Gravitational decoupling in $2+1$ dimensional space--times with cosmological term

Abstract

In this work we implement the Minimal Geometric Deformation method to obtain the isotropic sector and the decoupler matter content of any anisotropic solution of the Einstein field equations with cosmological constant in $2+1$ dimensional space--times. We obtain that the solutions of both sectors can be expressed analytically in terms of the metric functions of the original anisotropic solutions instead of formal integral as in its $3+1$ counterpart. As a particular example we study a regular black hole solution and we show that, depending on the sign of the cosmological constant, the solutions correspond to regular black holes violating the null energy condition or to a non--regular black hole without exotic hair. The exotic/non--exotic and the regular/non--regular black hole dualities are discussed.