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

TL;DR

This paper develops second order infinitesimal structures in the $L^1$-$L^ Infty$-geometry of Teichm"uller space, providing model spaces and affirming key folklore results about the real-analyticity of certain maps and metrics.

Contribution

It formulates second order infinitesimal structures on Teichm"uller space and proves the real-analyticity of the map from quadratic differentials to the tangent bundle and of the Teichm"uller metric.

Findings

The map from holomorphic quadratic differentials to the tangent bundle is a real-analytic diffeomorphism on each stratum.

The Teichm"uller metric is real-analytic on the image of each stratum.

A new duality between the Teichm"uller metric and the $L^1$-norm function at the infinitesimal level.

Abstract

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.