Timeline for Teichmuller Space of a Disk with Holes and Boundary Punctures
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2017 at 7:08 | comment | added | QGravity | @PaulPlummer, I understand, but I need a more explicit construction of it. For example, how can I phrase explicitly that restriction in terms of an equation for Fenchel-Nielsen coordinates on the Teichmuller space of the double surface? An equation like $f(\ell,\theta;\cdots)=0$ in which $\cdots$ shows possible derivatives of FN coordinates. etc | |
| Apr 23, 2017 at 19:38 | comment | added | user35370 | The restriction is that it is invariant under the natural involution, so the subspace you are looking have your subsurface with the prescribed mirror symmetry | |
| Apr 23, 2017 at 18:52 | comment | added | QGravity | @PaulPlummer, The restriction that I am thinking about is the following: If we use the double of the surface, the Teichmuller space of the original surface is a subset of the Teichmuller space of the double surface. If this is the correct picture, this subset can be obtained using some constriants on the Teichmuller space of the double surface. This is the restriction that I am referring to. | |
| Apr 22, 2017 at 16:00 | comment | added | user35370 | This recent paper (which is what got me thinking about it again) on the arxiv, I think, implicitly answers all your questions except for F-N coordinates. If I remember correctly (don't have the book on hand) the Penner book discusses that. Basically there is nothing special about boundary punctures, and all reasonable definitions should work (not 100% sure about representation variety definition but I have a couple of guesses on how to get it to work) | |
| Apr 22, 2017 at 15:49 | comment | added | user35370 | I was actually recently thinking about this question recently, and considering writing a more detailed answer. I am not sure what you restrictions you are thinking are necessary, the above definition is well defined, no restrictions necessary. People have been studying open Riemann surfaces for a while, I am no expert so I just cited a definition I have seen in multiple places, but the "real definition" should be about conformal equivalence, and the punctures, boundary or not, should be considered as marked points. This doubling procedure just brings us to the setting people are use to. | |
| Apr 22, 2017 at 7:01 | comment | added | QGravity | @PaulPlummer, Thank you for the answer. However, is it possible to think about an open Riemann surface without resorting to its double? The doubled surface makes it extremely complicated. Then what condition should I put on the Teichmuller space of the doubled surface to get the Teichmuller space of the original surface? These questions make me feel that the concept of double might not be the right setup to study open Riemann surface. | |
| Mar 26, 2017 at 19:07 | comment | added | user35370 | You might want to look at Decorated Teichmuller Theory by Penner. It looks like he defines this to be the subspace of Teichmuller space of the double invarient under the natural involution. In this case all the questions should be answerable by looking at the double. | |
| Mar 9, 2017 at 9:59 | history | asked | QGravity | CC BY-SA 3.0 |