In this paper, by using Belyi maps and dessin d’enfants, we construct some concrete examples of Strebel differentials with four double poles of residues 1, 1, 1, 1 on the Riemann sphere. We also prove that they have either two double zeroes or four simple zeroes. In particular, we show that they have two double zeroes if and only if their poles are coaxial; in such cases we also obtain their explicit expressions. On the other hand, for those differentials with four non-coaxial poles and whose metric ribbon graphs have edges of rational lengths, we characterize them optimally in terms of Belyi maps in the sense that the Belyi maps used here have minimal degree, and work out the explicit expressions of the five simplest ones among them. As applications, we obtain some explicit cone spherical metrics on the Riemann sphere.

中文翻译：

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通过Riemann球面上的Belyi映射构造Strebel微分

在本文中，通过使用Belyi映射和dessin d'enfant，我们构造了一些黎斯特球面上带有四个双极点残基1,1、1、1的Strebel微分的具体例子。我们还证明它们具有两个双零或四个简单零。特别是，我们证明了当且仅当它们的极点是同轴的时，它们才具有两个双零。在这种情况下，我们还可以获得它们的显式表达式。另一方面，对于那些具有四个非同轴极点且其度量带状图具有合理长度的边的微分，我们从Belyi映射的角度对其进行最佳表征，因为此处使用的Belyi映射具有最小的程度，并计算出其中五个最简单的表达式的明确表达。作为应用程序，我们在黎曼球面上获得一些显式圆锥球面度量。