---
OA_place: publisher
OA_type: hybrid
_id: '18112'
abstract:
- lang: eng
  text: 'It is conjectured that the only integrable metrics on the two-dimensional
    torus are Liouville metrics. In this paper, we study a deformative version of
    this conjecture: we consider integrable deformations of a non-flat Liouville metric
    in a conformal class and show that for a fairly large class of such deformations,
    the deformed metric is again Liouville. The principal idea of the argument is
    that the preservation of rational invariant tori in the foliation of the phase
    space forces a linear combination on the Fourier coefficients of the deformation
    to vanish. Showing that the resulting linear system is non-degenerate will then
    yield the claim. Since our method of proof immediately carries over to higher
    dimensional tori, we obtain analogous statements in this more general case. To
    put our results in perspective, we review existing results about integrable metrics
    on the torus.'
acknowledgement: I am very grateful to Vadim Kaloshin for suggesting the topic, his
  guidance during this project, and many helpful comments on an earlier version of
  the manuscript. Moreover, I would like to thank Comlan Edmond Koudjinan and Volodymyr
  Riabov for interesting discussions. Partial financial support by the ERC Advanced
  Grant ‘RMTBeyond’ No. 101020331 is gratefully acknowledged. This project received
  funding from the European Research Council (ERC) ERC Grant No. 885707.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Sven Joscha
  full_name: Henheik, Sven Joscha
  id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
  last_name: Henheik
  orcid: 0000-0003-1106-327X
citation:
  ama: Henheik SJ. Deformational rigidity of integrable metrics on the torus. <i>Ergodic
    Theory and Dynamical Systems</i>. 2025;45(2):467-503. doi:<a href="https://doi.org/10.1017/etds.2024.48">10.1017/etds.2024.48</a>
  apa: Henheik, S. J. (2025). Deformational rigidity of integrable metrics on the
    torus. <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University Press.
    <a href="https://doi.org/10.1017/etds.2024.48">https://doi.org/10.1017/etds.2024.48</a>
  chicago: Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on
    the Torus.” <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University
    Press, 2025. <a href="https://doi.org/10.1017/etds.2024.48">https://doi.org/10.1017/etds.2024.48</a>.
  ieee: S. J. Henheik, “Deformational rigidity of integrable metrics on the torus,”
    <i>Ergodic Theory and Dynamical Systems</i>, vol. 45, no. 2. Cambridge University
    Press, pp. 467–503, 2025.
  ista: Henheik SJ. 2025. Deformational rigidity of integrable metrics on the torus.
    Ergodic Theory and Dynamical Systems. 45(2), 467–503.
  mla: Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the
    Torus.” <i>Ergodic Theory and Dynamical Systems</i>, vol. 45, no. 2, Cambridge
    University Press, 2025, pp. 467–503, doi:<a href="https://doi.org/10.1017/etds.2024.48">10.1017/etds.2024.48</a>.
  short: S.J. Henheik, Ergodic Theory and Dynamical Systems 45 (2025) 467–503.
corr_author: '1'
date_created: 2024-09-22T22:01:43Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2026-04-07T12:37:10Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1017/etds.2024.48
ec_funded: 1
external_id:
  isi:
  - '001308182000001'
file:
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  creator: dernst
  date_created: 2025-01-13T08:51:40Z
  date_updated: 2025-01-13T08:51:40Z
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intvolume: '        45'
isi: 1
issue: '2'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '02'
oa: 1
oa_version: Published Version
page: 467-503
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
  call_identifier: H2020
  grant_number: '885707'
  name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
  eissn:
  - 1469-4417
  issn:
  - 0143-3857
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
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    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Deformational rigidity of integrable metrics on the torus
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 45
year: '2025'
...