Timeline for Disk with punctures and convex geodesical hull of the punctures isomorphic?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 24, 2019 at 17:55 | comment | added | Alexandre Eremenko | "Space of surfaces X"="space of n-tuples of points on the unit circle" = "(n-1)simplex". Similarly for Y. | |
| Jul 24, 2019 at 7:30 | comment | added | giulio bullsaver | Thank you for the notational clarification, I was not aware of it. Regarding the topological reasons, I am not sure I understood it. Are you describing the map from X to Y? In my first comment I meant a different thing: whether the map from the "space of surfaces X" to the "space of surfaces Y" is 1-1. Alternatively, can I think of every disk with marked points Y to be biholomorphic to the geodesical convex hull of some points on the unit disk? | |
| Jul 24, 2019 at 5:47 | comment | added | Alexandre Eremenko | The word "puncture" means a hole. Marked points INSIDE the Riemann surface are called punctures. But marked points on the boundary are not punctures. Anyway, the "topological reasons" mentioned in the previous comment is the following theorem: If you have a continuous map of a simplex into itself, sending each face (of any dimension) to itself, then this map is surjective. | |
| Jul 24, 2019 at 5:13 | comment | added | giulio bullsaver | oh yes apologies. My interest is to study the moduli space of the disk with marked points (sometimes called punctures) using the Fenchel-Nielsen coordinates. I noted that I can compute explicitly these coordinates for Y (since I have its metric by restricting the Poincaré metric of the disk to Y) and I was wondering if they were those of X as well (of which I cannot compute explicitly the metric). I obtained a simple and nice formula expressing the coordinates of Y in terms of the points (punctures/marked points) of X. | |
| Jul 23, 2019 at 17:47 | comment | added | Alexandre Eremenko | This map is certainly surjective, by topological reasons, but injectivity I don't know how to prove, though it is plausible. (If I unnderstood you correctly. You use some weird terminology, calling marking boundary points "punctures" etc.) | |
| Jul 22, 2019 at 17:37 | comment | added | giulio bullsaver | Interesting. Let me step back a little bit then. Clearly we can associate the surface Y to the surface X, is this procedure invertible in a unique way? That is, given a disk with punctures Y is there a unique choice of points on the boundary on the unit disk (up to biholomorphisms of the disk) such that Y arise as the convex hull of those points? | |
| Jul 22, 2019 at 15:48 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |