Reverse Isoperimetric Conjecture as a Noether-Charge Stability Theorem

Abstract

The reverse isoperimetric conjecture asserts that, at fixed thermodynamic volume, Schwarzschild--AdS black holes maximize entropy. We prove that this statement is the fixed-volume form of a boundary-completed Noether-charge stability theorem. The essential observation is that the bulk Hollands--Wald canonical energy is not the full entropy Hessian: along exact stationary black-hole families it vanishes, and the missing curvature is supplied by a constrained asymptotic charge Hessian. Combining this boundary term with bulk canonical-energy positivity gives entropy concavity on admissible fixed-volume components, while zero-energy rigidity determines the equality sector. The theorem reproduces the Einstein-gravity area-volume inequality and extends naturally to Wald entropy in higher-derivative theories. Known violations are thereby reinterpreted as failures of compactness, positivity, or rigidity rather than failures of the variational mechanism.