Timeline for Isoperimetric inequality in CAT(0) surfaces
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| May 10, 2017 at 7:21 | comment | added | Mikhail Katz | As far as the isoperimetric disk with prescribed center of mass and maximal area, can one get upper bounds on the geodesic curvature of its boundary? This might be enough even if the curvature of the boundary is not constant. | |
| May 9, 2017 at 19:26 | comment | added | Ivan Izmestiev | I have no explicit example... Osserman cites an article of Erhard Schmidt stating that there are metrics that have no simple closed curves of constant curvature. | |
| May 9, 2017 at 17:49 | comment | added | Mikhail Katz | It seems plausible therefore that one should be able to close up the curve to get a loop at least of constant mean curvature (with a singularity at a single point). | |
| May 9, 2017 at 17:46 | comment | added | Mikhail Katz | I was thinking along the following lines: in a smooth surface one can start with a modification of the geodesic equation incorporating a term for constant geodesic curvature. Doing this in polar coordinates, I can solve the equation until the angle $\theta$ traces out a full circle from $0$ to $2\pi$. If the curve comes back too close to the origin, I can decrease the mean curvature; if the curve comes back too far from the origin, I can increase it... | |
| May 9, 2017 at 17:39 | comment | added | Mikhail Katz | Interesting. Do you have a counterexample in mind? In the plane it certainly does. | |
| May 9, 2017 at 17:19 | comment | added | Ivan Izmestiev | Well, if you prescribe the center of mass, the boundary of the disk will not have constant geodesic curvature. | |
| May 9, 2017 at 17:01 | comment | added | Mikhail Katz | Ivan, thanks, but we are interested in optimal disks with prescribed center of mass in an Alexandrov surface of CAT(0) type, rather than in the Euclidean isoperimetric inequality in more general spaces. | |
| May 9, 2017 at 15:38 | history | edited | Ivan Izmestiev | CC BY-SA 3.0 |
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| May 9, 2017 at 15:38 | comment | added | Ivan Izmestiev | Yes, I meant constant geodesic curvature. I edited the answer. | |
| May 9, 2017 at 11:59 | comment | added | Mikhail Katz | I don't think you mean "every component of the boundary must be geodesic anyway". Do you mean "of constant mean curvature"? | |
| May 9, 2017 at 11:06 | comment | added | Mikhail Katz | I am familiar with Alexandrov's article. Unfortunately he does not address the issue of the disk of maximal area. | |
| May 8, 2017 at 16:12 | history | edited | Ivan Izmestiev | CC BY-SA 3.0 |
A reference to Burago-Zalgaller added.
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| May 8, 2017 at 15:57 | history | edited | Ivan Izmestiev | CC BY-SA 3.0 |
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| May 8, 2017 at 15:49 | history | answered | Ivan Izmestiev | CC BY-SA 3.0 |