Higher dimensional black hole initial data with prescribed boundary metric

Abstract

We obtain higher dimensional analogues of the results of Mantoulidis and Schoen in [8]. More precisely, we show that (i) any metric $g$ with positive scalar curvature on the $3$-sphere $S^3$ can be realized as the induced metric on the outermost apparent horizon of a $4$-dimensional asymptotically flat manifold with non-negative scalar curvature, whose ADM mass can be arranged to be arbitrarily close to the optimal value specified by the Riemannian Penrose inequality; (ii) any metric $g$ with positive scalar curvature on the $n$-sphere $S^n$, with $n\ge 4$, such that $(S^n,g)$ isometrically embeds into ${\mathbb{R}}^{n+1}$ as a star-shaped hypersurface, can be realized as the induced metric on the outermost apparent horizon of an $(n+1)$-dimensional asymptotically flat manifold with non-negative scalar curvature, whose ADM mass can be made to be arbitrarily close to the optimal value.

Theorem 1.1 significantly strengthened, the assumption on positive Ricci curvature relaxed to positive scalar curvature

Cited by in corpus (3)

• Asymptotically flat extensions of CMC Bartnik data
• Asymptotically hyperbolic extensions and an analogue of the Bartnik mass
• Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature