Annales de l’institut Fourier (AIF)

Coine, Clément

Le Merdy, Christian

Sukochev, Fedor

When do triple operator integrals take value in the trace class?

Annala, Toni M.

Oriented Borel–Moore homologies of toric varieties

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Formes modérément ramifiées de polydisques fermés et de dentelles

Wang, Yilin

A note on Loewner energy, conformal restriction and Werner’s measure on self-avoiding loops

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Phung, Trong-Thuc

Tô, Tât-Dat

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Drummond-Cole, Gabriel C.

Horel, Geoffroy

Homotopy transfer and formality

Guo, Hao

Hochs, Peter

Mathai, Varghese

Equivariant Callias index theory via coarse geometry

Gordon, Carolyn S.

Rossetti, Juan Pablo

Université Grenoble Alpes
ANNALES DE
L’INSTITUT FOURIER
Carolyn S. Gordon & Juan Pablo Rossetti
Corrigendum to “Boundary volume and length spectra of
Riemannian manifolds: What the middle degree Hodge
spectrum doesn’t reveal”
Tome 71, n
o
6 (2021), p. 2647-2648.
https://doi.org/10.5802/aif.3470
Article mis à disposition par ses auteurs selon les termes de la licence
Creative Commons attribution – pas de modification 3.0 France
http://creativecommons.org/licenses/by-nd/3.0/fr/
C EN T R E
MER S ENN E
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Centre Mersenne pour l’édition scientique ouverte
www.centre-mersenne.org e-ISSN : 1777-5310
Ann. Inst. Fourier, Grenoble
71, 6 (2021) 2647-2648
CORRIGENDUM TO “BOUNDARY VOLUME AND
LENGTH SPECTRA OF RIEMANNIAN MANIFOLDS:
WHAT THE MIDDLE DEGREE HODGE SPECTRUM
DOESN’T REVEAL”
Annales de l’institut Fourier, vol. 53 (2003), n°7, 2297–2314
by Carolyn S. GORDON & Juan Pablo ROSSETTI
Abstract. — We correct errors concerning the question of whether orbifolds
and manifolds can be distinguished by their spectra.
Résumé. — Nous corrigeons certaines erreurs concernant la question de savoir
si les orbifolds et les variétés peuvent être distingués au moyen de leurs spectres.
(i). — Two compact Riemannian manifolds or orbifolds are said to be
p-isospectral if their Hodge Laplacians acting on p-forms have the same
spectrum. Let O be the orbifold given by the quotient of the torus T =
Z2\R2 by the involution ρ induced from the map − Id of R2. Theorem 3.1
of [2] asserts that O is 1-isospectral to a cylinder, a Klein bottle and a
Möbius strip. As shown in the proof of Theorem 3.1, the 1-spectrum of
each of the latter three surfaces coincides with the 0-spectrum of T , while
the 1-spectrum of O consists of two copies of the spectrum of the Laplacian
∆T of T acting on the space of ρ-anti-invariant smooth functions on T .
The error is in the statement that the spectra of ∆T on the space of ρ anti-
invariant functions and the space of ρ-invariant functions coincide. They
in fact differ by the presence of the zero eigenvalue in the latter. Thus the
1-spectrum of the orbifold differs from that of the three surfaces precisely
by the absence of the zero eigenvalue.
The analogous error appears in the Remark following Theorem 3.2.
Keywords: isospectral, Laplacian, orbifold.
2020 Mathematics Subject Classification: 58J53, 53C20.
2648 Carolyn S. GORDON & Juan Pablo ROSSETTI
All other examples in [2] of 2m-dimensional orbifolds that arem-isospectral
to manifolds, in particular the examples in Theorem 3.2, remain valid. As
an aside, we note that one can enlarge the collection of manifolds and
orbifolds that are m-isospectral to those in Theorem 3.2(i) by taking the
quotient of Z2m\R2m by any of the involutions τk ◦ Lb, where τk is de-
fined as in the proof of the theorem and Lb is translation by an element of
( 12 Z)
2m not in Z2m.
(ii). — In the statement of Proposition 3.4(i), “strata” should be re-
placed by “primary stratum”. See [1] for the definition of “primary” and
further explanation. Proposition 3.4(ii) should read: If N is a manifold such
that O and N have a common finite Riemannian cover, then N and O can-
not be isospectral. The statement and proof also remain valid in the case of
a common infinite cover that is a Riemannian homogeneous space; indeed
homogeneity implies that the integrands yielding the invariants ak are con-
stants depending only on the geometry of the cover in a neighborhood of
an arbitrary point.
BIBLIOGRAPHY
[1] E. B. Dryden, C. S. Gordon, S. J. Greenwald & D. L. Webb, “Erratum to
“Asymptotic expansion of the heat kernel for orbifolds””, Mich. Math. J. 66 (2017),
no. 1, p. 221-222.
[2] C. S. Gordon & J. P. Rossetti, “Boundary volume and length spectra of Riemann-
ian manifolds: what the middle degree Hodge spectrum doesn’t reveal”, Ann. Inst.
Fourier 53 (2003), no. 7, p. 2297-2314.
Manuscrit reçu le 23 octobre 2020,
accepté le 7 janvier 2021.
Carolyn S. GORDON
Department of Mathematics
Dartmouth College
Hanover, NH 03755 (USA)
csgordon@dartmouth.edu
Juan Pablo ROSSETTI
FAMAF
Univ. Nac. Córdoba
5000-Córdoba (Argentina)
jprossetti@unc.edu.ar
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