The topology of the moduli space for Lamé functions of degree m is determined: this is a Riemann surface which consists of two connected components when m ≥ 2; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn’s polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic singularity with the angle 2π(2m + 1) on a torus. We show that the degeneration locus of such metrics is a union of smooth analytic curves and we enumerate these curves.

中文翻译：

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Lamé 函数的模空间和第二类阿贝尔微分

度数的 Lamé 函数的模空间拓扑米确定：这是一个黎曼曲面，由两个连通分量组成，当米 ≥ 2; 我们找到了这些分量的欧拉特征和属。作为推论，我们证明了 Maier 关于 Cohn 多项式次数的猜想。这些结果是在这些黎曼曲面的几何描述的帮助下获得的，作为某些奇异平面三角形的模空间的商。应用于研究具有一个圆锥奇点的恒定正曲率与角度的度量2π(2米 + 1)在一个圆环上。我们证明了这些度量的退化轨迹是平滑解析曲线的并集，我们列举了这些曲线。