I don't know much about the Teichmüller space, so maybe the question I ask is well known; still I can not find the answer by myself...

Let $\Sigma$ be a closed surface. Let $g_m$ be a sequence of (smooth) Riemannian metrics on $\Sigma$, such that the distances $d_{g_m}$ converge uniformly to $d_g$ ($g$ is a Riemannian metric on $\Sigma$). Each metric $g_m$ defines a point $X_m$ in the Teichmüller space. Do we have the convergence $X_m \rightarrow X$ (where $X$ is the point in the Teichmüller space associated to $g$) ?

If we do not have such a convergence in the Teichmüller space, is it true in the moduli space ?

Another way of seeing this question is the following : is the map $g \mapsto [g]$, from the set of (smooth) Riemannian metrics to the Teichmüller space, continuous (for the appropriate topology) ?