Scalar Curvature, Isoperimetric Collapse and General Relativity in the Constant Mean Curvature Gauge

We discuss a set of relations, stated in the form of results, conjectures and problems, between the L-g(2)-norm of the Ricci curvature of a 3-manifold, the scalar curvature and the volume radius. We illustrate the scope of these relations with potential applications to the Einstein Constant Mean Curvature flow (or GR seen as a geometric flow of constant mean curvature), but we believe the framework has it own geometric interest.