二次微分轨迹弧的Loewner驱动函数
The Loewner driving function of trajectory arcs of quadratic differentials
摘要：
我们得到一个一阶微分方程的弦Loewner微分方程的驱动函数的情况下，该区域是由一条曲线，这是一个轨迹弧的某些二次微分。特别是这包括的情况下，当曲线是一个路径上的正方形，三角形或六边形晶格在上半平面，或实际上，在任何域的边界上的晶格。我们还演示了如何使用它来计算驱动函数的数值。Loewner微分方程的其他变体的等价结果也得到：多缝弦Loewner微分方程和径向Loewner微分方程。该方法也适用于其他版本的Loewner微分方程。我们的公式的证明使用了一个推广的施瓦茨-Christoffel映射到域的轨道弧的旋转一个给定的二次微分，是在其本身的权利感兴趣。
英文摘要：
We obtain a first order differential equation for the driving function of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of certain quadratic differentials. In particular this includes the case when the curve is a path on the square, triangle or hexagonal lattice in the upper halfplane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained: Multiple slits in the chordal Loewner differential equation and the radial Loewner differential equation. The method also works for other versions of the Loewner differential equation. The proof of our formula uses a generalization of Schwarz-Christoffel mapping to domains bounded by trajectory arcs of rotations of a given quadratic differential that is of interest in its own right.
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