A Schottky decomposition theorem for complex projective structures

TL;DR

This paper proves that any complex projective structure on a closed surface can be decomposed into simpler parts with well-behaved geometric properties, connecting to known construction methods.

Contribution

It establishes a decomposition theorem for complex projective structures, demonstrating they can be assembled from components with injective developing maps and discrete holonomy, extending previous work.

Findings

Decomposition into pairs of pants and cylinders with injective developing maps

Every projective structure can be constructed via Gallo, Kapovich, and Marden's method

Existence of admissible loops for grafting on (S, C)

Abstract

Let S be a closed orientable surface of genus at least two, and let C be an arbitrary (complex) projective structure on S. We show that there is a decomposition of S into pairs of pants and cylinders such that the restriction of C to each component has an injective developing map and a discrete and faithful holonomy representation. This decomposition implies that every projective structure can be obtained by the construction of Gallo, Kapovich, and Marden. Along the way, we show that there is an admissible loop on (S, C), along which a grafting can be done.