Isometric embeddings into the Minkowski space and new quasi-local mass

Understand

The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity.

• The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds.
• An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries.
• This approach has been taken by Brown-York and Liu-Yau (see also related works) to define such notions using the isometric embedding theorem into the Euclidean three-space.