Tangent spaces to the Teichmueller space from the energy-conscious perspective

TL;DR

This paper explores the tangent space structure of Teichmüller space for compact Riemann surfaces, connecting different descriptions via harmonic vector fields inspired by harmonic map theory, and applies this to universal Teichmüller curves.

Contribution

It introduces a novel connection between tangent space descriptions using harmonic vector fields and demonstrates their role in describing universal Teichmüller curves.

Findings

Harmonic vector fields link holomorphic quadratic differentials and cohomology descriptions.

A new framework for understanding tangent spaces via harmonic maps.

Application to universal Teichmüller curve connections.

Abstract

Usually, the description of tangent spaces to the Teichmueller space $\mathscr{T}(\Sigma_{g})$ of a compact Riemann surface $\Sigma_{g}$ of genus $g \geq 2$ (which we can identify with the quotient space $\mathbb{H}^{2} / \Gamma_{g}$ of the upper half plane $\mathbb{H}^{2}$ by a discrete cocompact subgroup $\Gamma_{g}$ of $\mathrm{PSL}(2, \mathbb{R})$) comes in two different flavours: the space of holomorphic quadratic differentials on $\Sigma_{g}$ which are holomorphic sections of the tensor square of the canonical line bundle of $\Sigma_{g}$ and the first cohomology group $H^{1}(\Gamma_{g}; \mathfrak{g})$ of the fundamental group $\Gamma_{g}$ of $\Sigma_{g}$ with coefficients in the vector space $\mathfrak{g}$ of Killing vector fields on $\mathbb{H}^{2}$ (or on $\mathbb{D}$), a.k.a the Lie algebra of $\mathrm{PSL}(2, \mathbb{R})$. In this article, we are concerned with connecting the above-mentioned descriptions using the notion of a harmonic vector field on the upper half plane $\mathbb{H}^{2}$ (equivalently, on $\mathbb{D}$) that takes inspiration from the theory of harmonic maps between compact hyperbolic Riemann surfaces. As an application, we also show that how a harmonic vector field on $\mathbb{H}^{2}$ (or on $\mathbb{D}$) describes a connection on the universal Teichmueller curve.