Asymptotically flat extensions with charge

Abstract

The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [14] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, à la Mantoulidis-Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.

On the charged Riemannian Penrose inequality with charged matter

“…These extensions are exactly isometric to Schwarzschild manifolds outside of a compact set, so this combined with the above construction shows the following: if g is a metric on the 2-sphere S 2 and the operator L = −∆ g + 1 2 R(g) has positive first eigenvalue, then for any Q > 0 and ε > 0 there exists a charged asymptotically flat manifold with boundary isometric to (S 2 , g), total charge Q and mass m < |S 2 |g 16π 1/2 + ε. This may be compared to a recent result of Alaee, Cabrera Pacheco, and Cederbaum [1] which establishes the following: under the same eigenvalue hypothesis on g as used in [20], and for Q not too large relative to |S 2 |, one can construct asymptotically flat manifolds with boundary isometric to (S 2 , g), total charge Q, vanishing charge density, and mass m <…”

On the charged Riemannian Penrose inequality with charged matter

“…This should be compared to the work of Mantoulidis and Schoen [17], where they showed that given and metric on the 2-sphere satisfying a certain stability condition one can construct asymptotically flat manifolds with ADM mass arbitrarily close to the optimal mass given by the Riemannian Penrose inequality. These extensions are exactly isometric to Schwarzschild manifolds outside of a compact set, so this combined with the above construction shows the following: if g is a metric on the 2-sphere S 2 and the operator L = −∆ g + 1 2 R(g) has positive first eigenvalue, then for any Q > 0 and ε > 0 there exists a charged asymptotically flat manifold with boundary isometric to (S 2 , g), total charge Q and mass m < |S 2 |g 16π 1/2 + ε. This may be compared to a recent result of Alaee, Cabrera Pacheco, and Cederbaum [1] which establishes the following: under the same eigenvalue hypothesis on g as used in [17], and for Q not too large relative to |S 2 |, one can construct asymptotically flat manifolds with boundary isometric to (S 2 , g), total charge Q, vanishing charge density, and…”