Timeline for For what metrics are circles solutions of the isoperimetric problem?
Current License: CC BY-SA 3.0
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| Jul 20, 2012 at 15:31 | comment | added | Robert Bryant | @alvarezpaiva: Thanks for the response. If you are still interested in the question that you asked in your first comment above, i.e., how one can determine whether a given third order differential equation $y''' = F(x,y,y',y'')$ characterizes the solutions of an isoparametric problem in the plane, I can explain that if you want. There are known conditions, starting with the condition that $F$ be at most quadratic in $y''$ and going on from there. One can describe these in terms of Cartan's invariants for these equations under point transformations, and I can do this, too, if you like. | |
| Jul 5, 2012 at 20:50 | comment | added | alvarezpaiva | @Robert: Thanks for the answer, it's very pretty. It was a good idea to "uncouple" the length and the area element! | |
| Jul 5, 2012 at 15:59 | comment | added | Robert Bryant | @alvarerpaiva: See my comment above for the Finsler case. It turns out that there are nontrivial Finsler solutions. | |
| Jul 5, 2012 at 15:57 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added information about the Finsler metric-measure case
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| Jun 26, 2012 at 21:18 | comment | added | alvarezpaiva | Thanks again. I had indeed the Finsler setting in mind, although intuitively it seems to me that if all sufficiently small circles are solutions of the isoperimetric problem, then the metric would be Riemannian and even conformal to the standard one. The idea being that the circles would approach the isoperimetrices of the unit tangent discs. | |
| Jun 26, 2012 at 18:45 | comment | added | Robert Bryant | These conditions can certainly be read off of Chern's work on third-order equations in the plane, but I think that that approach is overkill. Also, I think it would be an interesting problem to generalize this to the Finsler setting, or rather, the Finsler-area setting. I.e., which pairs of Finsler structures and area forms on the plane have the property that the solutions of the isoperimetric problem are the standard circles? This should be a straightforward calculation using Cartan's machinery, but it might be interesting, especially if it turned out that there are non-Riemannian solutions. | |
| Jun 26, 2012 at 15:31 | comment | added | alvarezpaiva | Thanks Robert. Baule's paper looks interesting. It then seems likely that something non-trivial can be said about metrics that share the same solutions of the isoperimetric problem. Do you know if there are conditions on a third order differential equation on the plane (I'm thinking here of the work of Cartan and Chern on the invariants for these equations) so that its solutions are the constant curvature curves of some Riemannian metrics? | |
| Jun 26, 2012 at 15:23 | vote | accept | alvarezpaiva | ||
| Jun 26, 2012 at 11:59 | history | answered | Robert Bryant | CC BY-SA 3.0 |