Geometric singularities and Hodge theory
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912555618795520 |
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| author | Bandara, Lashi Habib, Georges |
| author_facet | Bandara, Lashi Habib, Georges |
| contents | We consider smooth vector bundles over smooth manifolds equipped with non-smooth geometric data. For nilpotent differential operators acting on these bundles, we show that the kernels of induced Hodge-Dirac-type operators remain isomorphic under uniform perturbations of the geometric data. We consider applications of this to the Hodge-Dirac operator on differential forms induced by so-called rough Riemannian metrics, which can be of only measurable coefficient in regularity, on both compact and non-compact settings. As a consequence, we show that the kernel of the associated non-smooth Hodge-Dirac operator with respect to a rough Riemannian metric remains isomorphic to smooth and singular cohomology when the underlying manifold is compact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01170 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric singularities and Hodge theory Bandara, Lashi Habib, Georges Differential Geometry Functional Analysis Primary: 47B02, 46C07, 58A14, 57R57, Secondary: 47A08, 58A10 We consider smooth vector bundles over smooth manifolds equipped with non-smooth geometric data. For nilpotent differential operators acting on these bundles, we show that the kernels of induced Hodge-Dirac-type operators remain isomorphic under uniform perturbations of the geometric data. We consider applications of this to the Hodge-Dirac operator on differential forms induced by so-called rough Riemannian metrics, which can be of only measurable coefficient in regularity, on both compact and non-compact settings. As a consequence, we show that the kernel of the associated non-smooth Hodge-Dirac operator with respect to a rough Riemannian metric remains isomorphic to smooth and singular cohomology when the underlying manifold is compact. |
| title | Geometric singularities and Hodge theory |
| topic | Differential Geometry Functional Analysis Primary: 47B02, 46C07, 58A14, 57R57, Secondary: 47A08, 58A10 |
| url | https://arxiv.org/abs/2407.01170 |