What is known about the almost complex structure on the Teichmüller space in Fenchel–Nielsen coordinates?

Question

There has been a question on the same subject, but I'm asking about something more specific.

In the Fenchel–Nielsen coordinates, the Teichmüller space of genus $g$ is represented as $\mathbb{R}^{3g-3}\times\mathbb{R}^{3g-3}_+$ where the last $3g-3$ coordinates are the lengths $(\ell_1,\ldots,\ell_{3g-3})$ and the first $3g-3$ coordinates are the twist parameters $(\theta_1,\ldots,\theta_{3g-3})$. The Teichmüller space also has a natural (almost) complex structure defined by, e.g., multiplication by $i$ on Beltrami differentials. On $\mathbb{R}^{3g-3}\times\mathbb{R}^{3g-3}_+$, this is just a square matrix $J$ of rank $6g-6$ that varies with the base point.

What do we know about the coefficients of $J$? For example, let's write $$J\frac{\partial}{\partial\theta_j}=\sum_k\left(a_{jk}\frac{\partial}{\partial\theta_k}+b_{jk}\frac{\partial}{\partial\ell_k}\right),$$ then are the $b_{jk}$'s independent of the $\theta_k$'s?

Thanks for reading.

(Edit: I changed the last part of the question.)

Answer 1

There is a close relationship between infinitesimal Fenchel-Nielsen deformations of hyperbolic surfaces and the Weil-Petersson geometry of Teichmüller space. Since the Weil-Petersson Kähler metric is compatible with the complex structure on Teichmüller space, this will unwind to quickly give some information about the complex structure in relation to Fenchel-Nielsen deformations. Below is an example of something that follows quickly from the literature.

Let $g_{\text{WP}}$ denote the Weil-Petersson metric and $\omega_{\text{WP}}$ be its Kähler form. Wolpert proved (among a range of other related things: those two papers would be good places to look for matters related to your question) that if we let $t_\alpha$ denote the vector field on $\mathcal{T}(S)$ induced by the Fenchel-Nielsen deformation about $\alpha$ and let $l_\alpha: \mathcal{T}(S) \to \mathbb{R}$ denote the length of $\alpha$, we have

$$ \omega_{\text{WP}}(t_\alpha, \cdot) = dl_\alpha, \qquad t_\alpha^* = -idl_\alpha, $$

where $\,^*$ denotes the dual with respect to the Weil-Petersson metric.

Since the Weil-Petersson metric is Kähler and compatible with the almost complex structure on Teichmüller space, we have $g_{\text{WP}}(u,v) = \omega_{\text{WP}}(u, Jv)$ and this yields information about how $J$ acts on tangent vectors.

For one simple example, one may verify that if $\alpha, \beta$ have intersection number $0$, then $dl_\alpha(t_\beta) = 0$. This follows from a much more general formula that one proves with a first variation formula argument e.g. Lemma 3.2 here. So for disjoint $\alpha, \beta$, we have $$g_{\text{WP}}(t_\alpha , J(t_\beta)) = \omega_{WP}(t_\alpha, J^2 t_\beta) = - dl_\alpha (t_\beta) = 0.$$