Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature
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arXiv
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866913485731921920 |
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| author | Guo, Hao |
| author_facet | Guo, Hao |
| contents | We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that G-equivariant Callias-type operators are self-adjoint, regular, and hence equivariantly invertible at infinity. Such operators explicitly arise from a pairing of the Dirac operator with the equivariant Higson corona. We apply the theory developed herein to obtain an obstruction to positive scalar curvature metrics on non-cocompact manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_05558 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature Guo, Hao Differential Geometry K-Theory and Homology Operator Algebras 58B34 (Primary), 58J20, 19K35, 19L47, 19K56, 47C15 (Secondary) We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that G-equivariant Callias-type operators are self-adjoint, regular, and hence equivariantly invertible at infinity. Such operators explicitly arise from a pairing of the Dirac operator with the equivariant Higson corona. We apply the theory developed herein to obtain an obstruction to positive scalar curvature metrics on non-cocompact manifolds. |
| title | Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature |
| topic | Differential Geometry K-Theory and Homology Operator Algebras 58B34 (Primary), 58J20, 19K35, 19L47, 19K56, 47C15 (Secondary) |
| url | https://arxiv.org/abs/1803.05558 |