Timeline for The existence of a harmonic diffeomorphism on a punctured surface
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 1 at 13:15 | comment | added | Yu Feng | Got it, thank you. | |
| Dec 1 at 12:06 | comment | added | Moishe Kohan | I guess you real question is: Given a flat fiber bundle $F\to E\to B$ and a section $s$ of this bundle, how does one obtain an equivariant map from the universal cover of $B$ to the fiber $F$. In your special case, $F$ happens to be the universal cover of $B$ and the holonomy representation $h$ of your flat bundle is discrete and faithful. Can you now put these two things together to get a map $B\to F/h(\pi_1(B))$? By the way, it is a terrible idea to treat Teichmuller space as the space of metrics on a fixed surface. Sadly, some people do this. | |
| Dec 1 at 11:58 | comment | added | Yu Feng | Thank you for your comment. The section s is a section of a fiber bundle over X whose fibers are isomorphic to the upper half-plane. I would like to know how to obtain a map from X to X using the section s. | |
| Nov 30 at 16:18 | comment | added | Moishe Kohan | In any case, the strongest existence results for harmonic diffeomorphisms are due to Vlad Markovi\'c who proved that every quasiconformal homeomorphism between Riemann surfaces (possibly of infinite type) is homotopic to a harmonic quasiconformal diffeomorphism; the uniqueness in this theorem was known earlier (I think, by Li and Tam). | |
| Nov 30 at 16:02 | comment | added | Moishe Kohan | You can get a harmonic map, but you would not know if it is a diffeomorphism (without quoting a suitable uniqueness theorem). Is this what you wanted to know? | |
| Nov 30 at 11:29 | history | asked | Yu Feng | CC BY-SA 4.0 |