Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass

Abstract

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. 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. 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 $n\ge 2$. We study in detail the sub-extremality of these manifolds, consider the so far unstudied case of extremality in extensions with electric charge and allow more general conditions for the metric of our extensions. 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.

References in corpus (20)

• On the Riemannian Penrose inequality in dimensions less than 8
• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• Present status of the Penrose inequality
• On the Penrose Inequality for Charged Black Holes
• An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality
• Penrose type inequalities for asymptotically hyperbolic graphs
• Mass rigidity for hyperbolic manifolds
• On the Bartnik mass of apparent horizons
• On a Penrose Inequality with Charge
• On the charged Riemannian Penrose inequality with charged matter
• Geometrical inequalities bounding angular momentum and charges in General Relativity
• Near-equality of the Penrose Inequality for rotationally symmetric Riemannian manifolds
• On the mass aspect function and positive energy theorems for asymptotically hyperbolic manifolds
• Asymptotically hyperbolic extensions and an analogue of the Bartnik mass
• Rigidity in the Positive Mass Theorem with Charge
• Smoothing the Bartnik boundary conditions and other results on Bartnik's quasi-local mass
• On perturbations of the Schwarzschild anti-de Sitter spaces of positive mass
• Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions
• Sobolev stability of the PMT and RPI using IMCF
• On the conformal scalar curvature equation and related problems