Uniqueness of minimizers along the E_min(L) and E_min(A) curves

Ascertain whether, for each fixed value of the minimal surface size (L in AdS4 or A in AdS5), there exists a unique time-symmetric, vacuum Einstein initial data set (with the specified AdS boundary conditions) that realizes the minimum energy value E_min(L) or E_min(A).

Background

The numerical procedure samples large classes of time-symmetric initial data to approximate the minimum-energy curves E_min(L) (AdS4) and E_min(A) (AdS5). Static solutions are known to extremize energy and correspond to turning points on these curves. However, the structure of minimizers at generic points on the curves is not established.

The authors explicitly pose the uniqueness of the minimizing solution at each point on the curves as an open question, contrasting with the standard positive energy theorem which ensures uniqueness of the ground state in its setting.