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Yemon Choi
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one A question respecton part of An"An introduction to teichmuller spacesspaces" by Imayoshi-Taniguchi

I was reading the part of fenchelon Fenchel-Nielsen coordinates, the proof of theon page 64, I don't understand when they say:

"Since $\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$ all $g_k$ can be glued together into marking-preserving homeomorphism, say $h$ of $R_{t_1}$ onto $R_{t_2}$. Since $h$ is holomorphic on $R_{t_1}$ except for a finite number of analytic curves, so is $h$ on the entire $R_{t_1}$ by Painleve's theorem

Since $\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$, all $g_k$ can be glued together into marking-preserving homeomorphism, say $h$ of $R_{t_1}$ onto $R_{t_2}$. Since $h$ is holomorphic on $R_{t_1}$ except for a finite number of analytic curves, so is $h$ on the entire $R_{t_1}$ by Painleve's theorem...

My questions are:

  • how do different twisting parameters determine different points in the Teichmüller space.?

  • iI would like to know some reference for this Painleve's theorem.

one question respect of An introduction to teichmuller spaces by Imayoshi-Taniguchi

I was reading the part of fenchel-Nielsen coordinates, the proof of the page 64, I don't understand when they say:

"Since $\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$ all $g_k$ can be glued together into marking-preserving homeomorphism, say $h$ of $R_{t_1}$ onto $R_{t_2}$. Since $h$ is holomorphic on $R_{t_1}$ except for a finite number of analytic curves, so is $h$ on the entire $R_{t_1}$ by Painleve's theorem

My questions are:

  • how different twisting parameters determine different points in the Teichmüller space.

  • i would like know some reference for this Painleve's theorem.

A question on part of "An introduction to teichmuller spaces" by Imayoshi-Taniguchi

I was reading the part on Fenchel-Nielsen coordinates, the proof on page 64, I don't understand when they say:

Since $\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$, all $g_k$ can be glued together into marking-preserving homeomorphism, say $h$ of $R_{t_1}$ onto $R_{t_2}$. Since $h$ is holomorphic on $R_{t_1}$ except for a finite number of analytic curves, so is $h$ on the entire $R_{t_1}$ by Painleve's theorem...

My questions are:

  • how do different twisting parameters determine different points in the Teichmüller space?

  • I would like to know some reference for this Painleve's theorem.

Source Link

one question respect of An introduction to teichmuller spaces by Imayoshi-Taniguchi

I was reading the part of fenchel-Nielsen coordinates, the proof of the page 64, I don't understand when they say:

"Since $\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$ all $g_k$ can be glued together into marking-preserving homeomorphism, say $h$ of $R_{t_1}$ onto $R_{t_2}$. Since $h$ is holomorphic on $R_{t_1}$ except for a finite number of analytic curves, so is $h$ on the entire $R_{t_1}$ by Painleve's theorem

My questions are:

  • how different twisting parameters determine different points in the Teichmüller space.

  • i would like know some reference for this Painleve's theorem.