Timeline for Minimal surface on $R^3$ with with non Euclidean metric
Current License: CC BY-SA 4.0
13 events
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| Jun 27, 2023 at 0:26 | comment | added | Daniel Asimov | Naruto: There is no standard definition of "surfaces in the most general sense". It is necessary to specify exactly what kind of surfaces you are talking about. That means what differentiability you are assuming, if any, and whether your surfaces are embedded, immersed, or just mapped in with no conditions about injectivity or local injectivity. (I presume from your comment that they are just mapped in, but that doesn't specify the differentiability class.) | |
| Jun 21, 2023 at 16:25 | comment | added | Daniel Asimov | If a smoothly embedded circle is unknotted but remains close enough to a double-covering of the unit circle in the xy-plane, then the unique area minimizer could be a smoothly embedded Möbius band, not a disk. | |
| Jun 21, 2023 at 4:58 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing
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| Jun 21, 2023 at 4:02 | comment | added | Naruto | Daniel Asimov: I mean surfaces in most general sense (allowing singularity). However, I would be mostly interested in the set of "nice enough" curves, and so all area-minimizer will be smooth embedded disks. | |
| Jun 21, 2023 at 3:45 | vote | accept | Naruto | ||
| Jun 20, 2023 at 23:23 | comment | added | Daniel Asimov | Naruto: Are you using the word "surface" to mean specifically an embedded (or immersed) surface? Or something else? | |
| Jun 20, 2023 at 23:10 | answer | added | Otis Chodosh | timeline score: 4 | |
| Jun 20, 2023 at 20:40 | comment | added | Naruto | Regarding the second question, I would love to see your argument on it. I think that the neighborhood of $g_E$ that guarantees the uniqueness of the area-minimizer depends on each $\gamma$ (in my case, circle). My second question here is that is there such a uniform neighborhood that works for all circles? | |
| Jun 20, 2023 at 20:35 | comment | added | Naruto | Thank you for the detailed answer. You are right, I forgot the crucial condition is that the area-minimizing surface bounded by $\gamma$ has no non-trivial Jacobi field. The theorem I was talking about is theorem 3 in here: ams.org/journals/tran/1984-283-01/S0002-9947-1984-0735418-6. | |
| Jun 20, 2023 at 20:31 | comment | added | Otis Chodosh | On the other hand, since the disk in $R^3$ is strictly stable, the second part of your question is true (if you perturb $g_E$ to $g$) then there's still a unique minimizer). (I can add a proof if you want). | |
| Jun 20, 2023 at 20:29 | comment | added | Otis Chodosh | This is analogous to the situation for geodesics in a manifold: If $q\in \textrm{Cut}(p)$ but there's a unique minimizing geodesic from $p$ to $q$ then there's a normal Jacobi field on the geodesic vanishing at the endpoints. In some situations this means that when $q$ is perturbed (within the cut locus) then there are two minimizing geodesics. This can happen on the paraboloid (cf. math.stackexchange.com/q/1663734) | |
| Jun 20, 2023 at 20:29 | comment | added | Otis Chodosh | Can you give a reference for this "theorem"? I do not think it is true in general. The point is that if the area-minimizer bounded by $\gamma$ has a non-trivial Jacobi field (with Dirichlet boundary conditions) then it is possible to bifurcate two area-minimizers as you perturb. | |
| Jun 20, 2023 at 20:09 | history | asked | Naruto | CC BY-SA 4.0 |