Abstract
Superfluid $^{3}\mathrm{He}$ in very narrow slab and cylindrical geometries is studied using the Ginzburg-Landau approach. It is found that, in the case of very narrow slabs, the effect of the boundary is to favor the formation of the A phase. At lower temperatures, this A phase is unstable against a deformed B phase. Both states are locally stable and can be supercooled or superheated. The phase diagram for $^{3}\mathrm{He}$ in a narrow slab resembles that of $^{3}\mathrm{He}$ in a magnetic field. The superfluid densities along the channel for both diffusive and specular boundary conditions are computed. Similar results are obtained for a cylindrical geometry. In addition, we present an analytic scheme for determining the order parameter in other geometries in the ``very strongly confined'' limit.
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