Are Bergman metrics on compact Riemann surfaces continuous on Teichmüller space?

Question

Let $R$ be a compact Riemann surface of genus $\geq 1$, and let $\omega_1,\ldots,\omega_g$ be holomorphic one forms that form the dual basis of canonical homology basis. Let $(\pi)_{ij}$ be the resulting period matrix. The Bergman metric is the Riemannian metric over $R$ that can be defined in a conformal chart by the metric $K(z)|dz|^2$, with
$$K(z)=\sum_{j=1}^g \omega_i (z) (\operatorname{Im}(\pi)^{-1})_{ij}\overline{\omega_j(z)}$$

Question Are these Riemannian metrics continuous over Teichmüller space?

Of course, it is well-known that $\pi$ is continuous (it is indeed holomorphic).

My primary reference is S. Nag, Riemann surfaces and their Jacobians: a toolkit Indian J. Pure Appl. Math.24(1993), no.12, 729–745.

Answer 1

Yes, these Riemannian metrics are continuous. In fact, much better is true: they are real-analytic on the Teichmüller universal curve $\mathscr{C}(R)$. This is a direct consequence of Theorem III of Bers' lovely paper "Holomorphic differentials as functions of moduli" (1961). I sketch how the story goes below.

Of course, saying something coherent here requires that we specify a meaning of "closeness" of points chosen from representatives of varying points in Teichmuller space $\mathcal{T}(R)$. Bers' perspective in this paper is extremely useful for doing this while paying attention to the complex structure of $\mathcal{T}(R)$.

The start to Bers' perspective goes like this. Let $\tau$ be a complex parameter for a Bers embedding $U$ of $\mathcal{T}(S)$ and let $a_1, ..., a_g, b_1, ..., b_g$ be a standard generating list for $\pi_1(R)$. Theorem I of Bers' paper is that there are holomorphic families of Möbius transformations $A_1(\tau), ..., A_g(\tau), B_1(\tau), ..., B_g(\tau)$ generating quasi-Fuchsian groups $G(\tau)$. Furthermore, the domains of discontinuity $\Omega(\tau) \subset \mathbb{CP}^1$ for $G(\tau)$ are chosen to have nicely varying boundary and to be so that $\Omega(\tau)/G(\tau)$ (with the associated marking) represents $\tau \in \mathcal{T}(R)$.

The point is then to consider holomorphic functions on the domain $M = \{\tau \in \mathcal{T}(R), z \in \Omega(\tau)\}$ in $U \times \mathbb{CP}^1$. Holomorphic $q$-adic differentials on a representative of $\tau \in \mathcal{T}(R)$ can be seen as holomorphic functions $f(z)dz^q$ on $\Omega(\tau)$ satisfying the appropriate equivariance condition. Let $W_q$ be the space of holomorphic functions on $\Omega(\tau)$ be denoted by $W_q(\tau)$. Let the space of holomorphic functions on all of $M$ so whose restrictions to $\Omega(\tau)$ are in $W_q(\tau)$ for every $\tau \in \mathcal{T}(R)$.

Let $p_i(\tau,z)$ be the functions on $M$ that restrict to the functions representing lifts of $\omega_i(\tau,z)$ to $\Omega(\tau)$. Bers proves (Theorem III) that $p_i(\tau,z)$ are elements of $W_1$ and in particular globally holomorphic on $M$. (While we're already here, I'll remark that Bers proves a similar extension theorem in general: for any $q \geq 1$, any $f \in W_q(\tau)$ is the restriction of an element of $W_q$).

Once one has this and the holomorphicity of $\pi$, the desired result is immediate.