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. In a more general setting, we take a path of $C{P}^1$ -structures $C_t(t⩾0)$ on $S$ that leaves every compact subset in its deformation space, such that the holonomy of $C_t$ converges in the ${\mathrm{PSL}}_{2}C$ -character variety as $t\to \infty$ . Then, it is well known that the complex structure $X_t$ of $C_t$ also leaves every compact subset in the Teichmüller space of $S$ . In this paper, under an additional assumption that $X_t$ is pinched along a loop $m$ on $S$ , we describe the limit of $C_t$ from different perspectives: namely, in terms of the developing maps, holomorphic quadratic differentials, and pleated surfaces. The holonomy representations of $C{P}^1$ -structures on $S$ are known to be nonelementary (i.e., strongly irreducible and unbounded). We also give a rather exotic example of such a path $C_t$ whose limit holonomy is the trivial representation.