Geometric Rigidity via Harmonic Twisted Spinors

Mathematics > Differential Geometry [Submitted on 17 Jun 2026] Title:Geometric Rigidity via Harmonic Twisted Spinors View PDF HTML (experimental)Abstract:We study Gromov's exact-lift two-form method in scalar-curvature geometry. For a closed Riemannian spin manifold carrying a homologically non-trivial closed two-form whose lift to the universal cover is exact, we prove the sharp hyperbolic scalar-curvature comparison with the bottom of the spectrum of the universal Riemannian covering. The two-form enters through Gromov's twisted \(L^2\)-index, which produces harmonic spinors for a family of small unitary twists. We analyze the equality case by interpreting the refined Kato equality defect conformally and use the harmonic spinors to construct a parallel spinor with respect to a suitable conformally related metric. This yields that the original metric is Einstein. In the positive-spectrum case, this method implies that the universal cover is real hyperbolic. Current browse context: math.DG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) 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.