(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?

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