Extremal length is a classical tool in 1-dimensional complex analysis for building conformal invariants. We propose a higher-dimensional generalization for complex manifolds and provide some ideas on how to estimate and calculate it. We also show how to formulate certain natural geometric inequalities concerning moduli spaces in terms of a complex analogue of the classical Riemannian notion of systole.

中文翻译：

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更高尺寸下的极长长度和复杂的收缩不等

极值长度是一维复杂分析中用于构建共形不变量的经典工具。我们提出了对复杂流形的高维概括，并提供了一些有关如何估计和计算它的想法。我们还展示了如何根据经典的收缩力的黎曼概念的复杂模拟来表达关于模空间的某些自然几何不等式。