  In recent years black hole configurations with non standard topology or with
non-standard asymptotic have gained considerable attention. In this article we
carry out numerical investigations aimed to find periodic coaxial
configurations of co-rotating 3+1 vacuum black holes, for which existence and
uniqueness has not yet been theoretically proven. The aimed configurations
would extend Myers/Korotkin-Nicolai's family of non-rotating (static) coaxial
arrays of black holes. We find that numerical solutions with a given value for
the area A and for the angular momentum J of the horizons appear to exist only
when the separation between consecutive horizons is larger than a certain
critical value that depends only on A and |J|. We also establish that the
solutions have the same Lewis's cylindrical asymptotic as Stockum's infinite
rotating cylinders. Below the mentioned critical value the rotational energy
appears to be too big to sustain a global equilibrium and a singularity shows
up at a finite distance from the bulk. This phenomenon is a relative of
Stockum's asymptotic's collapse, manifesting when the angular momentum (per
unit of axial length) reaches a critical value compared to the mass (per unit
of axial length), and that results from a transition in the Lewis's class of
the cylindrical exterior solution. This remarkable phenomenon seems to be
unexplored in the context of coaxial arrays of black holes. Ergospheres and
other global properties are also presented in detail.
