  We contribute to an original problem studied by Hamilton and others, in order
to understand the behaviour of maximal solutions of the Ricci flow both in
compact and non-compact complete orientable Riemannian manifolds of finite
volume. The case of dimension two has peculiarities, which force us to use
different ideas from the corresponding higher dimensional case. We show the
existence of connected regions with a connected complementary set (the
so-called "separating regions"). In dimension higher than two, the associated
problem of minimization is reduced to an auxiliary problem for the
isoperimetric profile. This is possible via an argument of compactness in
geometric measure theory. Indeed we develop a definitive theory, which allows
us to circumvent the shortening curve flow approach of previous authors at the
cost of some applications of geometric measure theory and Ascoli-Arzela's
Theorem.
