Flat grafting deformations of quadratic differentials on surfaces

TL;DR

This paper introduces flat grafting as a new deformation technique for quadratic differentials on surfaces, enabling path construction between different differentials and revealing rigidity properties related to curve lengths.

Contribution

It defines flat grafting for quadratic differentials, explores its properties, and demonstrates its role in connecting different differentials and analyzing their geometric structures.

Findings

Flat grafting maps are generic in the strata structure.

Flat grafting preserves parallel measured foliations.

Paths connecting any pair of quadratic differentials can be constructed using flat grafting.

Abstract

In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and preserve parallel measured foliations. We use flat grafting to construct paths connecting any pair of quadratic differentials. In other words, we characterize cone point splitting deformations. The slices of quadratic differentials closed under flat grafting maps with a fixed direction arise naturally and we prove rigidity properties with respect to the lengths of closed curves.