On some prerequisites for J-holomorphic curves and Gromov-Witten invariants

Question

I'm currently reading through J-holomorphic curves and Quantum cohomology by McDuff and Salamon, and I've been facing some unfamiliarity issues with respect to the PDE and functional analysis tools being used in there.

Say for example in understanding the local behaviour of J-holomorphic curves, we use the generalised Cauchy-Riemann equation to establish a few results about how "good" co-ordinates could be chosen, and some theorems akin to unique continuation, a variant of identity theorem (lemma 2.2.3 in the book) etc., So mostly I've been able to get along with some intuition borrowed from how Holomorphic curves are supposed to behave.

Now, my primary question is, how do I get a better intuition for geometric behaviour of J-holomorphic curves?

Also in a more broader sense are there any other fields which are a little more visual(familiar) in flavour, like say Riemannian geometry or complex geometry from where I could borrow some more intuition, especially in the context of say when I'm faced by a PDE describing a curve/surface I need some pointers on what are some primary questions about the behaviour of curve that I should start asking, (I do understand questions that one asks become very PDE specific, but I'm looking for any class of reasonable questions that one tries to ask with PDE that encounters in these areas of geometry)

Now I do understand that we get into the business of Moduli spaces of J-holo curves, in the pursuit of studying global invariants of symplectic manifolds, and things quickly tend to get more operator theoretic and all, and so essentially Riemannian geometry which uses curvature as a primary tool loses its relevance, but even then, say if there is some intuition from Riemannian geometry that I could use eventually, what kind of results in Riemannian geometry should I be looking at?

Also to get to understanding computation of Gromov-Witten invariants and proofs of results like Non-squeezing theorem (which for example uses theory of Minimal surfaces, but in a very basic sense), what is the amount of Riemannian geometry/complex geometry/variational principles (geometric analysis) do I need in my toolbox ? Would be helpful if someone could suggest a few references for the same.

I did do variational principles at a very basic level, as much as suggested in Ana Cannas Da Silva. I hope my question doesn't come off as too vague, would be happy to clarify more specifically. The essential idea being what can I borrow from Riemannian/Complex geometry/ Geometric analysis be it in terms of direct results or plain intuition, that would help me pursue this book in a more meaningful way.

Answer 1

I never got further than visualizing pseudoholomorphic curves as just plane algebraic curves, drawing their real points, but then sometimes correcting a little by remembering their complex points as much as I can. You can also work out how the Fubini-Study metric on the projective plane restricts to a quadric curve. As that curve bubbles into a pair of lines, the quadric gets a thin neck near that point where the bubbling is happening, and away from there builds up two bubbles. Gauss--Bonnet convinces you that, since the two bubbles are becoming two unit spheres, the curvature is building up near the neck into large negative curvature.