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

TL;DR

This paper constructs and analyzes asymptotically hyperbolic and Euclidean charged manifolds in higher dimensions, exploring their geometric properties, stability, and implications for the Riemannian Penrose Inequality and Bartnik mass.

Contribution

It generalizes previous constructions to higher dimensions with electric charge, studying extremality, stability, and providing unified insights into Bartnik mass estimates.

Findings

Constructed charged extensions in higher dimensions.

Studied sub-extremality and extremality cases.

Provided evidence for instability of the generalized Penrose Inequality.

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 $n\geq2$. 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.