论文标题
从曲线到电流
From curves to currents
论文作者
论文摘要
已知许多自然实现曲线的自然实现功能可以连续延伸至更大的大地电流空间。例如，相对于固定双曲线度量的长度扩展是地球电流发展的一个激励示例。我们在曲线函数上给出了一个简单的标准，该标准可以保证向大地电流的连续扩展。我们标准的主要条件是平滑属性，它在Anosov表示的翻译长度的收缩期研究中发挥了作用。很容易看出，对于几乎所有已知的连续功能的示例，我们的标准都可以满足地测量电流的连续功能，例如非物质弯曲的长度或表面组的稳定长度，同时也适用于诸如极端长度之类的新示例。我们使用此扩展名来获得最大长度的新曲线计数结果。
Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all the known examples of continuous functions on geodesic currents, such as non-positively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.
