  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. However,
there exist surfaces in the Minkowski space whose quasilocal mass is strictly
positive. It appears that the momentum information needs to be accounted for to
reconcile the difference. In order to fully capture this information, we use
isometric embeddings into the Minkowski space as references. In this article,
we first prove an existence and uniqueness theorem for such isometric
embeddings. We then solve the boundary value problem for Jang's equation as a
procedure to recognize such a surface in the Minkowski space. In doing so, we
discover new expression of quasi-local mass. The new mass is positive when the
ambient space-time satisfies the dominant energy condition and vanishes on
surfaces in the Minkowski space. It also has the nice asymptotic behavior at
spatial and null infinity. Some of these results were announced in [29].
