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

## TL;DR

This paper analyzes the degeneration of $CP^1$-structures on surfaces, focusing on neck-pinching phenomena and their limits in the character variety, providing detailed descriptions and examples of trivial holonomy limits.

## Contribution

It characterizes neck-pinching degenerations of $CP^1$-structures on surfaces and describes their limits using developing maps, quadratic differentials, and pleated surfaces, including trivial holonomy cases.

## Key findings

- Describes limits of $CP^1$-structures under neck-pinching.
- Provides explicit examples with trivial holonomy.
- Connects degenerations to developing maps and pleated surfaces.

## Abstract

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 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 limit holonomy is the trivial representation in the character variety.