from being round. The resulting manifolds, called extensions, are geometrically not ``close'' to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality. Their construction was later adapted to n+1 dimensions by Cabrera Pacheco and Miao, suggesting instability of the higher dimensional Riemannian Penrose Inequality. In recent papers by Alaee, Cabrera Pacheco, and Cederbaum and by Cabrera Pacheco, Cederbaum, and McCormick, a similar construction was performed for asymptotically Euclidean, electrically charged initial data sets and for asymptotically hyperbolic Riemannian manifolds, respectively, obtaining 3-dimensional extensions that suggest instability of the Riemannian Penrose Inequality with electric charge and of the conjectured asymptotically hyperbolic Riemannian Penrose Inequality in 3 dimensions. This paper combines and generalizes all the aforementioned results by constructing suitable asymptotically hyperbolic or asymptotically Euclidean extensions with electric charge in n+1 dimensions for n greater or equal to 2.
Besides suggesting instability of a naturally conjecturally generalized Riemannian Penrose Inequality, the constructed extensions give insights into an ad hoc generalized notion of Bartnik mass, similar to the Bartnik mass estimate for minimal surfaces proven by Mantoulidis and Schoen via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.
Alejandro Peñuela Diaz
Olivia Vičánek Martínez