Timeline for Analytic curve on Riemann surface
Current License: CC BY-SA 3.0
5 events
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| Feb 7, 2013 at 12:56 | comment | added | Lasse Rempe | @Zoltan: Since your complex structure is smooth, it would seem that the identity is automatically a diffeomorphism between your surface S with the original structure, and with the new complex structure? Also, you have not answered my question: what do you mean by the curve being analytic? Let me make it more precise. The unit circle is an analytic curve. Take a diffeomorphism of the sphere that maps the unit circle to itself, but not in a real-analytic manner. Pull back the usual complex structure using this diffeomorphism. Does this satisfy your condition or not? | |
| Jan 31, 2013 at 14:38 | comment | added | Zoltan Lengyel | (almost) complex structure is a smooth field of linear operators on tangent spaces squaring to minus identity. If you trace a curve on a surface and ask whether it is analytic the question does not make sense unless you fix a complex structure. Suppose you fix one and you trace an analytic curve with respect to it. Is this curve analytic with respect to all other complex structure? Probably not. And this is my question: to know what kind of deformations of the complex structure preserve the property of analycity. I still do not understand Aleksey's argument: why the diffeomorphism h exists? | |
| Jan 31, 2013 at 13:05 | comment | added | Lasse Rempe | @Zoltan: You should probably clarify what exactly you mean by complex structures here: just a collection of charts? The structure given by a Beltrami differential? Etc. More crucially: what does "$C$ remains analytic" mean? That the map $\gamma$ is analytic, or that the curve has some analytic parametrization? In the former case, the answer is essentially trivial, as noted by Aleksey: the identity map should restrict to be analytic on the curve $C$. In the latter case, I doubt there's a good general answer you can expect - e.g. any homeomorphism preserving $C$ will give such a structure. | |
| Jan 29, 2013 at 18:12 | comment | added | Zoltan Lengyel | Thank you. Just I do not understand one thing: A neighbourhood of $C$ has topology of annulus. I thought that you may equip annulus with two complex structures which are not related by a diffeomorphism. Could you comment, please? Zoltan | |
| Jan 29, 2013 at 15:51 | history | answered | Aleksey | CC BY-SA 3.0 |