Axisymmetric, extremal horizons in the presence of a cosmological constant

TL;DR

This paper classifies all axisymmetric near-horizon geometries with a cosmological constant on a 2-sphere, linking them to extremal Kerr-(anti-)de Sitter horizons and identifying special degenerate cases.

Contribution

It provides a complete classification of axisymmetric solutions with a cosmological constant, including degenerate horizons, and relates them to Petrov type D solutions.

Findings

Derived all solutions on a topological 2-sphere

Established correspondence with extremal Kerr-(anti-)de Sitter horizons

Identified a triply degenerate horizon solution

Abstract

All axisymmetric solutions to the near-horizon geometry equation with a cosmological constant defined on a topological $2$-sphere were derived. The regularity conditions preventing cone singularity at the poles were accounted for. The one-to-one correspondence of the solutions with the extremal horizons in the Kerr-(anti-)de Sitter spacetimes was found. A solution corresponding to the triply degenerate horizon was identified and characterized. The solutions were also identified among the solutions to the Petrov type D equation.