Regular decoupling sector and exterior solutions in the context of MGD

We implement the Gravitational Decoupling through the Minimal Geometric Deformation method and explore its effect on exterior solutions by imposing a regularity condition in the Tolman–Oppenheimer–Volkoff equation of the decoupling sector. We obtain that the decoupling function can be expressed formally in terms of an integral involving the $g_{tt}$ component of the metric of the seed solution. As a particular example, we implement the method by using the Schwarzschild exterior as a seed and we obtain that the asymptotic behavior of the extended geometry corresponds to a manifold with constant curvature.