Timeline for For what metrics are circles solutions of the isoperimetric problem?
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| Jun 29, 2012 at 22:13 | comment | added | Robert Bryant | @Vladimir: You don't want to require that 'isoperimetric' curves be closed, since, as you know, there will be very few of these on a general surface, especially one for which the Gauss curvature has no critical points. Instead, you should define 'isoperimetric' locally, as in Dido's Problem: A curve $C$ is 'isoperimetric' if each point of $C$ lies in a small open arc $A\subset C$ such that all small area-neutral deformations compactly supported within $A$ do not decrease the length of $A$. In the case of a Riemannian surface, this is equivalent to having constant geodesic curvature. | |
| Jun 29, 2012 at 16:36 | comment | added | Vladimir S Matveev | Juan Carlos, if all the curves of constant geodesic curvature of a metric are circles, then the geodesics are circles as well which implies (in dimension 2) that the metric has constant curvature by the classical result of Segre, see the link below ams.org/mathscinet/search/… | |
| Jun 29, 2012 at 6:17 | comment | added | alvarezpaiva | Thanks Vladimir. I have yet to understand Baule's paper, but it is a bit unsettling. | |
| Jun 28, 2012 at 16:22 | history | answered | Vladimir S Matveev | CC BY-SA 3.0 |