Adiabatic limits on Riemannian Sol-manifolds

TL;DR

This paper derives an asymptotic formula for the spectrum distribution of the Laplace operator on compact Riemannian Sol-manifolds in the adiabatic limit, focusing on foliations generated by a left-invariant flow.

Contribution

It provides the first explicit asymptotic spectral formula for Laplacians on Sol-manifolds in the adiabatic limit, linking geometry and spectral theory.

Findings

Asymptotic spectrum distribution formula derived

Spectral behavior characterized in adiabatic limit

Connection established between foliation dynamics and spectrum

Abstract

We obtain an asymptotic formula for the spectrum distribution function of the Laplace operator on a compact Riemannian Sol-manifold in the adiabatic limit determined by a one-dimensional foliation defined by the orbits of a left-invariant flow.