Width, Largeness and Index Theory
In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands × [−1, 1], and on a conjecture o...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211092 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Width, Largeness and Index Theory. Rudolf Zeidler. SIGMA 16 (2020), 127, 15 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862608142433517568 |
|---|---|
| author | Zeidler, Rudolf |
| author_facet | Zeidler, Rudolf |
| citation_txt | Width, Largeness and Index Theory. Rudolf Zeidler. SIGMA 16 (2020), 127, 15 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands × [−1, 1], and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on × ℝ. We show that there is a more general geometric statement underlying both of them, implying a quantitative negative upper bound on the infimum of the scalar curvature of a complete metric on × ℝ if the scalar curvature is positive in some neighborhood. We study (A^-)iso-enlargeable spin manifolds and related notions of width for Riemannian manifolds from an index-theoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties, and width.
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| first_indexed | 2026-03-14T05:22:33Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211092 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T05:22:33Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zeidler, Rudolf 2025-12-23T13:13:41Z 2020 Width, Largeness and Index Theory. Rudolf Zeidler. SIGMA 16 (2020), 127, 15 pages 1815-0659 2020 Mathematics Subject Classification: 58J22; 19K56; 53C21; 53C23 arXiv:2008.13754 https://nasplib.isofts.kiev.ua/handle/123456789/211092 https://doi.org/10.3842/SIGMA.2020.127 In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands × [−1, 1], and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on × ℝ. We show that there is a more general geometric statement underlying both of them, implying a quantitative negative upper bound on the infimum of the scalar curvature of a complete metric on × ℝ if the scalar curvature is positive in some neighborhood. We study (A^-)iso-enlargeable spin manifolds and related notions of width for Riemannian manifolds from an index-theoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties, and width. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), ProjectID 427320536– SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics–Geometry–Structure. Moreover, part of the research pertaining to this article was conducted while the author was employed at the University of Göttingen, funded through the DFG RTG 2491 Fourier Analysis and Spectral Theory. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Width, Largeness and Index Theory Article published earlier |
| spellingShingle | Width, Largeness and Index Theory Zeidler, Rudolf |
| title | Width, Largeness and Index Theory |
| title_full | Width, Largeness and Index Theory |
| title_fullStr | Width, Largeness and Index Theory |
| title_full_unstemmed | Width, Largeness and Index Theory |
| title_short | Width, Largeness and Index Theory |
| title_sort | width, largeness and index theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211092 |
| work_keys_str_mv | AT zeidlerrudolf widthlargenessandindextheory |