Timeline for Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates
Current License: CC BY-SA 3.0
8 events
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| Sep 13, 2016 at 5:51 | comment | added | Ian Agol | @QGravity: I suppose that's true, but those statements require proof, which I believe depends on already knowing the complex structure on Teichmüller space, so you have to be sure that your reasoning is not circular. Just knowing the formula for the Weil-Petersson metric and Wolpert's form, you can probably get an almost complex structure. But one must prove it is integrable. | |
| Sep 13, 2016 at 5:39 | comment | added | QGravity | According to Wolpert's magic formula, the Weil-Petersson Kahler form $\omega_{\bf{WP}}$ is known in terms of Fenchel-Nielsen coordinates. This means that the formula $\omega(X,Y)\equiv g_{\bf{WP}}(I\cdot X,Y)$ in which $g_{\bf{WP}}$ is the Weil-Petersson metric in terms of Fenchel-Nielsen coordinates and $X$ and $Y$ are two vector fields, can be used to get the complex structure I. Isn't it correct? | |
| Sep 11, 2016 at 9:37 | comment | added | QGravity | Thank you! If it is possible for you, please check my question: mathoverflow.net/questions/249586/… | |
| Sep 9, 2016 at 3:05 | comment | added | Ian Agol | I googled this topic, and found this paper: arxiv.org/abs/math/0505530 | |
| Sep 8, 2016 at 21:45 | comment | added | QGravity | Thank you! I just have a question: The determinant of the operators on the Riemann surface is defined by the determinant of the corresponding Laplacian which in turn can be written in terms of Selberg zeta function. Is there a notion of holomorphic factorization of Selberg zeta function with respect to the complex structure that you described? In the Selberg zeta function, everything is real, so I was wondering how one can define determinant of Dirac operator directly in terms of Selberg zeta function not through its corresponding Laplacian using this holomorphic factorization? | |
| Sep 8, 2016 at 16:47 | comment | added | Ian Agol | I think this is discussed in Chapter 6 of Hubbard's book: matrixeditions.com/TeichmullerVol1.html | |
| Sep 8, 2016 at 16:44 | comment | added | QGravity | Could you please suggest a good reference for this? | |
| Oct 4, 2013 at 22:41 | history | answered | Ian Agol | CC BY-SA 3.0 |