Scalar curvature, isoperimetric collapse and General Relativity in the Constant Mean Curvature gauge

TL;DR

This paper explores the relationships between Ricci curvature, scalar curvature, and volume radius in 3-manifolds, with potential applications to Einstein's equations in General Relativity, especially in the Constant Mean Curvature gauge.

Contribution

It introduces a set of relations, conjectures, and problems linking curvature measures and volume in 3-manifolds, with implications for geometric flows in General Relativity.

Findings

Relations between Ricci curvature and scalar curvature established

Potential applications to Einstein Constant Mean Curvature flow discussed

Framework has intrinsic geometric interest

Abstract

We discuss a set of relations, set in the form of results, conjectures and problems, between the L^{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.