  We consider the steady-state fingering instability of an elastic membrane
separating two fluids of different density under external pressure in a
rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are
considered, and the role of membrane tension is detailed in each case. Both
systems exhibit a centrifugally-driven Rayleigh-Taylor--like instability when
the density of the inner fluid exceeds that of the outer one, and this
instability competes with the restoring forces arising from curvature and
tension, thereby setting the finger scale. Numerical continuation is used to
compute not only strongly nonlinear primary finger states up to the point of
self-contact but also secondary branches of mixed modes and circumferentially
localized folds as a function of the rotation rate and the externally imposed
pressure. Both reflection-symmetric and symmetry-broken chiral states are
computed. The results are presented in the form of bifurcation diagrams. The
ratio of system scale to the natural length scale is found to determine the
ordering of the primary bifurcations from the unperturbed circle state as well
as the solution profiles and onset of secondary bifurcations.
