Spherically symmetric black holes in $f(R)$ gravity: Is geometric scalar hair supported ?

We discuss with a rather critical eye the current situation of black hole (BH) solutions in $f(R)$ gravity and shed light about its geometrical and physical significance. We also argue about the meaning, existence or lack thereof of a Birkhoff's theorem in this kind of modified gravity. We focus then on the analysis and quest of $non-trivial$ (i.e. hairy) $asymptotically\,\,flat$ (AF) BH solutions in static and spherically symmetric (SSS) spacetimes in vacuum having the property that the Ricci scalar does $not$ vanish identically in the domain of outer communication. To do so, we provide and enforce the $regularity\,\,conditions$ at the horizon in order to prevent the presence of singular solutions there. Specifically, we consider several classes of $f(R)$ models like those proposed recently for explaining the accelerated expansion in the universe and which have been thoroughly tested in several physical scenarios. Finally, we report analytical and numerical evidence about the $absence$ of $geometric\,\,hair$ in AFSSSBH solutions in those $f(R)$ models. First, we submit the models to the available no-hair theorems, and in the cases where the theorems apply, the absence of hair is demonstrated analytically. In the cases where the theorems do not apply, we resort to a numerical analysis due to the complexity of the non-linear differential equations. Within that aim, a code to solve the equations numerically was built and tested using well know exact solutions. In a future investigation we plan to analyze the problem of hair in De Sitter and Anti-De Sitter backgrounds.