Ambient geometry via min-max widths of embedded circles

[2607.12979] Ambient geometry via min-max widths of embedded circles Skip to main content Search Submit Donate Log in Mathematics > Differential Geometry arXiv:2607.12979 (math) [Submitted on 14 Jul 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: Ambient geometry via min-max widths of embedded circles Authors: Lucas Ambrozio , Ivan Miranda , Rafael Montezuma View PDF HTML (experimental) Abstract: We prove a lower bound for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles. The main tool is a method to induce a sweepout by pairs of points in an embedded circle from a given sweepout of the sphere by closed curves, so that the points of each pair of the induced sweepout lie close to two curves of the ambient sweepout that are close to each other. Moreover, considering a non-compact complete Riemannian manifold, we relate upper bounds for the min-max width of its embedded circles to the existence of continuous functions from the manifold to finite graphs with level sets that have uniformly bounded diameters, thus giving an estimate for its 1-width in Urysohn's quantitative dimension theory. In the specific case of the Euclidean plane, we bound from below the classical width of a simple closed curve by its min-max width. Finally, we prove related rigidity results characterizing the round spheres among Zoll spheres. Comments: 32 pages, 1 figure. Minor revision, references added. Submitted version Subjects: Differential Geometry (math.DG) Cite as: arXiv:2607.12979 [math.DG] (or arXiv:2607.12979v2 [math.DG] for this version) https://doi.org/10.48550/arXiv.2607.12979 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Rafael Montezuma [ view email ] [v1] Tue, 14 Jul 2026 17:21:48 UTC (54 KB) [v2] Sat, 22 Aug 2026 12:34:39 UTC (56 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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