Adiabatic Limits and Spectral Geometry of Foliations

TL;DR

This paper investigates the spectral asymptotics of the Laplace operator on differential forms in Riemannian foliated manifolds under adiabatic limits, deriving eigenvalue distribution formulas and exploring connections with noncommutative geometry.

Contribution

It provides a new asymptotic formula for eigenvalue distribution of the Laplacian in the adiabatic limit on foliated manifolds, linking spectral theory and noncommutative geometry.

Findings

Derived asymptotic eigenvalue distribution formula

Established relationships with leafwise Laplacian spectral theory

Connected spectral geometry with noncommutative geometry of foliations

Abstract

We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundle-like metric in the case when the metric is blown up in directions normal to the leaves of the foliation. The asymptotical formula for the eigenvalue distribution function is obtained. The relationships with the spectral theory of leafwise Laplacian and with the noncommutative spectral geometry of foliations are discussed.