  A generalization of the Euler-Plateau problem to account for the energy
contribution due to twisting of the bounding loop is proposed. Euler-Lagrange
equations are derived in a parameterized setting and a bifurcation analysis is
performed. A pair of dimensionless parameters govern bifurcations from a flat,
circular ground state. While one of these is familiar from the Euler-Plateau
problem, the other encompasses information about the ratio of the twisting to
bending rigidities, twist, and size of the loop. For sufficiently small values
of the latter parameter, two separate groups of bifurcation modes are
identified. On the other hand, for values greater than the critical twist
arising in Michell's problem of the bifurcation of a twisted annular ring, only
one bifurcation mode exists. Bifurcation diagrams indicate that a loop with
greater twisting rigidity shows more resistance to transverse buckling.
However, a twisted and closed filament spanned by a surface endowed with
uniform surface tension buckles at a twist less than the critical value for an
elastic ring.
