In addition to conformal weldings , simple curves growing in the upper half plane generate driving functions and hitting times through Loewner's differential equation. While the Loewner transform and its inverse have been carefully examined, less attention has been paid to the maps . We study the continuity properties of these latter transformations and show that uniform driver convergence implies uniform hitting time convergence and uniform welding convergence, even when the corresponding curves do not converge. Welding convergence implies neither hitting time nor driver convergence, while hitting time convergence implies driver convergence in (at least) the case of constant drivers. As an application, we show that a curve of finite Loewner energy can be well approximated by an energy minimizer that matches ’s welding on a sufficiently-fine mesh.

中文翻译：

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Loewner 方程中的发球杆、击球时间和焊接

除了保形焊接之外，简单曲线在上半平面生长产生驱动函数和击球次数通过 Loewner 微分方程。当 Loewner 变换及其逆已被仔细检查，但对地图的关注较少。我们研究了后面这些变换的连续性特性，并表明均匀的驱动器收敛意味着均匀的命中时间收敛和均匀的焊接收敛，即使相应的曲线不收敛也是如此。焊接收敛既不意味着命中时间也不意味着驱动器收敛，而命中时间收敛意味着（至少）在恒定驱动器的情况下驱动器收敛。作为一个应用程序，我们展示了一条曲线有限 Loewner 能量可以通过匹配的能量最小化器很好地近似在足够细的网格上焊接。