论文标题
在三个球中的谐波托里的空间
The space of equivariant harmonic tori in the 3-sphere
论文作者
论文摘要
在本文中，我们给出了从2个毛线到3杆的模量谐波图的模量空间的明确参数化。正如希钦（Hitchin）所证明的那样，其光谱数据描述了一个2腹肌的谐波图，该频谱数据由一对差异曲线以及一对差分和线束组成。光谱数据的空间自然是光谱曲线空间上的纤维束。对于均匀的托里，光谱曲线的空间是圆盘，束很微不足道。对于具有一维不变性组的Tori，我们列举了光谱曲线空间的路径连接成分，并表明它们是“螺旋体”或Annuli，并且它们密集地散落了参数空间。光谱数据的模量空间的束结构在Annuli组件上是不平凡的。在两种情况下，光谱数据分别仅需要基本和椭圆函数，我们在每个阶段都提供明确的公式。均匀的托里和Delaunay圆柱体的高斯图被用作说明性示例。
In this paper we give an explicit parametrisation of the moduli space of equivariant harmonic maps from a 2-torus to the 3-sphere. As Hitchin proved, a harmonic map of a 2-torus is described by its spectral data, which consists of a hyperelliptic curve together with a pair of differentials and a line bundle. The space of spectral data is naturally a fibre bundle over the space of spectral curves. For homogeneous tori the space of spectral curves is a disc and the bundle is trivial. For tori with a one-dimensional invariance group, we enumerate the path connected components of the space of spectral curves and show that they are either `helicoids' or annuli, and that they densely foliate the parameter space. The bundle structure of the moduli space of spectral data over the annuli components is nontrivial. In the two cases, the spectral data require only elementary and elliptic functions respectively and we give explicit formulae at every stage. Homogeneous tori and the Gauss maps of Delaunay cylinders are used as illustrative examples.
