Timeline for Explicit check of the invariance of the Weil-Petersson form
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 16, 2021 at 10:02 | comment | added | giulio bullsaver | Thanks, I had the same feeling too. | |
| Aug 16, 2021 at 8:06 | comment | added | Timothy Budd | If I'm not mistaken, you have $\ell''(\ell,\tau) = \ell'(\ell,\tau + (k+\tfrac12) \ell)$ when the boundary components are of equal length, where $k$ is an integer parametrizing the possible choices of the curve 13|24. This is because interchanging the boundaries 2 & 3 is like performing half a Dehn twist on the 14|23 curve. | |
| Aug 15, 2021 at 10:51 | comment | added | giulio bullsaver | But then, is there a formula that gives both $\ell'$ and $\ell''$ using a single $\tau$? After all, if $\ell$ and $\tau$ are fixed one should be able to write down the lengths of all geodesics in the sphere | |
| Aug 15, 2021 at 5:06 | comment | added | Timothy Budd | You cannot simply permute the boundary components, because you would have to transform $\tau$ accordingly as it is tied to the boundary components. | |
| Aug 14, 2021 at 20:05 | comment | added | giulio bullsaver | I was thinking of formula 5.12, which gives $\ell'(\ell,\tau)$, where $\ell'$ is the length of the curve 12|34 and $\ell$ that of the curve 14|23. What I meant is that the formula seems to give the same result for the curves 12|34 and 13|24, because it is symmetric in the four boundary components and I would imagine that the formula giving the length $\ell''$ of the curve 13|24 is given by the same formula upon permuting the boundary labels? | |
| Aug 14, 2021 at 5:51 | comment | added | Timothy Budd | Can you elaborate? What formulae are you referring to? $\ell'(\ell, \tau)$ should depend non-trivially on $\tau$. | |
| Aug 13, 2021 at 12:55 | comment | added | giulio bullsaver | Another doubt: if you set the boundary lengths to 0 (or just equal) in the formula for the sphere, the formula becomes completely symmetric in i=1,2,3,4. But then it says that the length of the curve separating 12|34 and 13|24 are equal which doesn't make sense (for instance you could send them both to 0 simultaneously although they intersect). How is it possible? | |
| Oct 23, 2020 at 7:40 | history | answered | Timothy Budd | CC BY-SA 4.0 |