Elastic Anisotropies and Long-Ranged Interactions in Solid Membranes

I study the effects of elastic anisotropy and long-ranged interactions on the fluctuations of solid membranes. Elastic anisotropies prove to be irrelevant in the absence of long-ranged interactions. Long-ranged interactions which fall off with distance r like r^-σ are relevant only for sufficiently small σ. In real (two-dimensional) membranes with dipolar interactions (σ=3), the Lam\'e coefficients μ and λ vanish like [ln(1/q)]^-1, as wave vector q→0. For larger but still relevant σ's, μ∝q^η_⊥∝λ with η_⊥=2σ-3D, exactly, for D-dimensional membranes.