2024/06/11 by Hideki Miyachi, Miyachi, Hideki
Mathematics · Physics and Astronomy · #32G15 #32Q45 #32V20 #53B05 #53G60 #57M50 #58A05 #58A30. Secondary: 32U15 #Advanced Differential Geometry Research #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Primary: 32G05
paper · pdf · doi:10.48550/arxiv.2406.07776
openalex publication_date 2024/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The L1-L^∞ geometry is the Finsler geometry of the Teichmüller space by the Teichmüller metric and the L1-norm function of holomorphic quadratic differentials. In this paper, aiming to develop the L1-L^∞-geometry and the differential geometry on the Teichmüller space, we formulate the second order infinitesimal structures (the infinitesimal structures on the (co)tangent bundles) over the Teichmüller space. We will give model spaces of the second order infinitesimal spaces. By applying our formulation, we give affirmative answers to two folklore. We first show that the map from the space of holomorphic quadratic differentials to the tangent bundle defined by Teichmüller Beltrami differentials is a real-analytic diffeomorphism on every stratum in the space of holomorphic quadratic differentials. Second, we show that the Teichmüller metric is real-analytic on the image of each stratum. We also observe a new duality between the Teichmüller metric and the L1-norm function at the infinitesimal level.