[{"page":"467-503","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.","department":[{"_id":"LaEr"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"date_created":"2024-09-22T22:01:43Z","file_date_updated":"2025-01-13T08:51:40Z","abstract":[{"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.","lang":"eng"}],"file":[{"file_size":659100,"checksum":"650fe115d998fe0ac3a8d0c7519447c8","access_level":"open_access","date_created":"2025-01-13T08:51:40Z","date_updated":"2025-01-13T08:51:40Z","relation":"main_file","file_name":"2025_ErgodicTheory_Henheik.pdf","content_type":"application/pdf","file_id":"18828","creator":"dernst","success":1}],"article_type":"original","title":"Deformational rigidity of integrable metrics on the torus","author":[{"first_name":"Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","orcid":"0000-0003-1106-327X","last_name":"Henheik","full_name":"Henheik, Sven Joscha"}],"citation":{"ista":"Henheik SJ. 2025. Deformational rigidity of integrable metrics on the torus. Ergodic Theory and Dynamical Systems. 45(2), 467–503.","short":"S.J. Henheik, Ergodic Theory and Dynamical Systems 45 (2025) 467–503.","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>","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>.","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>.","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>","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."},"oa":1,"scopus_import":"1","oa_version":"Published Version","OA_place":"publisher","year":"2025","has_accepted_license":"1","intvolume":"        45","ec_funded":1,"publisher":"Cambridge University Press","isi":1,"related_material":{"record":[{"status":"public","id":"19540","relation":"dissertation_contains"}]},"project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","grant_number":"101020331"},{"name":"Spectral rigidity and integrability for billiards and geodesic flows","call_identifier":"H2020","grant_number":"885707","_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A"}],"publication_identifier":{"eissn":["1469-4417"],"issn":["0143-3857"]},"volume":45,"day":"01","language":[{"iso":"eng"}],"publication_status":"published","corr_author":"1","publication":"Ergodic Theory and Dynamical Systems","external_id":{"isi":["001308182000001"]},"doi":"10.1017/etds.2024.48","month":"02","ddc":["510"],"issue":"2","quality_controlled":"1","date_updated":"2026-04-07T12:37:10Z","_id":"18112","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","OA_type":"hybrid","date_published":"2025-02-01T00:00:00Z","article_processing_charge":"Yes (via OA deal)"}]