Timeline for Minimal graph over convex domain is area-minimizing
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 29, 2019 at 4:01 | comment | added | Anton Petrunin | @RBega2 Don't you get zero? | |
| Nov 28, 2019 at 23:52 | comment | added | RBega2 | @AntonPetrunin You can also take the orientation double cover as this is homologous to twice the minimal graph and so the calibration argument implies that the area of the double cover is greater than twice that of the graph. | |
| Nov 28, 2019 at 17:50 | comment | added | Anton Petrunin | @MohammadGhomi to make it orientable one has to work a bit (more than you did in your answer). For example after mapping the surface into the cylinder you might pass to the "upper boundary" of set bounded by the obtained surface with self-intersections. | |
| Nov 28, 2019 at 15:22 | comment | added | RBega2 | @MohammadGhomi The issue is to establish the existence of a minimizer in the class of of all orientable and non-orientable surfaces (otherwise there might be non-orientable competitors that don't satisfy any equation). You can do this by working with mod 2 currents, but its not exactly elementary... | |
| Nov 28, 2019 at 14:14 | comment | added | Mohammad Ghomi | I think orientability follows from embededness as I explain in the comment below. | |
| Nov 28, 2019 at 3:45 | comment | added | Anton Petrunin | @RBega2 Yes calibration works for oriented surface. The proof I see requires modifying the surface into oriented one with less area and then applying calibration. I did not see such proof written. (Maybe I missed a simpler argument.) | |
| Nov 28, 2019 at 2:04 | comment | added | RBega2 | That being said, the CM argument uses a calibration so doesn't that restrict its applicability to oriented surfaces? | |
| Nov 28, 2019 at 1:59 | comment | added | RBega2 | If you look below Corollary 1.2 in Colding Minicozzi's book they prove what you are asking by observing projecting to the cylinder is distance non-increasing map as the cylinder is a convex region). | |
| Nov 27, 2019 at 21:12 | answer | added | Mohammad Ghomi | timeline score: 1 | |
| Nov 27, 2019 at 5:52 | history | edited | Anton Petrunin | CC BY-SA 4.0 |
added 42 characters in body
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| Nov 27, 2019 at 1:06 | comment | added | Alexandre Eremenko | Probably the "minimal graph" is defined as a solution of the non-linear PDE which is called the "minimal surface equation in non-parametric form"? | |
| Nov 26, 2019 at 23:28 | comment | added | Joseph O'Rourke | A naive question: What is a "minimal graph"? | |
| Nov 26, 2019 at 22:29 | history | asked | Anton Petrunin | CC BY-SA 4.0 |