Existence of solutions saturating the minimum-energy bound away from extrema

Establish the existence (and, if possible, construct) time-symmetric, vacuum Einstein initial data sets with the given AdS boundary conditions that saturate the lower bounds E ≥ E_min(L) and E ≥ E_min(A) at non-extremal points on the curves, i.e., for values of L or A not corresponding to static solutions.

Background

While static solutions are known to extremize the energy and thus saturate the bounds at turning points of the curves, it is unclear whether there exist solutions that attain the numerically inferred minimum energy at generic (non-extremal) values of L or A.

The authors explicitly note that they have not shown the existence of solutions that saturate the bound except at the extrema, highlighting a gap between numerical lower bounds and rigorous existence of minimizers across the full domain.