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

Abstract

We characterize a certain neck-pinching degeneration of (marked) $C{P}^1$- structures on a closed oriented surface S of genus at least two. Namely, we consider a path $C_t$ of $C{P}^1$-structures on S leaving every compact subset in the deformation space of (marked) $C{P}^1$-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.

To appear in Journal of Topology; 66 pages, 24 figures