  We investigate Sobolev and Hardy inequalities, specifically weighted
Minerbe's type estimates, in noncompact complete connected Riemannian manifolds
whose geometry is described by an isoperimetric profile. In particular, we
assume that the manifold satisfies the $p$-hyperbolicity property, stated in
terms of a necessary integral Dini condition on the isoperimetric profile. Our
method seems to us to combine sharply the knowledge of the isoperimetric
profile and the optimal Bliss type Hardy inequality depending on the geometry
of the manifold. We recover the well known best Sobolev constant in the
Euclidean case.
