The $L^1$-$L^\infty$-geometry of Teichmüller space -- Second order infinitesimal structures

Abstract

The $L^1$-$L^\infty$ geometry is the Finsler geometry of the Teichmüller space by the Teichmüller metric and the $L^1$-norm function of holomorphic quadratic differentials. In this paper, aiming to develop the $L^1$-$L^\infty$-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 $L^1$-norm function at the infinitesimal level.