Triangulations and Volume Form on Moduli Spaces of Flat Surfaces
Triangulations and Volume Form on Moduli Spaces of Flat Surfaces
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DOI：
10.1007/s00039-010-0056-9
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发表时间：
2010-02
影响因子：
2.2
通讯作者：
Duc-Manh Nguyen
中科院分区：
文献类型：
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作者：
Duc-Manh Nguyen
在本文中，我们研究了具有锥奇点的平坦表面的模空间，验证了以下性质：表面上存在不相交测地树的并集，使得补集是平移表面。这些空间可以被视为平移表面的模空间在平面空间中的变形。我们证明这样的空间是由一组适当不连续地作用的平面复仿射流形的商，并保持平行体积形式。平移表面可以被认为是具有擦除森林的平面的特殊情况，在这种情况下，事实证明我们的体积形式与通常的体积形式（通过周期映射定义）在乘法常数范围内一致。我们还证明了具有规定锥角至位似的 n 穿刺球上平面度量结构的模空间的类似结果。当所有角度都小于 2π 时，已知（参见[T]）该模空间是复双曲轨道折叠。在这种特殊情况下，我们证明我们的体积形式导出了一个等于复双曲体积形式直至乘法常数的体积形式。
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.