The universality of islands outside the horizon

We systematically calculate the quantum extremal surface (QES) associated with Hawking radiation for general $D$-dimensional ($D\geq2$) asymptotically flat (or AdS) eternal black holes using the island formula. We collect the Hawking radiation particles by a non-gravitational bath and find that a QES exists in the near-horizon region outside the black hole when $c\cdot G_{(D)}$ is smaller enough where $c$ is the central charge of the conformal matter and $G_{(D)}$ the $D$-dimensional Newton constant. The locations of the QES in these backgrounds are obtained and the late-time radiation entropy saturates the two times of black hole entropy. Finally, we numerically check that the no island configuration exists once $c\cdot G_{(D)}$ exceeds a certain upper bound in two-dimensional generalized dilaton theories (GDT). When $c\cdot G_{(D)}$ is close to the upper bound, the backreaction of the matter field on the background can not be neglected. We also consider the conditions of existence of the island configuration with the backreaction and prove that the upper bound also exists for the Witten black hole and Weyl-related Witten black hole.