Single-rotating five-dimensional near-horizon extremal geometry in general relativity

The geometries with $\mathrm{SL}(2,R)$ and some axial U(1) isometries are called "near-horizon extremal geometries" and are found usually, but not necessarily, in the near-horizon limit of the extremal black holes. We present a new member of this family of solutions in five-dimensional Einstein-Hilbert gravity that has only one nonzero angular momentum. In contrast with the single-rotating Myers-Perry extremal black hole and its near-horizon geometry in five dimensions, this solution may have a nonvanishing and finite entropy. Although there is a uniqueness theorem that prohibits the existence of such single-rotating near-horizon geometries in five-dimensional general relativity, this solution has a curvature singularity at one of the poles, which breaks the smoothness conditions in the theorem.