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We focus on a special class of $D$-dimensional ``warped membranes'' embedded in a $d-$dimensional space with $d\\ge D+1$ and a preferred height profile characterized by quenched random Gaussian variables $\\{h_\\alpha({\\bf q})\\}$, $\\alpha=D+1,\\ldots, d$, in Fourier space with zero mean and a power law variance $\\overline{ h_\\alpha({\\bf q}_1) h_\\beta({\\bf q}_2) } \\sim \\delta_{\\alpha, \\beta} \\, \\delta_{{\\bf q}_1, -{\\bf q}_2} \\, q_1^{-d_h}$. 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