2019/09/30 by Ser-Wei Fu, Christopher J. Leininger, Fu, Ser-Wei +1 · 1 citation
Mathematics · #57K20 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1909.13760
openalex publication_date 2019/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 < 3 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 > 2 and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if q > 2 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.