The $L^1$-$L^\infty$-geometry of Teichm\"uller space -- Second order infinitesimal structures

The $L^1$-$L^\infty$ geometry is the Finsler geometry of the Teichm\"uller space by the Teichm\"uller 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 Teichm\"uller space, we formulate the second order infinitesimal structures (the infinitesimal structures on the (co)tangent bundles) over the Teichm\"uller 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 Teichm\"uller Beltrami differentials is a real-analytic diffeomorphism on every stratum in the space of holomorphic quadratic differentials. Second, we show that the Teichm\"uller metric is real-analytic on the image of each stratum. We also observe a new duality between the Teichm\"uller metric and the $L^1$-norm function at the infinitesimal level.