Timeline for Parametrizations of the Moduli Space of Riemann Surfaces
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 28, 2017 at 8:56 | comment | added | QGravity | @Cusp Thanks for the comment, it does not explain the details of other parametrizations. It just mentions that we need a generalization of McShane identity for bordered surfaces which is required for her recursive procedure for integration over the moduli space. If ${m_a}_{a=1}^{3g-3+n}$ is a parametrization of moduli space, what are the problems with this parametrization? Why is it difficult to integrate over the moduli space using these coordinates? This sort of general questions that I am looking for. | |
| Aug 12, 2017 at 15:16 | comment | added | Cusp | I would suggest you to have a look at page 3 (after the table), 4 and 5 of the paper: math.stonybrook.edu/~mlyubich/Archive/Geometry/… it give the computation for simple case and tells you what are the difficulties in general case. | |
| Aug 12, 2017 at 15:10 | comment | added | Cusp | One of the problem is that moduli space is not compact (although the fixed point set of the mapping class groups (MCG) on modulie space is finite - the action of MCG on the Teichmuller space is properly discontinuous). Another problem is the explicit relation between the parametrization of moduli space and the volume form of the moduli space. Wolpert's formula (it's actually called Wolpert's magical formula) give explicit description of volume form of the moduli space under Weil-Peterssen metric with respect to Fenchel-Nielsen coordinate as you mentioned. | |
| Aug 12, 2017 at 9:19 | comment | added | QGravity | 2) The explicit volume form in other parameterizations is not known unlike the case of Fenchel-Nielsen coordinates for which Wolpert's form exists; 3) The moduli space is not a manifold (is the set of fixed points of the mapping class group a dense subset of the moduli space?), therefore the doing an explicit integration seems to be difficult. These are the questions that I am looking for answers. | |
| Aug 12, 2017 at 9:12 | comment | added | QGravity | @Cusp Thank you for the comment. Wolpert's book only talks about Fenchel-Nielsen coordinates and is very useful for that. What I am looking for is a thorough exposition of different parametrizations of the moduli space and pros and cons of each of them. As someone who is working on physics, I still don't know why the computation of the volume of moduli space requires the machinery developed by Mirzakhani? There are several things that I can think about: 1) The fundamental domain of the action of the mapping class group inside the Teichmuller space is not known explicitly; | |
| Aug 10, 2017 at 16:17 | comment | added | Cusp | I am sure you are well aware of Mirzakhani's paper. Apart from that, I would suggest the book "Families of Riemann Surfaces and Weil-Petersson Geometry" by Wolpert. | |
| Jun 15, 2017 at 19:48 | history | asked | QGravity | CC BY-SA 3.0 |