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How would I characterize such things? Is the minimal spanning (real) surface of a (real) curve in $S^3$ contained entirely in that $S^3$?

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I presume you are choosing an almost complex structure on $T^*S^3$ that is compatible with the canonical symplectic form? (This includes the complex structure coming from the holomorphic structure on the affine quadric.)

In that case, you need to specify a bit more about what kinds of curves you are looking for. If the curves only have boundary components and all these boundary components are mapped to the 0 section, then they are all constant. Indeed, the canonical symplectic form is exact (with primitive by the canonical 1-form on $T^*S^3$), and this primitive vanishes on the $0$-section. Therefore, the symplectic area of any holomorphic curve is $0$. Since the symplectic form is compatible with the almost complex structure, we also have that this area \[ \int u^*\omega = \int |du|^2 d vol. \]

In order to get non-trivial holomorphic curves, you need to allow punctures. This picture becomes technical fairly quickly, so before I explain more, you should specify what you are looking for exactly.

Update with punctured holomorphic curves: You can think of the above as some kind of analogue of looking for holomorphic sections of a bundle. There are plenty of bundles without sections, but that have meromorphic sections. Punctured curves are then some analogue of meromorphic sections.

More precisely, you want to think of $T^*S^3$ as being composed of a compact piece, whose boundary is the unit cotangent bundle (with its canonical contact structure - denote this by $Y = S^3 \times S^2$ since $S^3$ is paralellizable), and a piece that looks like $\mathbb{R}^+ \times Y$. If you choose a contact form on $Y$, you can now ask to have an almost complex structure in this region $\mathbb{R}^+ \times Y$ that is invariant under the $\mathbb{R}$ action and that sends the generator of this action to the Reeb vector field on $Y$. Now, with a suitable notion of energy, curves defined on punctured domains can converge to periodic orbits of this Reeb vector field as they go to infinity (see the Hofer-Kriener survey paper I mentioned or take a look at the difficult papers by Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder on compactness or by Eliashberg-Givental-Hofer on SFT).

This family of ideas allows one to study string topology... see for instance Cieliebak-Latschev (it has something like string topology in the title) or Seidel's paper on "a biased view of symplectic cohomology".

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I came to the conclusion you describe shortly after asking this question. I would like to hear what you have to say about the case with punctures if you would, however. – wzzx Mar 20 '13 at 3:50
I recommend a nice survey article (though a little obsolete) by Hofer and Kriener. I'll update what I wrote with a few more details. – Sam Lisi Mar 20 '13 at 10:29

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