Mechanical Properties of Warped Membranes

We explore how a frozen background metric affects the mechanical properties of planar membranes with a shear modulus. We focus on a special class of "warped membranes" with a preferred random height profile characterized by random Gaussian variables $h({\bf q})$ in Fourier space with zero mean and variance $<|h({\bf q})|^2> \sim q^{-d_h}$ and show that in the linear response regime the mechanical properties depend dramatically on the system size $L$ for $d_h \ge 2$. Membranes with $d_h=4$ could be produced by flash polymerization of lyotropic smectic liquid crystals. Via a self consistent screening approximation we find that the renormalized bending rigidity increases as $\kappa_R \sim L^{(d_h-2)/2}$ for membranes of size $L$, while the Young and shear modulii decrease according to $Y_R,\ \mu_R \sim L^{-(d_h-2)/2}$ resulting in a universal Poisson ratio. Numerical results show good agreement with analytically determined exponents.