The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity

We prove the spacetime Penrose Inequality for smooth three-dimensional initial-data exteriors with one spherical asymptotically anti-de Sitter end, satisfying the stated decay and integrability assumptions, the dominant energy condition, and a compact weakly future outer trapped boundary. We assume that the hyperbolic metric four-flux is future timelike; its Lorentz norm is the mass, and the area is the infimum over full enclosing cuts. On the connected outermost, outer area-minimizing horizon subclass, equality characterizes the original data as a spacelike hypersurface in Schwarzschild–anti-de Sitter spacetime with matching metric mass. No maximality, evolution, or auxiliary solvability assumption is used.