Spectral theory of a class of nilmanifolds attached to clifford modules

TL;DR

This paper analyzes the spectral properties of sub-Laplacians on pseudo H-type nilmanifolds, revealing isospectral pairs that are not diffeomorphic and exploring heat trace expansions.

Contribution

It provides a method to construct multiple isospectral but non-diffeomorphic nilmanifolds and examines their spectral and heat trace properties.

Findings

Constructed pairs of isospectral non-diffeomorphic nilmanifolds.

Showed these pairs are also isospectral for the Laplacian.

Identified nilmanifolds with coinciding heat trace expansions up to infinite order.

Abstract

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-diffeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-diffeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.