On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

We study a family of geometric variational functionals introduced by Hamilton, and considered later Daskalopulos, Sesum, Del Pino Hsu, in order to understand the behavior maximal solutions Ricci flow both compact noncompact complete Riemannian manifolds finite volume. The case dimension two has some peculiarities, which force us use different ideas from corresponding higher-dimensional case. Under natural restrictions, we investigate sufficient necessary conditions allow show existence connected regions with complementary set (the so-called “separating regions”). In higher than two, associated problem minimization is reduced an auxiliary for isoperimetric profile (with investigation minimizers). This possible via argument compactness measure theory valid volume manifolds. Moreover, that minimum separating achieved region. requires techniques proof. present results develop definitive theory, allows circumvent shortening curve approach above mentioned authors at cost applications Ascoli-Arzela's Theorem.