Uniqueness of static vacuum asymptotically flat black holes and equipotential photon surfaces in $n+1$ dimensions à la Robinson

In this paper, we combine and generalize to higher dimensions the approaches to proving the uniqueness of connected (3+1)-dimensional static vacuum asymptotically flat black hole spacetimes by Müller zum Hagen–Robinson–Seifert and by Robinson. Applying these techniques, we prove and/or reprove geometric inequalities for connected (n + 1)-dimensional static vacuum asymptotically flat spacetimes with either black hole or equipotential photon surface or specifically photon sphere inner boundary. In particular, assuming a natural upper bound on the total scalar curvature of the boundary, we recover and extend the well-known uniqueness results for such black hole and equipotential photon surface spacetimes. We also relate our results and proofs to existing results, in particular to those by Agostiniani–Mazzieri and by Nozawa–Shiromizu–Izumi–Yamada.