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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 =4-D expansion about the upper critical dimension Dₔ₂=4 is performed; exponents characterizing the renormalization of the Lam\'e coefficients and, and the rigidity k, are computed to O () ; and exact equations connecting these exponents are derived. Near D=4, fluctuations increase the rigidity, and so tend to stabilize the flat phase.
Aronovitz et al. (Mon,) studied this question.