The Riemannian Horowitz-Myers conjecture

Let $(M,g)$ be a complete asymptotically Poincaré-Einstein (APE) $n$-manifold with $n\ge 3$, compact, totally geodesic, possibly empty boundary, and flat toroidal conformal infinity. Suppose that the scalar curvature $R_g$ satisfies

$$R_g+n(n-1)\ge 0.$$

Riemannian Horowitz-Myers conjecture. The Wang mass-energy $m$ of $(M,g)$ satisfies $m\ge m_0$, where $m_0$ is the mass of the Horowitz-Myers geon having least mass among all Horowitz-Myers geons with the same conformal boundary at infinity.

If true, this would provide evidence for an AdS/CFT correspondence without supersymmetry and might relate the confining phase of QCD to general relativity in asymptotically anti-de Sitter spacetimes. The conjecture remains unproved, with little progress since it was first posed; it concerns precisely manifolds to which the Witten spinorial method does not apply.