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

Usually, the description of tangent spaces to the Teichmueller space $\mathcal{T}({\mathrm{\Sigma }}_g)$ of a compact Riemann surface ${\mathrm{\Sigma }}_g$ of genus $g \geq 2$(whichwecanidentifywiththequotientspace$$\mathbb{H}$^{2} / \Gamma_{g}$oftheupperhalfplane$$\mathbb{H}$^{2}$ by a discrete cocompact subgroup ${\mathrm{\Gamma }}_g$ of $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$) comes in two different flavours: the space of holomorphic quadratic differentials on ${\mathrm{\Sigma }}_g$ which are holomorphic sections of the tensor square of the canonical line bundle of ${\mathrm{\Sigma }}_g$ and the first cohomology group $H^{1}(\Gamma_{g}; \mathfrak{g})$ofthefundamentalgroup$\Gamma_{g}$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 $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$. 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.