Timeline for Plateau's Problem from an annulus
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 8, 2023 at 20:37 | comment | added | Pietro Majer | Struwe has a chapter on (the parametric approach to the) the Plateau problem in his book Variational Methods, and in particular a very clear exposition of the case of the annulus and Douglas’ result. | |
| Jan 8, 2023 at 19:28 | answer | added | gaoqiang | timeline score: 2 | |
| Nov 5, 2020 at 23:37 | vote | accept | Totoro | ||
| Oct 10, 2020 at 17:12 | comment | added | Totoro | @RBega2 Thank you. | |
| Oct 10, 2020 at 16:06 | comment | added | RBega2 | @Totoro The article "Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds" by Jurgen Jost is I believe the most modern treatment. Struwe has a book on Plateau's problem that might also cover it (it's been a while since I looked at his book). | |
| Oct 10, 2020 at 15:50 | comment | added | Totoro | @RBega2 Is there any reference for the Douglas' condition? Thanks. | |
| Oct 10, 2020 at 14:51 | comment | added | RBega2 | The Douglas' condition says that if there is an annulus with less area the the sum of the areas of the two minimizing disks, then there is an area minizimizing annulus -- this is the most general situation, but also not so easy to check. Sometimes you can make an argument using a barrier or topological condition (e.g., if the two curves are homotopic but not null-homotopic). | |
| Oct 10, 2020 at 9:21 | answer | added | Robert Bryant | timeline score: 14 | |
| Oct 10, 2020 at 4:19 | history | edited | Totoro | CC BY-SA 4.0 |
added 50 characters in body
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| Oct 10, 2020 at 4:18 | comment | added | Totoro | @AlexandreEremenko Yes. The problem only concerns the case when at least one such map $u$ exists. | |
| Oct 10, 2020 at 3:46 | comment | added | Alexandre Eremenko | Under the conditions that you stated there may be no annulus bounded by $\gamma_1\cup\gamma_2$. For example when your two curves are boundaries of disjoint disks in a torus. | |
| Oct 10, 2020 at 1:00 | history | asked | Totoro | CC BY-SA 4.0 |