Timeline for Unique Beltrami Differential of the form $k\frac{\bar{q}}{q}$?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 3, 2012 at 4:59 | comment | added | BrainDead | Right, by punctured disk, I meant the strip [0,1] x [0,infinity] in the upperhalf plane. And the problem is that $\phi = q^2$ does not have a finite $L^1$ norm. Theorem 3 in Chapter 9.3 (pg.183) seems to be exactly what I need. | |
| Apr 3, 2012 at 3:44 | comment | added | Misha | @BrainDead: You are welcome. Judging by your description, your $q$ is holomorphic on the upper half-plane (composition of two holomorphic functions), so Theorem 5 would apply. Maybe the problem is that your quadratic differential is unbounded: I forgot if Theorem 5 has boundedness requirement on $q$. | |
| Apr 3, 2012 at 2:32 | comment | added | BrainDead | Thank you for the reference. I'm actually looking at a beltrami differential of a very particular form: mu(z) = k \bar{q}/q, where q is the Koebe function precomposed with the map exp(2pi i z). (So it is a beltrami differential on a punctured disk.) Theorem 5 doesn't really apply in my case, but Chapter 9 of the reference you gave me (which contains the pages you gave) seems to be the right place to start. Again, thank you very much for your answer. | |
| Mar 30, 2012 at 12:20 | history | edited | Misha | CC BY-SA 3.0 |
added 106 characters in body
|
| Mar 30, 2012 at 5:33 | history | answered | Misha | CC BY-SA 3.0 |