Dynamically generated initial non-collapse scale

Establish an independent mechanism relating a dynamical quantity to the initial non-collapse scale, thereby deriving a stronger spatial non-collapse result than the finite-time estimate obtained from prescribed initial non-collapse and curvature and kinematical bounds.

Background

The paper proves a pre-barrier scale-invariant lower bound for the volume of spatial geodesic balls along a hypersurface-orthogonal timelike geodesic congruence. This bound requires an initial scale-invariant volume lower bound, together with upper bounds on timelike Ricci curvature, expansion, and shear.

The authors emphasize that the curvature and kinematical bounds alone do not produce a universal positive non-collapse constant from arbitrary initial data. The unresolved direction is therefore to identify an additional dynamical mechanism that connects the evolution or other dynamical quantities of the congruence to the initial non-collapse scale, allowing a stronger theorem with less dependence on an independently imposed initial condition.