  We establish some a priori geometric relations on stable minimal surfaces
lying inside three-manifolds with scalar curvature uniformly bounded below. The
relations are based on a slight generalization of a formula due to Castillon.
We apply it to prove non-local rigidity results in the particular sense that
they express how local isoperimetric properties in some region affect the local
isoperimetric properties in any other region. We present applications to
understand the notion of isoperimetric collapse on three-manifolds with scalar
curvature uniformly bounded below.
