Equality and rigidity in the spacetime Penrose inequality

Equality in the spacetime Penrose inequality identifies the original initial data as a spacelike slice of a Schwarzschild–Tangherlini exterior, smoothly attached to its horizon, under the stated outermostness and decay hypotheses. We prove this in every spatial dimension n ≥ 3 for a strong-decay exterior class and for weaker decay in dimensions three and four. In dimension three the finite-component theorem forces a single spherical horizon and gives qualitative strictness for disconnected boundaries.