Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds

[2402.02398] Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2402.02398 (math) [Submitted on 4 Feb 2024 ( v1 ), last revised 11 Apr 2026 (this version, v4)] Title: Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds Authors: Pengshuai Shi View a PDF of the paper titled Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds, by Pengshuai Shi View PDF HTML (experimental) Abstract: On complete non-compact manifolds with bounded sectional curvature, we consider a class of self-adjoint Dirac-type operators called Dirac-Schrödinger operators. Assuming two Dirac-Schrödinger operators coincide at infinity, by previous work, one can define their relative eta invariant. A typical example of Dirac-Schrödinger operators is the (twisted) spin Dirac operators on spin manifolds which admit a Riemannian metric of uniformly positive scalar curvature. In this case, using the relative eta invariant, we get a geometric formula for the spectral flow on non-compact manifolds, which induces a new proof of Gromov-Lawson's result about compact area enlargeable manifolds in odd dimensions. When two such spin Dirac operators are the boundary restriction of an operator on a manifold with non-compact boundary, under certain conditions, we obtain an index formula involving the relative eta invariant. This generalizes the Atiyah-Patodi-Singer index theorem to non-compact boundary situation. As a result, we can use the relative eta invariant to study the space of uniformly positive scalar curvature metrics on some non-compact connected sums. Comments: Theorem 1.2 was strengthened with a different proof (in Section 4), miscellaneous other revisions. Published version Subjects: Differential Geometry (math.DG) Cite as: arXiv:2402.02398 [math.DG] (or arXiv:2402.02398v4 [math.DG] for this version) https://doi.org/10.48550/arXiv.2402.02398 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1090/tran/9602 Focus to learn more DOI(s) linking to related resources Submission history From: Pengshuai Shi [ view email ] [v1] Sun, 4 Feb 2024 08:23:12 UTC (31 KB) [v2] Wed, 14 Feb 2024 11:19:18 UTC (31 KB) [v3] Wed, 29 Jan 2025 09:33:02 UTC (31 KB) [v4] Sat, 11 Apr 2026 14:06:36 UTC (37 KB) Full-text links: Access Paper: View a PDF of the paper titled Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds, by Pengshuai Shi View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2024-02 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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