Timeline for Why do people study curve shortening flows?
Current License: CC BY-SA 3.0
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| Jul 8, 2019 at 14:37 | comment | added | Hollis Williams | Does anyone have the original reference for the paper where the isoperimetric inequality was proved on the plane using curve shortening flow? | |
| Apr 4, 2017 at 2:30 | history | edited | Alec Payne | CC BY-SA 3.0 |
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| Apr 4, 2017 at 1:55 | comment | added | Alec Payne | @PaulBryan, thanks for the references. | |
| Apr 4, 2017 at 1:40 | comment | added | Paul Bryan | @AlecPayne See Felix Schulze's paper arxiv.org/abs/math/0606675 for partial results in higher dimensions. Taking the convex hull in higher dimensions may worsen the isoperimetric ratio, but taking the mean-convex hull improves it. The problem is as you say, that singularities may form for mean-convex mean curvature flow and he deals with this by level set methods but with a dimensions restriction of $n\leq 7$. | |
| Apr 4, 2017 at 1:30 | history | edited | Alec Payne | CC BY-SA 3.0 |
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| Apr 4, 2017 at 1:25 | comment | added | Alec Payne | @IgorRivin You're absolutely correct, but it does go to show how natural curve-shortening flow is! I should also say that the convexity of an isoperimetric domain is only trivial on $\mathbb{R}^2$. It's non-trivial in higher dimensions or if you're not in Euclidean space. Perhaps this argument could suggest a higher-dimensional argument that is less trivial. Though, as far as I understand--correct me if I'm wrong here, mean curvature flow has not had much success with regards to isoperimetry. | |
| Apr 4, 2017 at 1:21 | comment | added | Paul Bryan | @IgorRivin By distance comparison arguments it is not hard to prove the main CSF result for arbitrary embedded curves. I think that proof is MUCH easier than Gage-Hamilton. | |
| Apr 4, 2017 at 1:20 | comment | added | Paul Bryan | Gage observed the monotonicity of the isoperimetric ratio. | |
| Apr 4, 2017 at 1:08 | comment | added | Igor Rivin | This is a very nice argument, but, to be devil's advocate, it requires more regularity than many others, and perhaps more significantly, it is a triviality that an isoperimetric curve should be convex, so we can restrict to those, in which case the result is MUCH easier (Gage, not Grayson). | |
| Apr 4, 2017 at 0:54 | history | answered | Alec Payne | CC BY-SA 3.0 |