  The B-phase of superfluid $^3$He is a 3D time-reversal invariant (TRI)
topological superfluid with an isotropic energy gap separating the ground-state
and bulk continuum states. We report calculations of surface spectrum, spin-
and mass current densities originating from the Andreev surface states for
confined $^3$He-B. The surface states are Majorana Fermions with their spins
polarized transverse to their direction of propagation along the surface. The
negative energy states give rise to a ground-state helical spin current
confined on the surface. The spectral functions reveal the subtle role of the
spin-polarized surface states in relation to the ground-state spin current. By
contrast, these states do not contribute to the $T=0$ mass current. Superfluid
flow through a channel of confined $^3$He-B is characterized by the flow field,
$p_s$. The flow field breaks 2D rotational symmetry and time reversal (T).
However, the Bogoliubov-Nambu Hamiltonian remains invariant under $T$ combined
with a 180 degree rotation about the normal to the film. As a result the
B-phase in the presence of a superflow remains a topological phase with a
gapless spectrum of Majorana modes on the surface. Thermal excitation of the
Doppler shifted Majorana branches leads to a power law suppression of the
superfluid mass current, providing a direct signature of the Majorana branches
of surface excitations in the fully gapped 3D topological superfluid, $^3$He-B.
Results are reported for the superfluid fraction (mass current) and helical
spin current for confined $^3$He-B, including the temperature dependences, as
well as dependences on confinement, interactions between quasiparticles and
pressure.
