NDLI: Triangulations and Volume Form on Moduli Spaces of Flat Surfaces

Content Provider	Springer Nature Link
Author	Nguyen, Duc Manh
Copyright Year	2010
Abstract	In this paper, we study the moduli spaces of flat surfaces with cone singularities verifying the following property: there exists a union of disjoint geodesic tree on the surface such that the complement is a translation surface. Those spaces can be viewed as deformations of the moduli spaces of translation surfaces in the space of flat surfaces. We prove that such spaces are quotients of flat complex affine manifolds by a group acting properly discontinuously, and preserving a parallel volume form. Translation surfaces can be considered as a special case of flat surfaces with erasing forest, in this case, it turns out that our volume form coincides with the usual volume form (which are defined via the period mapping) up to a multiplicative constant. We also prove similar results for the moduli space of flat metric structures on the n-punctured sphere with prescribed cone angles up to homothety. When all the angles are smaller than 2π, it is known (cf. [T]) that this moduli space is a complex hyperbolic orbifold. In this particular case, we prove that our volume form induces a volume form which is equal to the complex hyperbolic volume form up to a multiplicative constant.
Ending Page	228
Page Count	37
Starting Page	192
File Format	PDF
ISSN	1016443X
e-ISSN	14208970
Journal	Geometric and Functional Analysis
Issue Number	1
Volume Number	20
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2010-03-10
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Flat surface moduli space translation surface Analysis Conformal differential geometry Duality Topological nonlinear incidence structures
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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