Positive scalar curvature and an equivariant Callias-type index theorem for proper actions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866917762736062464 |
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| author | Guo, Hao Hochs, Peter Mathai, Varghese |
| author_facet | Guo, Hao Hochs, Peter Mathai, Varghese |
| contents | For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_07336 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Positive scalar curvature and an equivariant Callias-type index theorem for proper actions Guo, Hao Hochs, Peter Mathai, Varghese Differential Geometry K-Theory and Homology Operator Algebras For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption. |
| title | Positive scalar curvature and an equivariant Callias-type index theorem for proper actions |
| topic | Differential Geometry K-Theory and Homology Operator Algebras |
| url | https://arxiv.org/abs/2001.07336 |