2023/03/22 by Vincent Borrelli, Roland Denis, Borrelli, Vincent +7
Mathematics · #30F45 (Secondary) #53C21 #53C42 (Primary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2303.12449
openalex publication_date 2023/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We build an explicit C1 isometric embedding f∞:ℍ2→𝔼3 of the hyperbolic plane whose image is relatively compact. Its limit set is a closed curve of Hausdorff dimension 1. Given an initial embedding f0, our construction generates iteratively a sequence of maps by adding at each step k a layer of Nk corrugations. To understand the behavior of df_∞ we introduce a formal corrugation process leading to a formal analogue Φ∞:ℍ2→ L(ℝ2,ℝ3). We show a self-similarity structure for Φ∞. We next prove that df_∞ is close to Φ∞ up to a precision that depends on the sequence N_*:= (Nk)k. We then introduce the pattern maps \boldsymbolν∞Φ and \boldsymbolν∞, of respectively Φ∞ and df_∞, that together with df0 entirely describe the geometry of the Gauss maps associated to Φ∞ and df_∞. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of \boldsymbolν∞ towards \boldsymbolν∞Φ over circles of rational radii.