Microcanonical action and the entropy of Hawking radiation

The island formula---an extremization prescription for generalized entropy---is known to result in a unitary Page curve the entropy of Hawking radiation. This semiclassical formula has been derived Jackiw-Teitelboim (JT) gravity coupled conformal matter using ``replica trick'' evaluate Euclidean path integral. Alternatively, eternal Anti--de Sitter black holes, we derive from minimizing microcanonical action an entanglement wedge. on-shell is minus and arises saddle-point approximation (nonreplicated) When hole bath, islands emerge maximizing at fixed energy, consistent with formula. Our method applies JT as well other two-dimensional dilaton theories.