For an orientable surface of finite type equipped with a flat metric with holonomy of finite order $q$, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when $q\le 2$ is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core curves form an infinite diameter subset of the curve complex. In this paper we focus on the case $q\ge 3$ and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if $q\ge 3$ and the surface is fully punctured, then the embedded cylinder curves form a finite diameter subset of the curve complex. The same analysis shows that the embedded cylinder curves can only have infinite diameter when the metric has a very specific form. Using this we characterize precisely when the embedded cylinder curves accumulate on a point in the Gromov boundary.

对于配备有限阶完整度的平面度量的有限型可定向曲面$q$，最大嵌入圆柱体的集合可以是空的、非空的、有限的或无限的。的情况下$q\le \mathrm{2个}$被充分研究，因为这样的表面是（半）平移表面。不仅集合总是无限的，而且核心曲线形成曲线复合体的无限直径子集。在本文中，我们专注于案例$q\ge \mathrm{3个}$并构造示例来说明嵌入式圆柱曲线的一系列行为。我们证明如果$q\ge \mathrm{3个}$并且表面被完全刺穿，然后嵌入的圆柱曲线形成曲线复合体的有限直径子集。同样的分析表明，当度量具有非常特定的形式时，嵌入的圆柱曲线只能具有无穷大的直径。使用它，我们可以精确地描述嵌入的圆柱曲线何时在 Gromov 边界中的一个点上累积。