Let $B^4$ be the closed unit ball in $\mathbb{C}^2$ and $J$ an almost complex structure sufficiently closed to the standard complex structure on $\mathbb{C}^2$ in the $C^0$-topology. Let $u \colon S \rightarrow B^4$ be a smooth $J$-holomorphic curve transverse to $\partial B^4$. Can $u$ extend to a $\tilde{J}$-holomorphic curve in $\mathbb{C}P^2$? Here $\tilde{J}$ is an extension of $J$ on $\mathbb{C}P^2$.


Any hint and comment are really appreciated. Thank you in advance.



Extension of a holomorphic curve in $B^4$ to one in $\mathbb{C}P^2$
Yes, you can.


The "boundary" $\partial B \cap {\rm Im}(u)$ of $u$ is a link $K$ that is transverse to the contact structure $JTS^3\cap TS^3$, which is isotopic to the standard contact structure on $S^3$. John Etnyre and I [proved](http://arXiv.org/abs/2001.08978) that every transverse link in $(S^3,\xi\_{st})$ can be capped off by a smooth symplectic surface $\hat C$ in $\mathbb{CP}^2\setminus B^4$ (here $B^4$ is a Darboux ball). We call any such cap a *hat* (because "relative symplectic cap" was too long).


As Chris Gerig mentions in his comment above, you can extend $J$ to an almost complex structure $\tilde J$ on $\mathbb{CP}^2$ for which $C = \hat C \cup u(S)$ is pseudo-holomorphic.


The argument of the proof of the existence of $\hat C$ isn't very hard: one constructs a cobordism to a torus link using local moves on braids, and then caps it off with a piece of an algebraic curve.


