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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?

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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.

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