Timeline for Isometric embedding of the modular surface
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Jul 2, 2020 at 14:49 | history | edited | Ben McKay | CC BY-SA 4.0 |
added 158 characters in body
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| Jul 2, 2020 at 14:47 | comment | added | Ben McKay | @RobertBryant Ok, I will leave this up, but it is wrong for the reasons you point out. I think that Hilbert's result can be strengthened to prevent isometric immersions of sufficiently wide strips of the hyperbolic plane, but I am not sure if that is relevant. | |
| Jul 2, 2020 at 14:34 | comment | added | Robert Bryant | Actually, Hilbert's theorem only applies to immersions of the entire hyperbolic plane. I don't see how Hilbert's proof would generalize to cover the case of "immersions" that have conical points or 'spikes' (i.e., cusps). For example, the mean curvature need not be smooth or even continuous at the conical points. For comparison, consider that the flat plane modulo the involution $v\mapsto -v$ can be isometrically embedded as a cone in $\mathbb{R}^3$, but the mean curvature blows up at the vertex of the cone even though $K\equiv0$. | |
| Jul 2, 2020 at 13:24 | history | answered | Ben McKay | CC BY-SA 4.0 |