Ergoregions between two ergospheres

For a stationary spacetime metric, black holes are spatial regions which disturbances may not propagate out of. In our previous work an existence and regularity theorem was proven for black holes in two space dimensions, in the case where the boundary of the ergoregion is a simple closed curve surrounding a singularity. In this paper we study the case of an annular ergoregion, whose boundary has two components.