Scalar curvature and the multiconformal class of a direct product Riemannian manifold
- For a closed, connected direct product Riemannian manifold (M, g) = (M-1, g(1)) x ... x (M-l, g(l)), we define its multiconformal class [[g]] as the totality {integral(2)(1)g(1) circle plus center dot center dot center dot integral(2)(l)g(l)} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a positive function fi on the total space M. A multiconformal class [[ g]] contains not only all warped product type deformations of g but also the whole conformal class [(g) over tilde] of every (g) over tilde is an element of[[ g]]. In this article, we prove that [[g]] contains a metric of positive scalar curvature if and only if the conformal class of some factor (Mi, gi) does, under the technical assumption dim M-i = 2. We also show that, even in the case where every factor (M-i, g(i)) has positive scalar curvature, [[g]] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided l = 2 and dim M = 3.
Author details: | Saskia RoosORCiD, Nobuhiko Otoba |
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DOI: | https://doi.org/10.1007/s10711-021-00636-9 |
ISSN: | 0046-5755 |
ISSN: | 1572-9168 |
Title of parent work (English): | Geometriae dedicata |
Publisher: | Springer |
Place of publishing: | Dordrecht |
Publication type: | Article |
Language: | English |
Date of first publication: | 2021/07/06 |
Publication year: | 2021 |
Release date: | 2023/10/02 |
Tag: | Constant scalar curvature; Positive scalar curvature; The Yamabe; Twisted product; Umbilic product; Warped product; problem |
Volume: | 214 |
Issue: | 1 |
Number of pages: | 29 |
First page: | 801 |
Last Page: | 829 |
Funding institution: | DFG (Deutsche Forschungsgemeinschaft)German Research Foundation (DFG) [SFB 1085]; Hausdorff Center for Mathematics in Bonn |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access / Hybrid Open-Access |
License (German): | CC-BY - Namensnennung 4.0 International |