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

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.