Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces

[2412.12631] Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2412.12631 (math) [Submitted on 17 Dec 2024 ( v1 ), last revised 9 Apr 2026 (this version, v5)] Title: Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces Authors: Giovanni Catino , Luciano Mari , Paolo Mastrolia , Alberto Roncoroni View a PDF of the paper titled Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces, by Giovanni Catino and 3 other authors View PDF HTML (experimental) Abstract: In this paper we prove general criticality criteria for operators $\Delta + V$ on manifolds with more than one end, where $V$ bounds the Ricci curvature, and a related spectral splitting theorem extending Cheeger-Gromoll's one. Our results give new insight on Li-Wang's theory of manifolds with a weighted Poincaré inequality. We apply them to study stable and $\delta$-stable minimal hypersurfaces in manifolds with non-negative bi-Ricci or sectional curvature, in ambient dimension up to $5$ and $6$, respectively. In the special case where the ambient space is $\mathbb{R}^4$, we prove that a $1/3$-stable minimal hypersurface must either have one end or be a catenoid, and that proper, $\delta$-stable minimal hypersurfaces with $\delta > 1/3$ must be hyperplanes. Comments: 37pp. We strengthened a topological result, see Corollary 1.8 (which improves on previous Corollary 3.11, and corrects a typo in its statement). Package axessibility included to make the paper available to visually impaired people Subjects: Differential Geometry (math.DG) ; Analysis of PDEs (math.AP) Cite as: arXiv:2412.12631 [math.DG] (or arXiv:2412.12631v5 [math.DG] for this version) https://doi.org/10.48550/arXiv.2412.12631 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Luciano Mari [ view email ] [v1] Tue, 17 Dec 2024 07:46:11 UTC (54 KB) [v2] Wed, 8 Jan 2025 16:11:56 UTC (55 KB) [v3] Mon, 20 Jan 2025 17:41:57 UTC (58 KB) [v4] Thu, 6 Nov 2025 21:37:33 UTC (61 KB) [v5] Thu, 9 Apr 2026 08:15:37 UTC (63 KB) Full-text links: Access Paper: View a PDF of the paper titled Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces, by Giovanni Catino and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2024-12 Change to browse by: math math.AP 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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