Geometric inequalities for asymptotically anti-de Sitter spacetimes

The research proposed in this application focuses on the study of mass of asymptotically anti-de Sitter spacetimes, with main emphasis on proving certain geometric inequalities for this class of spacetimes.

Asymptotically anti-de Sitter spacetimes have very rich geometry at infinity which in many cases impedes the application of methods that were designed to deal with the setting of asymptotically flat spacetimes.

To circumvent this difficulty, we propose to study the so-called horospherical surfaces, i.e. constant mean curvature surfaces of codimension 2 modelled on horospheres of hyperbolic space.

These surfaces seem to be the most natural variational objects for this setting adequately capturing the mass of an asymptotically anti-de Sitter spacetime, which we plan to utilise for proving the related Positive Mass Conjecture.

The only available results in this setting require additional topological restrictions in dimensions other than 3+1, while we hope to cover dimensions up to 7+1 by using geometric measure theory.

Another goal is to use certain geometric evolution equations to prove the Penrose conjecture for asymptotically anti-de Sitter spacetimes foliatied by hypersurfaces of zero extrinsic curvature providing the first available result in this direction.

We are also concerned with a very fundamental question: are the conventional definitions of mass derived by physicists in 2000s the ultimate ones in the context of Penrose and Positive Mass Conjectures?