On the Kähler form of the moduli space of once punctured tori
On the Kähler form of the moduli space of once punctured tori
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一次刺穿环面模空间的凯勒形式
DOI：
10.1007/bf02564634
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发表时间：
1983
影响因子：
0.9
通讯作者：
S. Wolpert
中科院分区：
文献类型：
--
作者：
S. Wolpert
具有负欧拉特征的黎曼曲面R具有唯一的双曲度量。如果R在这个度量中有有限的面积，则R的Teichmiiller空间T（R）将是一个复流形。T（R）的复结构通过描述R上的全纯余切空间来刻画。全纯余切空间与R上的全纯二次微分空间Q（R）之间存在一个自然的一致性。因此，Q（R）上的厄米特结构自然会产生T（R）上的厄米特结构。一个例子是彼得森内积。给定~ 0，~ B Q（R），定义（q~，0）= IR~ x-2，其中h2是R的双曲面积元. T（R）上相应的厄米特结构是Weil-Petersson度量的厄米特结构。度量关于Teichmiiller模群是不变的，因此可以用来研究R的模空间的几何。Ahlfors和Weil建立了度规为K~ ihler。我们讨论度规的K~ ihler形式。一个关系之间存在的几何形状的to和向量场来自一个建设的芬克尔-尼尔森。Teichmiiller空间上的一个Fenchel-Nielsen向量场t（ot）与R的每一个闭测地线ot相关联。在手稿[10]中，量to（t（~ t），t（/3））被评估为α和/3的相交角的余弦之和。证明了向量场t（a）是Harniltonian的，因为t0在t（o~）的流下是不变的.形式to和向量场t（ot）是T（R）的辛几何的元素。几何是自然的意义上说，是不变的Teichmiiller模群。T（R）与模群的商是R的经典模空间。将K~ ihler形式投影到模空间上。上述讨论的最简单示例是在一次穿孔环面的情况下提供的。在第一节中，我们描述自然全局坐标
A Riemann surface R of negative Euler characteristic has a unique hyperbolic metric. Provided R has finite area in this metric the Teichmiiller space T (R) of R will be a complex manifold. The complex structure of T (R) is characterized by describing the holomorphic cotangent space at R. A natural identification exists of the holomorphic cotangent space and Q (R), the space of holomorphic quadratic differentials on R. Consequently a Hermitian structure on Q (R) naturally gives rise to one on T (R). An example is the Petersson inner product. Given~ 0,~ b Q (R) define (q~, O)= IR~ x-2 where h 2 is the hyperbolic area element of R. The corresponding Hermitian structure on T (R) is that of the Weil-Petersson metric. The metric is invariant with respect to the Teichmiiller modular group and hence can be used to study the geometry of the moduli space of R. Ahlfors and Weil established that the metric is K~ ihler. We are concerned with the K~ ihler form to of the metric. A relationship exists between the geometry of to and that of the vector fields derived from a construction of Fenchel-Nielsen. A Fenchel-Nielsen vector field t (ot) on Teichmiiller space is associated to each closed geodesic ot of R. In the manuscript [10] the quantity to (t (~ t), t (/3)) is evaluated as the sum of the cosines of the intersection angles of ot and/3. It is also shown that the vector fields t (a) are Harniltonian for to; to is invariant under the flow of t (o~). The form to and vector fields t (ot) are the elements of a symplectic geometry for T (R). The geometry is natural in the sense that to is invariant with respect to the Teichmiiller modular group. The quotient of T (R) by the modular group is the classical moduli space of R. The K~ ihler form to projects to the moduli space. The simplest example of the above discussion is provided in the case of the once punctured torus. In the first section we describe natural global coordinates