Alejandro Peñuela

A key test for any quasi-local energy in general relativity is that it be nonnegative and satisfy a rigidity property; if it vanishes, the region enclosed is flat. We show that the Hawking energy, when evaluated on its natural area-constrained critical surfaces, henceforth called "Hawking surfaces", satisfies both properties under the dominant energy condition. In the time-symmetric case, where Hawking surfaces coincide with area-constrained Willmore surfaces, we extend positivity and rigidity to include electric charge, a nonzero cosmological constant, and higher dimensions. In the fully dynamical (non-time-symmetric) case, we establish the first nonnegativity and rigidity theorems for the Hawking energy in this general setting. These results confirm the Hawking energy consistency with basic physical principles and address several longstanding ambiguities and criticisms.

Local foliations of area constrained Willmore surfaces on a 3-dimensional Riemannian manifold were constructed by Lamm, Metzger and Schulze , and Ikoma, Machiodi and Mondino, the leaves of these foliations are in particular critical surfaces of the Hawking energy in case they are contained in a totally geodesic spacelike hypersurface. We generalize these foliations to the general case of a non-totally geodesic spacelike hypersurface, constructing an unique local foliation of area constrained critical surfaces of the Hawking energy. A discrepancy when evaluating the so called small sphere limit of the Hawking energy was found by Friedrich, he studied concentrations of area constrained critical surfaces of the Hawking energy and obtained a result that apparently differs from the well established small sphere limit of the Hawking energy of Horowitz and Schmidt, this small sphere limit in principle must be satisfied by any quasi local energy. We confirm independently the discrepancy and explain the reasons for it to happen. We also prove that these surfaces are suitable to evaluate the Hawking energy in the sense of Lamm, Metzger and Schulze, and we find an indication that these surfaces may induce an excess in the energy measured.

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in 1991 to study local constant mean curvature foliations, we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. These foliations were used by Cederbaum and Sakovich to characterize the center of mass in general relativity. Second, we study local foliations of constant expansion surfaces.

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 \emph{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 Riemannian manifolds 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≥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.