Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass. (English) Zbl 1514.53075
Summary: In [Classical Quantum Gravity 32, No. 20, Article ID 205002, 16 p. (2015; Zbl 1326.83051)], C. Mantoulidis and R. 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 as discussed in [loc. cit.; A. J. Cabrera Pacheco and C. Cederbaum, Contemp. Math. 775, 1–30 (2021; Zbl 1495.53001)]. Their construction was later adapted to \(n + 1\) dimensions by A. J. Cabrera Pacheco and P. Miao [Math. Res. Lett. 25, No. 3, 937–956 (2018; Zbl 1401.83014)]. In recent papers by A. Alaee et al. [Adv. Theor. Math. Phys. 23, No. 8, 1951–1980 (2020; Zbl 07432486)] and by A. J. Cabrera Pacheco et al. [J. Geom. Phys. 132, 338–357 (2018; Zbl 1396.53057)], a similar construction was performed for 3-dimensional asymptotically Euclidean, electrically charged Riemannian manifolds and for asymptotically hyperbolic Riemannian manifolds, respectively.
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 \(n \geq 2\). 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 [loc. cit.] via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.
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 \(n \geq 2\). 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 [loc. cit.] via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 53Z05 | Applications of differential geometry to physics |
| 83C40 | Gravitational energy and conservation laws; groups of motions |
Keywords:
Riemannian manifolds; prescribed asymptotics; electric fields; geometric flows; explicit construction; gluingReferences:
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