Adiabatic limits and spectral sequences for Riemannian foliations

TL;DR

The paper analyzes the adiabatic limit of the Laplacian on differential forms over a closed manifold endowed with a foliation, by blowing up transverse directions. It connects small eigenvalues to the foliation’s differentiable spectral sequence $(E_k,d_k)$, and develops an $L^2$ spectral sequence $(\mathbf{E}_k,\mathbf{d}_k)$ that is isomorphic to $E_k$ for $k\ge2$ in the Riemannian case, enabling a Hodge-theoretic description via a nested sequence of spaces $\mathcal{H}_k$. A key result provides sharp bounds $\lambda_i^r(h)=O(h^{2k})$ for $i\le m_k^r$ and relates the growth of small eigenvalues to the dimensions of $E_k^r$, with a variational formula for the spectral distribution function guiding the analysis. The work extends Mazzeo–Melrose and Forman’s findings to arbitrary Riemannian foliations, using a combination of leafwise Hodge theory, $L^2$-methods, and direct-sum decompositions of spectral sequences. Altogether, the paper ties adiabatic spectral limits to both the topology of foliations and noncommutative geometric structures via a robust Hodge-theoretic framework.

Abstract

For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The number of ``small'' eigenvalues is given in terms of the differentiable spectral sequence of the foliation. The asymptotics of the corresponding eigenforms also leads to a Hodge theoretic description of this spectral sequence. This is an extension of results of Mazzeo-Melrose and R. Forman.

Theorems & Definitions (110)

• Theorem A
• Theorem B
• Corollary C
• Theorem D
• Theorem E
• Theorem 2.1: Álvarez-Kordyukov AlvKordy1
• Theorem 2.2: Álvarez-Kordyukov AlvKordy1
• Lemma 2.3
• proof
• Lemma 2.4