Adiabatic limits and spectral geometry of foliations

Let (M, F) be a closed foliated manifold, dimM = n, dimF = p, p+q = n, equipped with a Riemannian metric gM . We assume that the foliation F is Riemannian, and the metric gM is bundle-like. Let F = TF be the integrable distribution of tangent p-planes to F , and H = F⊥ be the orthogonal complement to F . The decomposition of TM into a direct sum, TM = F ⊕ H , induces a decomposition of the metric gM : gM = gF + gH . For any h > 0, let ∆h, h > 0, be the Laplace operator on differential forms defined by a metric gh on M , given by the formula gh = gF + h−2gH . The operator ∆h is an elliptic differential operator with the positive definite, scalar principal symbol, which is self-adjoint and has discrete spectrum in the Hilbert space L2(M, ΛT ∗M, gh). The main result of the paper is an asymptotical formula for the eigenvalue distribution function Nh(λ) of the operator ∆h: