Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912437382414336 |
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| author | Cecchini, Simone Hanke, Bernhard Schick, Thomas Schoenlinner, Lukas |
| author_facet | Cecchini, Simone Hanke, Bernhard Schick, Thomas Schoenlinner, Lukas |
| contents | Using the index theory for twisted Dirac operators acting on sections of Lipschitz bundles over non-compact manifolds, we prove Llarull-type comparison results in scalar curvature geometry. They apply to spin Riemannian manifolds with cone-type singularities and Lipschitz comparison maps to spheres.
We use the language of abstract cone operators which are introduced and studied in a general functional analytic setting and which may be of independent interest.
Applying our discussion to spherical suspensions of odd-dimensional closed manifolds, we generalize a Lipschitz rigidity result of the first three named authors from even to odd dimensions. Under stronger conditions, this has already been shown by Lee-Tam using geometric flows and by Baer using an upper estimate for the smallest Dirac eigenvalue. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_14054 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds Cecchini, Simone Hanke, Bernhard Schick, Thomas Schoenlinner, Lukas Differential Geometry Primary: 51F30, 53C23, 53C24, Secondary: 30C65, 53C27, 58J20 Using the index theory for twisted Dirac operators acting on sections of Lipschitz bundles over non-compact manifolds, we prove Llarull-type comparison results in scalar curvature geometry. They apply to spin Riemannian manifolds with cone-type singularities and Lipschitz comparison maps to spheres. We use the language of abstract cone operators which are introduced and studied in a general functional analytic setting and which may be of independent interest. Applying our discussion to spherical suspensions of odd-dimensional closed manifolds, we generalize a Lipschitz rigidity result of the first three named authors from even to odd dimensions. Under stronger conditions, this has already been shown by Lee-Tam using geometric flows and by Baer using an upper estimate for the smallest Dirac eigenvalue. |
| title | Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds |
| topic | Differential Geometry Primary: 51F30, 53C23, 53C24, Secondary: 30C65, 53C27, 58J20 |
| url | https://arxiv.org/abs/2505.14054 |