Multiple saddle connections on flat surfaces and the principal boundry of the moduli spaces of quadratic differentials

Howard Masur, Anton Zorich

Research output: Contribution to journalArticlepeer-review

27 Citations (Scopus)

Abstract

We describe typical degenerations of quadratic differentials thus describing "generic cusps" of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from "generic" degenerations is often negligible in problems involving information on compactification of the moduli space. However, even for a typical degeneration one may have several short loops on the Riemann surface which shrink simultaneously. We explain this phenomenon, describe all rigid configurations of short loops, present a detailed description of analogs of desingularized stable curves arising here, and show how one can reconstruct a Riemann surface endowed with a quadratic differential which is close to a "cusp" from the corresponding point at the principal boundary.

Original languageEnglish
Pages (from-to)919-987
Number of pages69
JournalGeometric and Functional Analysis
Volume18
Issue number3
DOIs
Publication statusPublished - Sep 2008
Externally publishedYes

Keywords

  • Flat structure
  • Moduli space of quadratic differentials
  • Saddle connection
  • Teichmüller geodesic flow
  • Thick part

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