In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary, we introduce several more general classes of quadratic differentials (called non-chaotic, gradient, and positive gradient) which possess certain properties of Strebel differentials and often appear in applications. We discuss the relation between gradient differentials and special signed measures supported on their set of critical trajectories. We provide a characterization of gradient differentials for which there exists a positive measure in the latter class.

中文翻译：

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二次微分和有符号测度

在本文中，受 Strebel 二次微分在无边界紧凑黎曼曲面上的经典概念的启发，我们介绍了几种更一般的二次微分类（称为非混沌、梯度和正梯度），它们具有 Strebel 微分的某些性质并且经常出现在应用程序中。我们讨论了梯度微分与其关键轨迹集所支持的特殊签名测度之间的关系。我们提供了梯度微分的特征，在后一类中存在正度量。