2015/06/29 by Guowu Yao, Yao, Guowu · 1 citation
Mathematics · Physics and Astronomy · #30C62 #Advanced Differential Geometry Research #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Primary 30C75 #math.CV #msc:30C62 #msc:30C75
paper · pdf · doi:10.48550/arxiv.1506.08769
25 pages
arxiv created 2015/06/29 · openalex publication_date 2015/06/29 · arxiv updated 2015/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In an infinite-dimensional Teichmüller space, it is known that the geodesic connecting two points can be unique or not. In this paper, we study the situation on the geodesic in the universal asymptotic Teichmüller space AT(Δ). We introduce the notion of substantial point and non-substantial point in AT(Δ). The set of all non-substantial points is open and dense in AT(Δ). It is shown that there are infinitely many geodesics joining a non-substantial point to the basepoint. Although we have difficulty in dealing with the substantial points, we give an example to show that there are infinitely many geodesics connecting certain substantial point and the basepoint. It is also shown that there are always infinitely many straight lines containing two points in AT(Δ). Moreover, with the help of the Finsler structure on the asymptotic Teichmüller space, a variation formula for the asymptotic Teichmüller metric is obtained.