Geometric constraints via Page curves: insights from island rule and quantum focusing conjecture*

Abstract To explore the inverse problem tied to the Page curve phenomenon and island paradigm, we investigate the geometric conditions underpinning black hole evaporation, where information is preserved and islands manifest, giving rise to the characteristic Page curve. Focusing on a broad class of static spherical symmetry black hole metrics in asymptotically Minkowski or (anti-)de Sitter spacetimes, we derive a pivotal constraint, the second derivative of the blacken factor for which the island exists, and reproduce the Page curve. Moreover, starting from the quantum focusing conjecture theory, we obtain another constraint on the blacken factor for which the theory can be satisfied: . In particular, by studying these two constraints, we find common properties. Specifically, we reveal that a universal criterion, manifested in the negativity of the second derivative of , i.e., , in proximity to the event horizon where , ensures the emergence of Page curves and follows the quantum focusing conjecture in a manner transcending specific theoretical models. Finally, we argue that the negativity of the second derivative of the blacken factor near the event horizon strongly indicates negative heat capacity, which implies that black holes with a negative heat capacity must have islands and satisfy the quantum focusing conjecture.