  We study the phase diagram of a three-component Fermi gas with weak
attractive interactions, which shows three superfluid and one normal phases. At
weak symmetry breaking between the components the existence of domain walls
interpolating between two superfluids introduces a new length scale much larger
than the coherence length of each superfluid. This, in particular, limits the
applicability of the local density approximation in the trapped case, which we
also discuss. In the same regime the system hosts soft collective modes with a
mass much smaller than the energy gaps of individual superfluids. We derive
their dispersion relations at zero and finite temperatures and demonstrate that
their presence leads to a significant enhancement of fluctuations near the
superfluid-normal transitions.
