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

## TL;DR

This paper applies the Minimal Geometric Deformation method to $2+1$ dimensional Einstein equations with a cosmological constant, deriving analytical solutions for isotropic and anisotropic sectors and exploring black hole properties.

## Contribution

It introduces an analytical approach to decouple anisotropic solutions in $2+1$ dimensions with a cosmological term, extending the MGD method beyond 3+1 dimensions.

## Key findings

- Solutions can be expressed analytically without integrals.
- Depending on the cosmological constant, solutions describe different black hole types.
- The study discusses dualities between regular and non-regular, exotic and non-exotic black holes.

## 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.