Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps
We consider a complete Riemannian manifold, which consists of a compact interior and one or more -cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here, is a function of the radial coordinate that describes the shape of such an end. Given an action by a com...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2023 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2023
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211920 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps. Peter Hochs and Hang Wang. SIGMA 19 (2023), 023, 32 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862567303498956800 |
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| author | Hochs, Peter Wang, Hang |
| author_facet | Hochs, Peter Wang, Hang |
| citation_txt | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps. Peter Hochs and Hang Wang. SIGMA 19 (2023), 023, 32 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider a complete Riemannian manifold, which consists of a compact interior and one or more -cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here, is a function of the radial coordinate that describes the shape of such an end. Given an action by a compact Lie group on such a manifold, we obtain an equivariant index theorem for Dirac operators, under conditions on . These conditions hold in the cases of cylindrical ends and hyperbolic cusps. In the case of cylindrical ends, the cusp contribution equals the delocalised -invariant, and the index theorem reduces to Donnelly's equivariant index theory on compact manifolds with boundary. In general, we find that the cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero.
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| first_indexed | 2026-03-13T10:15:08Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211920 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T10:15:08Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
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| spelling | Hochs, Peter Wang, Hang 2026-01-16T11:19:44Z 2023 Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps. Peter Hochs and Hang Wang. SIGMA 19 (2023), 023, 32 pages 1815-0659 2020 Mathematics Subject Classification: 58J20; 58D19 arXiv:2110.00390 https://nasplib.isofts.kiev.ua/handle/123456789/211920 https://doi.org/10.3842/SIGMA.2023.023 We consider a complete Riemannian manifold, which consists of a compact interior and one or more -cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here, is a function of the radial coordinate that describes the shape of such an end. Given an action by a compact Lie group on such a manifold, we obtain an equivariant index theorem for Dirac operators, under conditions on . These conditions hold in the cases of cylindrical ends and hyperbolic cusps. In the case of cylindrical ends, the cusp contribution equals the delocalised -invariant, and the index theorem reduces to Donnelly's equivariant index theory on compact manifolds with boundary. In general, we find that the cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero. We thank Mike Chen for a helpful discussion, and Christian B¨ar for pointing out a useful reference. We are grateful to the referees for several helpful comments and corrections. In particular, we thank the referee who pointed out an error in the previous version of [27], on which the current paper builds, which has since been fixed. PH is partially supported by the Australian Research Council, through the Discovery Project DP200100729. HW is supported by NSFC-11801178 and Shanghai Rising-Star Program 19QA1403200. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps Article published earlier |
| spellingShingle | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps Hochs, Peter Wang, Hang |
| title | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_full | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_fullStr | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_full_unstemmed | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_short | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_sort | spectral asymmetry and index theory on manifolds with generalised hyperbolic cusps |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211920 |
| work_keys_str_mv | AT hochspeter spectralasymmetryandindextheoryonmanifoldswithgeneralisedhyperboliccusps AT wanghang spectralasymmetryandindextheoryonmanifoldswithgeneralisedhyperboliccusps |