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 $\epsilon =4-D$ expansion about the upper critical dimension $D_{\mathrm{uc}}=4\mathrm{}$ is performed; exponents characterizing the renormalization of the Lamé coefficients $\lambda$ and $\mu$, and the rigidity $k$, are computed to $O\left(\epsilon \right)$; and exact equations connecting these exponents are derived. Near $D=4\mathrm{}$, fluctuations increase the rigidity, and so tend to stabilize the flat phase.