# Analytic curve on Riemann surface

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.
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 – Zoltan Lengyel Jan 29 '13 at 18:12
@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. – Lasse Rempe-Gillen Jan 31 '13 at 13:05