Geometric Rigidity via Almost-Harmonic Twisted Spinors

[2606.19567] Geometric Rigidity via Almost-Harmonic Twisted Spinors Skip to main content Search Submit Donate Log in Mathematics > Differential Geometry arXiv:2606.19567 (math) [Submitted on 17 Jun 2026 ( v1 ), last revised 26 Aug 2026 (this version, v3)] Title: Geometric Rigidity via Almost-Harmonic Twisted Spinors Authors: Francesco Bei , Simone Cecchini View PDF HTML (experimental) Abstract: We establish sharp scalar-curvature bounds and rigidity consequences of Gromov's exact-lift two-form method. Let $(M^n,g)$, $n\geq 4$ even, be a closed spin Riemannian manifold carrying a homologically $\hat{A}$-non-singular closed two-form $\omega$ whose lift to the universal cover $X$ is exact. Then $$\inf_M\scal_g\leq -\frac{4n}{n-1}\lambda_0(X).$$ Equality forces $g$ to be Einstein; if $\lambda_0(X)>0$, then $X$ is real hyperbolic, while if $\lambda_0(X)=0$ and $\int_M\omega^{n/2}\neq 0$, then $g$ is flat. The proof combines Gromov's twisted $L^2$-index with a conformal interpretation of the refined Kato equality and a recentering argument. The same method yields untwisted rigidity results when zero belongs to the spectrum of the Dirac operator on the universal cover, with applications to nonvanishing$\widehat A$-genus and enlargeability. Comments: Improved exposition; Comments are welcome! Subjects: Differential Geometry (math.DG) ; Geometric Topology (math.GT) Cite as: arXiv:2606.19567 [math.DG] (or arXiv:2606.19567v3 [math.DG] for this version) https://doi.org/10.48550/arXiv.2606.19567 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Francesco Bei [ view email ] [v1] Wed, 17 Jun 2026 20:09:54 UTC (30 KB) [v2] Thu, 30 Jul 2026 14:24:24 UTC (39 KB) [v3] Wed, 26 Aug 2026 19:23:43 UTC (41 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-06 Change to browse by: math math.GT References & Citations NASA ADS 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) Major funding support from

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