I know that Teichmüller $\mathcal{T}_g$ spaces support different metrics. One of them is the Bergman metric; which is a particular case of the Bergman metric on any domain of holomorphy. On the other hand $\mathcal{T}_g$ is a Stein manifold by Bers' work. So $\mathcal{T}_g$ is a domain of holomorphy and support the Kähler structure induce by $\mathbb{C}^{3g-3}$ since there exists an holomorphic embedding of $\mathcal{T}_g$ in $\mathbb{C}^{3g-3}$.

Now my question is: Is the Bergman metric equivalent to the Kähler metric induce by $\mathbb{C}^{3g-3}$ If so how can I prove it? Otherwise how can I prove the contrary?