{
  "id": "1907.00092",
  "version": "v1",
  "published": "2019-06-28T21:50:26.000Z",
  "updated": "2019-06-28T21:50:26.000Z",
  "title": "Neck-Pinching of $CP^1$-structures in the PSL(2,C)-character variety",
  "authors": [
    "Shinpei Baba"
  ],
  "comment": "53 pages, 22 figures",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "Let S be a closed oriented surface of genus at least two. We consider a path of $CP^1$-structures $C_t$ 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 Teichm\\\"uller 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 $C_t$ where the limit holonomy is the trivial representation in the character variety.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-28T21:50:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact subset",
      "holomorphic quadratic differentials",
      "neck-pinching",
      "character variety",
      "trivial representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}