Suppose there is a closed analytic curve $C$ on a Riemann surface $S$, that is the image of a map $\gamma$ from the equator $E$ of the Riemann sphere to the surface $S$ which is a restriction of a one-to-one complex analytic map $\Gamma$ of annulus $A(\supset E)$ into the surface $S$. Suppose that I change the complex structure on $S$: under which conditions the curve $C$ remains analytic?
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On a neighborhood of $C$, the two complex structures are related by a diffeomorphism $h$, $J'=h^*J$. If $C$ is analytic with respect to $J$, it is analytic with respect to $J'$ (in the sense you described at the beginning) if and only if the restriction of $h$ to $C$ is real-analytic with respect to the analytic structure induced by $J$. One direction is clear; the extension of the map to the annulus for the opposite direction is provided by (3.4) in math/1301.1074.
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$\begingroup$ 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 $\endgroup$ Commented Jan 29, 2013 at 18:12
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$\begingroup$ @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. $\endgroup$ Commented Jan 31, 2013 at 13:05
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$\begingroup$ (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? $\endgroup$ Commented Jan 31, 2013 at 14:38
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$\begingroup$ @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? $\endgroup$ Commented Feb 7, 2013 at 12:56