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Flat grafting deformations of quadratic differentials on surfaces

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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. The 1-parameter family obtained by flat grafting allows us to explicitly describe a path connecting any pair of quadratic differentials. 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.

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Acknowledgements

The author thanks Chris Leininger, Spencer Dowdall, Howard Masur, Jayadev Athreya, Matthew Stover, and many other people for helpful discussions over the development of the concept of flat grafting. Thanks to the anonymous referee whose comments improved the overall quality of the paper. The author would also like to thank UIUC, Temple University, and NTU for their excellent research environment.

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  1. National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan, ROC

    Ser-Wei Fu

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Fu, SW. Flat grafting deformations of quadratic differentials on surfaces. Geom Dedicata 214, 119–138 (2021). https://doi.org/10.1007/s10711-021-00607-0

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