Initial Data for Black Holes and Black Strings in 5d

We explore time-symmetric hypersurfaces containing apparent horizons of black objects in a 5d spacetime with one coordinate compactified on a circle. We find a phase transition within the family of such hypersurfaces: the horizon has different topology for different parameters. The topology varies from $S^3$ to $S^2 \times S^1$. This phase transition is discontinuous -- the topology of the horizon changes abruptly. We explore the behavior around the critical point and present a possible phase diagram.