A CP^1-structure on a surface is a certain locally homogeneous structure, and it can be regarded as a pair of a Riemann surface and a holomorphic quadratic differential (on it). A CP^1-structure also corresponds to a representation of the fundament group of the surface into PSL(2, C).

In this talk, we consider a one-parameter family of diverging CP^1-structures on a closed surface of genus at least two, and we describe its limit under the assumption that the family of their representations converges in the PSL(2, C)-character variety and that the family of their Riemann surface structures is asymptotically pinched along disjoint loops (in a coarse sense).