Abstract
This work systematically studies the island formula in the general asymptotically flat eternal black holes in generalized dilaton gravity theories or higher-dimensional spherical black holes. Under some reasonable and mild assumptions, we prove that (the boundary of) the island always appears barely outside the horizon in the late time of Hawking radiation, so the information paradox is resolved. In particular, we find a proper island in the Liouville black hole that solves the previous the puzzle.
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