Neck-Pinching of $CP^1$-structures in the $PSL(2,C)$-character variety
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866929593787613184 |
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| author | Baba, Shinpei |
| author_facet | Baba, Shinpei |
| contents | We characterize a certain neck-pinching degeneration of (marked) $CP^1$- structures on a closed oriented surface S of genus at least two. Namely, we consider a path $C_t$ of $CP^1$-structures on S leaving every compact subset in the deformation space of (marked) $CP^1$-structures on S, such that its holonomy converges in the PSL(2, C)-character variety. In this setting, it is known that the complex structure $X_t$ of $C_t$ also leaves every compact subset in the Teichmüller space.
In this paper, under the assumption that $X_t$ is pinched along a single loop m, we describe the limit of $C_t$ in terms of the developing maps, holomorphic quadratic differentials, and pleated surfaces.
Moreover, we give an example of such a path $C_t$ whose limit holonomy is the trivial representation in the character variety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_00092 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Neck-Pinching of $CP^1$-structures in the $PSL(2,C)$-character variety Baba, Shinpei Geometric Topology Differential Geometry We characterize a certain neck-pinching degeneration of (marked) $CP^1$- structures on a closed oriented surface S of genus at least two. Namely, we consider a path $C_t$ of $CP^1$-structures on S leaving every compact subset in the deformation space of (marked) $CP^1$-structures on S, such that its holonomy converges in the PSL(2, C)-character variety. In this setting, it is known that the complex structure $X_t$ of $C_t$ also leaves every compact subset in the Teichmüller space. In this paper, under the assumption that $X_t$ is pinched along a single loop m, we describe the limit of $C_t$ in terms of the developing maps, holomorphic quadratic differentials, and pleated surfaces. Moreover, we give an example of such a path $C_t$ whose limit holonomy is the trivial representation in the character variety. |
| title | Neck-Pinching of $CP^1$-structures in the $PSL(2,C)$-character variety |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/1907.00092 |