Timeline for Neighborhood theorem for conical Lagrangian
Current License: CC BY-SA 4.0
5 events
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| Dec 7, 2022 at 18:31 | comment | added | Robert Bryant | @Soham: Well, that's a good question. Certainly, the Legendrian isotopy class of the link is an invariant, but it is not clear to me that there is anything beyond that. In some sense, you are asking whether the 'natural' radial vector field on $CT$ can be extended to a conformally symplectic vector field on a neighborhood of the singular point, or something like that. I'll think about whether that can be made precise. In any case, Balarka Sen's cautionary counterexample for $n=1$ would have to be dealt with, maybe by requiring that $C$ be connected. | |
| Dec 7, 2022 at 17:41 | comment | added | Soham | Indeed my intention was to have $C$ go to $C'$, thanks for the example! I guess for such a local result, the assumption should incorporate the link information. Say for example $C$ and $C'$ have the same Legendrian isotopy class of the link near the conical point, would it be possible to map $C$ to $C'$ ? | |
| Dec 7, 2022 at 13:25 | comment | added | Balarka Sen | It also fails for n = 1 with this assumption. Take two configurations of four lines through the origin in (R^2, dx ^ dy) with different cross ratios. I'd hoped the weaker version is what OP meant by "is there a softer version..." | |
| Dec 7, 2022 at 13:10 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Clarified that I'm assuming that the symplectomorphism is supposed to take C to C'
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| Dec 7, 2022 at 10:28 | history | answered | Robert Bryant | CC BY-SA 4.0 |