Spectral theory of a class of nilmanifolds attached to Clifford modules

Wolfram Bauer; Kenro Furutani; Chisato Iwasaki; Abdellah Laaroussi

doi:10.1007/s00209-020-02525-5

Spectral theory of a class of nilmanifolds attached to Clifford modules

Wolfram Bauer^*
, Kenro Furutani
, Chisato Iwasaki
, Abdellah Laaroussi

^*Corresponding author for this work

Institute of Analysis

Research output: Contribution to journal › Article › Research › peer review

Abstract

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic 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-homeomorphic 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.

Original language	English
Pages (from-to)	557-583
Number of pages	27
Journal	Mathematische Zeitschrift
Volume	297
Issue number	1-2
DOIs	https://doi.org/10.1007/s00209-020-02525-5
Publication status	Published - 3 Apr 2020

Keywords

Heat kernel
Isospectral
Pseudo H-type group
Sub-Laplacian
Subriemannian manifold

ASJC Scopus subject areas

General Mathematics

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10.1007/s00209-020-02525-5Licence: CC BY 4.0

Cite this

@article{17793c8f801c4bdb81317eecd489d5e1,

title = "Spectral theory of a class of nilmanifolds attached to Clifford modules",

abstract = "We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic 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-homeomorphic 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.",

keywords = "Heat kernel, Isospectral, Pseudo H-type group, Sub-Laplacian, Subriemannian manifold",

author = "Wolfram Bauer and Kenro Furutani and Chisato Iwasaki and Abdellah Laaroussi",

note = "Funding Information: The first and the last named author have been supported by the priority program SPP 2026 geometry at infinity of Deutsche Forschungsgemeinschaft (project number BA 3793/6-1), the second named author was supported by the Grant-in-aid for Scientific Research (C) No. 17K05284, JSPS; the third named author was supported by the Grant-in-aid for Scientific Research (C) No. 24540189, JSPS.",

year = "2020",

month = apr,

day = "3",

doi = "10.1007/s00209-020-02525-5",

language = "English",

volume = "297",

pages = "557--583",

journal = "Mathematische Zeitschrift",

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T1 - Spectral theory of a class of nilmanifolds attached to Clifford modules

AU - Bauer, Wolfram

AU - Furutani, Kenro

AU - Iwasaki, Chisato

AU - Laaroussi, Abdellah

N1 - Funding Information: The first and the last named author have been supported by the priority program SPP 2026 geometry at infinity of Deutsche Forschungsgemeinschaft (project number BA 3793/6-1), the second named author was supported by the Grant-in-aid for Scientific Research (C) No. 17K05284, JSPS; the third named author was supported by the Grant-in-aid for Scientific Research (C) No. 24540189, JSPS.

PY - 2020/4/3

Y1 - 2020/4/3

N2 - We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic 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-homeomorphic 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.

AB - We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic 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-homeomorphic 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.

KW - Heat kernel

KW - Isospectral

KW - Pseudo H-type group

KW - Sub-Laplacian

KW - Subriemannian manifold

UR - https://www.scopus.com/pages/publications/85083167178

U2 - 10.1007/s00209-020-02525-5

DO - 10.1007/s00209-020-02525-5

M3 - Article

AN - SCOPUS:85083167178

SN - 0025-5874

VL - 297

SP - 557

EP - 583

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1-2

ER -

Spectral theory of a class of nilmanifolds attached to Clifford modules

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Spectral Analysis of Sub-Riemannian Structures (project in the Priority Programme 2026: Geometry at infinity)

Cite this

Spectral theory of a class of nilmanifolds attached to Clifford modules

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Projects

Spectral Analysis of Sub-Riemannian Structures (project in the Priority Programme 2026: Geometry at infinity)

Cite this