Geometric Constraints via Page Curves: Insights from Island Rule and Quantum Focusing Conjecture

Exploring 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 black hole metrics in asymptotically Minkowski or (anti-)de Sitter spacetimes, we derive a pivotal constraint on the blacken factor $f(r)$ for which the island exists and reproduce the Page curve. Specifically, we reveal that a sufficient yet not universally necessary criterion -- manifested in the negativity of the second derivative of $f(r)$, i.e. $f^{\prime \prime} (r)<0$, in proximity to the event horizon where $r \sim r_h+ {\cal O} (G_N)$, ensures the emergence of Page curves in a manner transcending specific theoretical models. This pivotal finding, supported by the tenets of the quantum focusing conjecture.
Comment: 21 pages, 2 figures, references added