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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
\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 $n\geq2$.
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.