The renormalization of volume and Chern-Simons invariant for hyperbolic 3-manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912445229957120 |
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| author | Lee, Dongha |
| author_facet | Lee, Dongha |
| contents | We renormalize the Chern-Simons invariant for convex-cocompact hyperbolic 3-manifolds by finding the asymptotics along an equidistance foliation. We prove that the metric Chern-Simons invariant has an exponentially divergent term given by the integral of the torsion 2-form with respect to a Weitzenböck connection. This produces the asymptotics of hyperbolic volume plus the metric Chern-Simons invariant, which is often called complex volume. The leading coefficient of the asymptotics introduces a complex-valued quantity consisting of mean curvature and torsion 2-form, which is defined on smooth surfaces embedded in a Riemann-Cartan 3-manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04776 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The renormalization of volume and Chern-Simons invariant for hyperbolic 3-manifolds Lee, Dongha Differential Geometry 58J28, 57K32 We renormalize the Chern-Simons invariant for convex-cocompact hyperbolic 3-manifolds by finding the asymptotics along an equidistance foliation. We prove that the metric Chern-Simons invariant has an exponentially divergent term given by the integral of the torsion 2-form with respect to a Weitzenböck connection. This produces the asymptotics of hyperbolic volume plus the metric Chern-Simons invariant, which is often called complex volume. The leading coefficient of the asymptotics introduces a complex-valued quantity consisting of mean curvature and torsion 2-form, which is defined on smooth surfaces embedded in a Riemann-Cartan 3-manifold. |
| title | The renormalization of volume and Chern-Simons invariant for hyperbolic 3-manifolds |
| topic | Differential Geometry 58J28, 57K32 |
| url | https://arxiv.org/abs/2310.04776 |