Entanglement islands for finite regions in Schwarzschild black holes

In this talk, I consider the island formation for entangling regions of finite size in the eternal Schwarzschild black hole and compare it with previously obtained results for semi-infinite regions. Comparison of the results for finite and semi-infinite regions reveals infrared anomalies in the regularization of the Cauchy surface. Different types of finite regions are considered, corresponding to different types of regularization of spacelike infinity. Depending on the size of finite regions, a qualitatively new behavior arises, such as discontinuous evolution of the entanglement entropy, the absence of island formation, the "discontinuity" of Cauchy surface, and asymmetric island solutions. I will also show that the island prescription does not solve the information paradox for some types of finite regions.