Extreme $\ell$-boson stars
  A new class of complex scalar field objects, which generalize the well known
boson stars, was recently found as solutions to the Einstein-Klein-Gordon
system. The generalization consists in incorporating some of the effects of
angular momentum, while still maintaining the spacetime's spherical symmetry.
These new solutions depend on an (integer) angular parameter $\ell$, and hence
were named $\ell$-boson stars. Like the standard $\ell=0$ boson stars these
configurations admit a stable branch in the solution space; however, contrary
to them they have a morphology that presents a shell-like structure with a
"hole" in the internal region. In this article we perform a thorough
exploration of the parameter space, concentrating particularly on the extreme
cases with large values of $\ell$. We show that the shells grow in size with
the angular parameter, doing so linearly for large values, with the size
growing faster than the thickness. Their mass also increases with $\ell$, but
in such a way that their compactness, while also growing monotonically,
converges to a finite value corresponding to about one half of the Buchdahl
limit for stable configurations. Furthermore, we show that $\ell$-boson stars
can be highly anisotropic, with the radial pressure diminishing relative to the
tangential pressure for large $\ell$, reducing asymptotically to zero, and with
the maximum density also approaching zero. We show that these properties can be
understood by analyzing the asymptotic limit $\ell\rightarrow\infty$ of the
field equations and their solutions. We also analyze the existence and
characteristics of both timelike and null circular orbits, especially for very
compact solutions.
