The electrically charged BTZ black hole with self (anti-self) dual Maxwell field

The Einstein-Maxwell equations with a negative cosmological constant Λ in 2 + 1 spacetime dimensions discussed by Bañados, Teitelboim and Zanelli are solved by assuming a self (anti-self) dual equation Er̂ = ±B^, which is imposed on the orthonormal basis components of the electric field Er̂ and the magnetic field B^. This solution describes an electrically charged extreme black hole with mass, angular momentum and electric charge Qe. Although the coordinate components of the electric field Er and the magnetic field B have singularities on the horizon at r = (4πGQe2|Λ|)12, the spacetime has the same value of constant negative curvature R = 6Λ of Bañados et al.