Let $f,g:(D^2,j\_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman similarity principle as in Floer--Hofer--Salamon):


**Lemma:** Suppose $f=g$ over a neighborhood of $0\in D^2$. Then $f=g$ everywhere.


Is there a similar unique continuation result for pseudo-holomorphic *submanifolds* (instead of *maps*)? Here is a precise formulation of what I expect might be true:


**Conjecture**: Let $U,V\subseteq(D^2)^\circ$ be two closed topological disks with smooth boundary, and let $\phi:U^\circ\to V^\circ$ be a biholomorphism (which necessarily extends continuously to a homeomorphism $U\to V$). Suppose $f=g\circ\phi$ over $U$. Then $\phi$ extends holomorphically to an open neighborhood of $U$ (and hence $f=g\circ\phi$ over this neighborhood by the unique continuation result for maps).


Is some statement along these lines known?



Does pseudo-holomorphic *submanifolds* satisfy unique continuation?
The desired statement is in fact a well known property of pseudo-holomorphic maps. Lemma 1.3.1 in [Singularities and positivity of
intersections of J-holomorphic curves](https://mathscinet.ams.org/mathscinet-getitem?mr=1274930) by Dusa McDuff says that for a pair of pseudo-holomorphic maps $u,v:(D^2,0)\to(M,p)$ with $du(0)\ne 0$, either $u(0)=v(0)$ is an isolated point of intersection, or $v$ factors through $u$. This (combined with, say the fact that critical points of pseudo-holomorphic maps are isolated, see Lemma 1.2.1 in the same paper) gives the desired result.


