Neck-Pinching of $CP^1$-structures in the $PSL(2,C)$-character variety

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 $C_t$ whose the limit holonomy is the trivial representation in the character variety.Comment: 52 pages, 22 figure