Analytic form and proof of the minimum-energy bounds in AdS4 and AdS5

Determine explicit analytic expressions for the minimum-energy functions E_min(L) for time-symmetric, vacuum Einstein initial data with negative cosmological constant and S1×S1 conformal boundary in AdS4 (where L is the length of the contractible minimal circle) and E_min(A) for time-symmetric, vacuum Einstein initial data with negative cosmological constant and S1×S2 conformal boundary in AdS5 (where A is the area of the contractible minimal 2-sphere), and prove rigorously that these functions provide lower bounds on the total energy for all such initial data (i.e., establish E ≥ E_min(L) and E ≥ E_min(A)).

Background

The paper introduces a new class of energy inequalities in asymptotically AdS spacetimes with a spatial circle at the conformal boundary, focusing on time-symmetric initial data. For AdS4 with S1×S1 boundary, the authors numerically construct a curve E_min(L) that bounds the energy in terms of the length L of a contractible minimal circle. For AdS5 with S1×S2 boundary, they analogously construct E_min(A) as a function of the area A of a contractible minimal sphere.

While these bounds are obtained numerically and exhibit characteristic features (e.g., cubic growth for large L in AdS4 and quadratic growth in A with linear s-dependence in AdS5), the analytic form of the functions remains undetermined. The authors explicitly identify deriving such analytic expressions and proving their validity as rigorous lower bounds as a central open problem.