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Motivated by recently discovered unusual properties of bulk nematicelastomers, we study a phase diagram of liquid-crystalline polymerized phantommembranes, focusing on in-plane nematic order. We predict that such membranesshould enerically exhibit five phases, distinguished by their conformationaland in-plane orientational properties, namely isotropic-crumpled,nematic-crumpled, isotropic-flat, nematic-flat and nematic-tubule phases. Inthe nematic-tubule phase, the membrane is extended along the direction of {\emspontaneous} nematic order and is crumpled in the other. The associatedspontaneous symmetries breaking guarantees that the nematic-tubule ischaracterized by a conformational-orientational soft (Goldstone) mode and theconcomitant vanishing of the in-plane shear modulus. We show that long-rangeorientational order of the nematic-tubule is maintained even in the presence ofharmonic thermal luctuations. However, it is likely that tubule's elasticproperties are ualitatively modified by these fluctuations, that can be studiedusing a nonlinear elastic theory for the nematic tubule phase that we derive atthe end of this paper.Comment: 12 pages, 4 eps figures. To appear in PR

Phases and transitions in phantom nematic elastomer membranes