Suppose $\Sigma$ be a punctured Riemann surface with punctured boundary, and $(M, J)$ be a $2n$-manifold with almost complex structure $J$. Let $L$ be a totally real submanifolds of $M$ in smooth category.
Then, my question is, what facts are known for the boundaries of J-holomorphic curves $u : \Sigma \to M$.
- The regularity of $u(\partial \Sigma)$. Does it need to be smooth? or just $C^1$?
- Local structure of $u(\partial\Sigma)$, possibly analogy of the Micallef-White theorem or similar things.
- Types of singuarlities on the boundary.
- The nature of J-holomorphic curve near some type of singularities?
Maybe the facts will also depend on the regularity, and additional properties of $L$, but I don't know this neither.
I've already found some papers about this(especially for J-holomorphic disc), but since I don't know what results are known, I would like to know recent developments and collections of results on this topic. I'm worried that this question is too big to ask here.
This question is asking kind of the converse of related:Which curves are boundary of pseudoholomorphic curves?
