  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.
