Symplectic submanifolds in $\mathbb{R}^{4}$

Question

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the numbers of holes and handles do exist?

Answer 1

None.

By Moser, the only invariants of a symplectic surface are its topological type and its total volume.

Suppose we can embed some surface of genus $g$ and $b$ boundary components as a symplectic manifold of area $A$. Then, you can embed it as a submanifold of arbitrary area by scaling $\mathbb{R}^4$.

Now suppose we can embed a surface of genus $G$ and any number of boundary components. Then, for any $g \le G$ and $b$ arbitrary, we can embed a symplectic surface of genus $g$ and $b$ boundary points by cutting out a sub-surface.

Finally, we observe that I can embed a surface of genus $G$ large by taking a degree $d$ holomorphic curve in $\mathbb{C}P^2$ ( so $G = \frac{1}{2} (d-1)(d-2)$ ) and deleting a small neighbourhood of the $\mathbb{C}P^1$ at $\infty$. Holomorphic implies symplectic, so we're done.

More generally, Gromov tells us that symplectic embeddings of codimension 2 open symplectic manifolds satisfy an h-principle. In other words, there are no obstructions other than homotopic ones.