Fluctuations of Solid Membranes

The free energy of a D-dimensional elastic solid embedded in a d-dimensional space and subject to an extrinsic bending energy is defined, and the solid's flat phase studied. An ε=4-D expansion about the upper critical dimension Duc=4 is performed; exponents characterizing the renormalization of the Lam\'e coefficients λ and μ, and the rigidity k, are computed to O(ε); and exact equations connecting these exponents are derived. Near $D=4$, fluctuations increase the rigidity, and so tend to stabilize the flat phase.