Thermal excitations of warped membranes

We explore thermal fluctuations of thin planar membranes with a frozen spatially varying background metric and a shear modulus. We focus on a special class of D-dimensional warped membranes embedded in a d-dimensional space with d >= D + 1 and a preferred height profile characterized by quenched random Gaussian variables {h alpha (q)}, alpha = D + 1,..., d, in Fourier space with zero mean and a power-law variance <(h(alpha) (q(1)) h(beta) (q(2)))over bar> similar to delta(alpha,beta)delta(q1,-q2) q(1)(-dh). The case D = 2, d = 3, with d(h) = 4 could be realized by flash-polymerizing lyotropic smectic liquid crystals. For D < max{4, d(h)} the elastic constants are nontrivially renormalized and become scale dependent. Via a self-consistent screening approximation we find that the renormalized bending rigidity increases for small wave vectors q as K-R similar to q(-eta f), while the in-hyperplane elastic constants decrease according to lambda(R), mu(R) similar to q(+eta u). The quenched background metric is relevant (irrelevant) for warped membranes characterized by exponent d(h) > 4 - eta ((F))(f) (d(h) < 4 - eta((F))(f)), where eta((F))(f) is the scaling exponent for tethered surfaces with a flat background metric, and the scaling exponents are related through eta(u) + eta(f) = d(h) - D (eta(u) + 2 eta(f) = 4 - D).