Let $\Sigma$ be a closed orientable surface of genus $g$. It is well known that every almost complex structure on a surface is induced by a complex atlas. Therefore, if we call $\mathcal{J}(\Sigma)$ the space of almost complex structures on $\Sigma$ (preserving some orientation), we can identify the Teichmuller space with $\mathcal{J}(\Sigma)/\text{Diff}^+_0 (\Sigma)$ (where $\text{Diff}^+_0 (\Sigma)$ are orientation preserving diffeos homotopic to the identity).
After playing with this identification for a while and doing plenty of computations, I come here with questions. A part of me says the answers are well-known (although I couldn't find references). Suppose you fix some complex structure on $\Sigma$:
Question 1: Let $g$ be a metric locally given by $g = e^{2\phi} |dz + \mu \bar{dz}|$. The conformal class of $g$ induces an almost complex structure $j$ via clockwise rotation by $\frac{\pi}{2}$ on an orthonormal basis. Is there a nice expression for $j$ in terms of $\mu$?
Question 2: Is it easy to see that the cotangent space of the Teichmuller space is the space of quadratic differentials using this point of view?
Let me give some extra info on the second one: it is easy to see that if one takes a variation of almost complex structures $j_t$ and differentiates, then $A = \frac{d}{dt}\vert_{t=0} j_t$ is complex antilinear for $j_0$, therefore $A = \nu dz \otimes \partial_{\bar{z}}$ (where we took complex charts for $j_0$). Obviously, the tangent space is way smaller because we still need to quotient by the subspace generated by the equivalence relation. How easy is it to see that it is enough to consider harmonic Beltrami differentials?
Thank you beforehand.
