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?



Metrics on Teichmüller spaces
These two metrics are not equivalent to each other:


1. The Bergman metric $d\_B$ on $T\_g$ is complete: this result is essentially due to Earle (who proved that Caratheodori metric $d\_C$ on $T\_g$ is complete, while $d\_C$ is bounded from above by $d\_B$), a "better" proof is due to B.-Y. Chen (2004) who proved that $d\_B$ is equivalent to the Teichmuller metric on $T\_g$ and the latter is complete. 
This proof is better since it provides a bilipschitz model for $d\_B$.
2. If you use the Bers embedding $T\_g\to {\mathbb C}^{3g-3}$ then the pull-back metric from ${\mathbb C}^{3g-3}$ is incomplete.


