  We present a rigorous theory of the inverse scattering transform (IST) for
the three-component defocusing nonlinear Schrodinger (NLS) equation with
initial conditions approaching constant values with the same amplitude as
$x\to\pm\infty$. The theory combines and extends to a problem with non-zero
boundary conditions three fundamental ideas: (i) the tensor approach used by
Beals, Deift and Tomei for $n$-th order scattering problems, (ii) the
triangular decompositions of the scattering matrix used by Novikov, Manakov,
Pitaevski and Zakharov for the $N$-wave interaction equations, and (iii) a
generalization of the cross product via the Hodge star duality, which, to the
best of our knowledge, is used in the context of the IST for the first time in
this work. The combination of the first two ideas allows us to rigorously
obtain a fundamental set of analytic eigenfunctions. The third idea allows us
to establish the symmetries of the eigenfunctions and scattering data. The
results are used to characterize the discrete spectrum and to obtain exact
soliton solutions, which describe generalizations of the so-called dark-bright
solitons of the two-component NLS equation.
