The Existence of Non-Equivariant Gromov Tori

[2608.14688] The Existence of Non-Equivariant Gromov Tori Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2608.14688 (math) [Submitted on 6 Aug 2026] Title: The Existence of Non-Equivariant Gromov Tori Authors: Grigoriy Yakovlev View a PDF of the paper titled The Existence of Non-Equivariant Gromov Tori, by Grigoriy Yakovlev View PDF HTML (experimental) Abstract: In this paper, we address the following question: if a flat torus $\mathbb{T}^n$ is isometrically and minimally embedded into a sphere $\mathbb{S}^N$, must its translation group extend to the isometry group of the ambient sphere? As shown by Robert Bryant, for $n=2$ the answer is positive. Furthermore, while Ying Lu, Peng Wang, and Zhenxiao Xie recently demonstrated that the answer is negative for immersions when $n \geq 3$, the question for embeddings remained open. This problem is deeply tied to the work of Mikhail Gromov and Anton Petrunin concerning optimal curvature bounds. Petrunin proved that any immersion of a torus into a unit ball must have a maximum normal curvature of at least $\sqrt{\frac{3n}{n+2}}$. This bound is attained, for example, by families of tori constructed by Gromov. We call the tori that attain this optimal bound "Gromov tori". In this work, we first demonstrate that any Gromov torus is intrinsically flat, lies within a sphere, and is minimal inside it. We then establish the necessary and sufficient conditions for defining these tori. Finally, we present our main result: for dimensions $n \ge 3$, there exists a non-equivariant embedded Gromov torus, which provides a definitive negative answer to the question above. Comments: 17 pages Subjects: Differential Geometry (math.DG) MSC classes: 53C42, 53C40 Cite as: arXiv:2608.14688 [math.DG] (or arXiv:2608.14688v1 [math.DG] for this version) https://doi.org/10.48550/arXiv.2608.14688 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Grigoriy Yakovlev [ view email ] [v1] Thu, 6 Aug 2026 17:58:28 UTC (16 KB) Full-text links: Access Paper: View a PDF of the paper titled The Existence of Non-Equivariant Gromov Tori, by Grigoriy Yakovlev View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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