Schwarz-Christoffel accessory parameter for quadrilaterals via isomonodromy
Schwarz-Christoffel accessory parameter for quadrilaterals via isomonodromy
复制标题
基于等单性的四边形的 Schwarz-Christoffel 辅助参数
DOI：
10.1088/1751-8121/ab9f71
复制
发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Anselmo T
中科院分区：
文献类型：
--
作者：
Anselmo T
我们发展了作者最近提出的利用Jimbo， Miwa和Ueno （JMU）定义的等同构tau函数来解决保角映射理论中的附属参数问题的建议。我们在这里关注Schwarz-Christoffel类型的映射：特别是从上半平面到四面多边形的映射，其中所有的边都是直线。我们证明了可以通过一个特殊的“零曲率极限”获得相关的辅助参数——多边形顶点的预像，其中一些边的曲率半径趋于零。我们将此过程应用于矩形域，其中JMU函数由黎曼函数的比值给出，称为皮卡德解，并采用零曲率极限来恢复Nehari使用完全不同的方法获得的辅助参数。然后，我们转向梯形，导出了在大小纵横比极限下的附件参数的新渐近公式。我们的工作为同构问题提供了一个新的几何视角，我们相信这提供了理论见解，同时也展示了保角映射中的经典问题如何受益于同构变形理论中出现的新思想。
We develop the recent proposal by the authors to exploit the isomonodromic tau function defined by Jimbo, Miwa and Ueno (JMU) to solve the accessory parameter problem in conformal mapping theory. We focus here on mappings of Schwarz–Christoffel type: in particular, the mapping from the upper half plane to a four-sided polygon where the sides are all straight lines. We show that one can obtain the relevant accessory parameters—the pre-image of the polygonal vertices—via a special'zero curvature limit'in which the radius of curvature of some of the edges tends to zero. We apply the procedure to rectangular domains where the JMU tau function is given by a ratio of Riemann theta functions, known as the Picard solution, and take the zero curvature limit to recover the accessory parameter obtained by Nehari using quite different methods. We then turn to trapezoids, deriving new asymptotic formulas for the accessory parameters in the limit of large and small aspect ratios. Our work lends a new geometrical perspective to problems of isomonodromy that we believe provides theoretical insight, while also showing how classical problems in conformal mapping can benefit from new ideas emerging from isomonodromic deformation theory.