  Let $\Lambda$ be an Artin algebra and let $\rm{Gprj}\mbox{-}\Lambda$ denote
the class of all finitely generated Gorenstein projective $\Lambda$-modules. In
this paper, we study the components of the stable Auslander-Reiten quiver of a
certain subcategory of the monomorphism category $\mathcal{S}({\rm
Gprj}\mbox{-}\Lambda)$ containing boundary vertices. We describe the shape of
such components. It is shown that certain components are linked to the orbits
of an auto-equivalence on the stable category
$\underline{\rm{Gprj}}\mbox{-}\Lambda$. In particular, for the finite
components, we show that under certain mild conditions their cardinalities are
divisible by $3$. We see that this three-periodicity phenomenon reoccurs
several times in the paper.
