Timeline for Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface
Current License: CC BY-SA 3.0
19 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 6, 2015 at 0:11 | vote | accept | Joseph O'Rourke | ||
| May 5, 2015 at 0:01 | answer | added | Joseph O'Rourke | timeline score: 2 | |
| May 4, 2015 at 19:51 | comment | added | Joseph O'Rourke | @Rbega: Thanks, very relevant (and indeed pretty images). | |
| May 4, 2015 at 16:54 | comment | added | Rbega | You might be interested in this cs.jhu.edu/~misha/MyPapers/SGP12.pdf paper which introduces a modification of the mean curvature flow (motivated heuristically by numerical analysis considerations). The authors claim that numerical evidence suggests that it doesn't form neckpinch singularities. However, this is not rigorously shown and the geometric meaning of the flow is a bit unclear. The article does have some pretty pictures. | |
| May 4, 2015 at 16:17 | comment | added | Joseph O'Rourke | @IanAgol: This is my fault, Ian; sorry. In fact I was confused about the difference, and will now correct. Thanks. | |
| May 4, 2015 at 16:16 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Changed title: Ricci => mean-curvature.
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| May 4, 2015 at 16:13 | comment | added | Ian Agol | @JosephO'Rourke: Did you mean to discuss Ricci flow? I was confused by the title of your question, since you actually seem to be looking for something related to mean curvature flow. | |
| May 4, 2015 at 12:47 | comment | added | Willie Wong | Hmm... I was being slightly imprecise in my previous comment. So lest you be misled: what I said about the inflation is more-or-less true (the width of the cylinder will be similar to, but not exactly the same as, the width of the head) if you assume the handle is sufficiently short and skinny. // Perhaps a better illustration of the problem is that the wIMCF allows changes in topology in its definition: if you start with two spheres, or a skinny torus, both will eventually become spheres under this flow. | |
| May 4, 2015 at 12:17 | comment | added | Joseph O'Rourke | @WillieWong: Thanks, Willie, for that detail. I now understand wIMCF much better. | |
| May 4, 2015 at 12:08 | comment | added | Willie Wong | The problem with the wIMCF is that the handle of the dumbbell will get immediately inflated to the cylinder the same width as the heads of the dumbbell. If you are happy with that then great. But as Otis noted, the definition of the wIMCF allows this kinds of jumps which is not very "flowy". | |
| May 4, 2015 at 11:47 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Citations for neck-pinch. Added links. Substituted "mean curvature flow" for Ricci flow in several spots. My confusion.
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| May 4, 2015 at 11:40 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Citations for neck-pinch. Added links.
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| May 4, 2015 at 10:42 | comment | added | Joseph O'Rourke | @OtisChodosh: Thank you! Inverse mean curvature flow---Just the type of work I was seeking. | |
| May 4, 2015 at 1:04 | comment | added | Otis Chodosh | Huisken--Ilmanen's (weak) inverse mean curvature flow projecteuclid.org/euclid.jdg/1090349447 preserves embeddedness, and the surface will remain connected as long as it starts connected, and it will become a (giant) round sphere for large time. It is unlikely to be a smooth flow for all time, but instead there should be some times when the surface jumps outwards... | |
| May 4, 2015 at 0:05 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Clarified in response to Steve's comment.
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| May 4, 2015 at 0:02 | comment | added | Joseph O'Rourke | @SteveHuntsman: Thanks for the reference. Dilation seems not to be quite what I have in mind, for it allows nonconvex shapes to self-intersect. I added a clarification that the surface should remain embedded. | |
| May 3, 2015 at 23:55 | comment | added | Steve Huntsman | en.wikipedia.org/wiki/Dilation_(morphology) | |
| May 3, 2015 at 23:41 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |