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.

中文翻译：

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三球体中等变调和环的空间

在本文中，我们给出了从 2 环面到 3 球面的等变调和映射的模空间的显式参数化。正如 Hitchin 所证明的那样，一个 2 环面的谐波映射是由它的光谱数据描述的，它由一条超椭圆曲线、一对微分和一个线丛组成。光谱数据空间自然是光谱曲线空间上的纤维束。对于齐次环面，谱曲线空间是一个圆盘，而丛是微不足道的。对于具有一维不变群的环面，我们列举了光谱曲线空间的路径连通分量，并表明它们是“螺旋面”或环面，并且它们密集地叶状化参数空间。环分量上光谱数据的模空间的丛结构是重要的。在这两种情况下，光谱数据分别只需要初等和椭圆函数，我们在每个阶段都给出了明确的公式。Delaunay 圆柱的齐次圆环图和高斯图用作说明性示例。