  We present a model for smectic elastomer membranes which includes elastic and
liquid crystalline degrees of freedom. Based on our model, we determined the
qualitative phase diagram of a smectic elastomer membrane using mean-field
theory. This phase diagram is found to comprise five phases, viz.
smectic-A--flat, smectic-A--crumpled, smectic-C--flat, smectic-C--crumpled and
smectic-C--tubule, where in the latter phase, the membrane is flat in the
direction of mesogenic tilt and crumpled in the perpendicular direction. The
transitions between adjacent phases are second order phase transitions. We
study in some detail the elasticity of the smectic-C--flat and the
smectic-C--tubule phases which are associated with a spontaneous breaking of
in-plane rotational symmetry. As a consequence of the Goldstone theorem, these
phases exhibit soft elasticity characterized by the vanishing of in-plane shear
moduli.
