Timeline for Solution of Plateau Problem for a simple, smooth closed curve on a Riemannian Manifold (Kahler) gives a surface that can be parametrized by a closed disk?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 17, 2011 at 13:06 | vote | accept | Italo | ||
| Jan 16, 2011 at 3:24 | comment | added | Deane Yang | Bill Thurston's answer (which is quite good) shows that my comment is not right. At best the approach outlined by me might work if the initial curve is "sufficiently flat", whatever that means. | |
| Jan 16, 2011 at 0:39 | comment | added | Italo | I edited the question, i hope it is more clear now | |
| Jan 16, 2011 at 0:38 | history | edited | Italo | CC BY-SA 2.5 |
Changed title and expanded the question
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| Jan 15, 2011 at 18:01 | comment | added | Rbega | Just to echo what has been said, you should be more precise in your question. Namely, given your $\gamma$ there is a solution that is an (immersed) minimal disk spanning $\gamma$ namely a Douglas-Rado solution (see Bill's great answer for some details). This is because the $\gamma$ you describe is spanned by some disk and hence by direct methods in the calculus of variations is spanned by a minimal disk (of course easier said then proved). However, there may be many other minimal surfaces (or weaker concepts) spanning the curve so it is not correct to speak of THE solution. | |
| Jan 15, 2011 at 15:30 | answer | added | Bill Thurston | timeline score: 13 | |
| Jan 14, 2011 at 16:53 | comment | added | Deane Yang | I think this is a good question. I haven't worked out the details myself, but I am pretty sure that you can prove this on any Riemannian manifold using standard tools in differential geometry. I'm hoping that a real expert will provide a reference, more explicit details, or a counterexample. The idea is to use the inverse function theorem applied to the appropriate functional to demonstrate the existence of a minimal surface parameterized by the closed disk. Then use a comparison argument to show existence and minimality. | |
| Jan 14, 2011 at 14:03 | history | asked | Italo | CC BY-SA 2.5 |