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(q) in Fourier space with zero mean and variance ~q(-d(h)) and show that in the linear response regime the mechanical properties depend dramatically on the system size L for d(h)≥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 κ(R)~L((d(h)-2)/2) for membranes of size L, while the Young and shear moduli decrease according to Y(R),μ(R)~L(-(d(h)-2)/2) resulting in a universal Poisson ratio. Numerical results show good agreement with analytically determined exponents.