Timeline for For what metrics are circles solutions of the isoperimetric problem?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 28, 2012 at 16:22 | answer | added | Vladimir S Matveev | timeline score: 2 | |
| Jun 26, 2012 at 15:23 | vote | accept | alvarezpaiva | ||
| Jun 26, 2012 at 11:59 | answer | added | Robert Bryant | timeline score: 21 | |
| Jun 26, 2012 at 11:10 | comment | added | alvarezpaiva | Well, the classic geometries (two dimensional, complete, constant curvature) have all the same circles. Yes, in particular, the question asks for what Riemannian metrics on the plane do the standard circles have constant geodesic curvature (which is the constant mean curvature in dimension two). I guess that this should mean that geodesics are circles and thus that the only such metrics are of constant curvature ... | |
| Jun 26, 2012 at 9:24 | comment | added | Deane Yang | So you are fixing one of three possible models of "standard circles" and asking what Riemannian metrics make them isoperimetric extremals? I believe that an extremal circle must have constant (extrinsic) curvature. So a related question seems to be: For which Riemannian metrics do the standard circles have constant curvature? Is anything known about this? Is this maybe equivalent to your question? | |
| Jun 26, 2012 at 9:15 | comment | added | alvarezpaiva | I mean this in the same sense that Hilbert's fourth problem asks for metrics whose geodesics are straight lines. The space (the plane, the sphere, the disc) is given and so are the usual circles, then one looks for metrics for which these circles solve the isoperimetric problem. | |
| Jun 26, 2012 at 9:09 | comment | added | Deane Yang | Could you say more by what you mean by a "standard circle" versus a "geodesic circle"? Are you assuming that the space has a "canonical" embedding into a Euclidean space of higher dimension? | |
| Jun 26, 2012 at 9:07 | history | asked | alvarezpaiva | CC BY-SA 3.0 |