Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Question

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?

Answer 1

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 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.

Answer 2

A partial answer is yes.

Given that the domain has a real analytic boundary the construction is standard. A Schwarz reflection can be performed locally near each boundary point, which together with normal analytic continuation produces the sought extension.

In general, I propose the following approach. Assume that $f(D^2) \subset g(D^2)$. The critical points of $g$ (and $f$) form a discrete set by the aforementioned similarity principle. Away from the critical points $\phi^{-1}(\mathrm{Crit}(g))$, we can write $\phi=g^{-1} \circ f$ and in this way obtain a unique continuation along small punctured discs covering the boundary of $U$. However, since $\phi$ is bounded along the boundary by assumption, the removal of singularities theorem can be applied to complete $\phi$ over the punctures.

Remark. At least in real dimension four, the assumption $f(D^2) \subset g(D^2)$ is not too severe since two unparametrised pseudo-holomorphic curves intersect in a discrete set, as was show in [McDuff; The Local Behaviour of J-holomorphic Curves in Almost Complex 4-manifolds].

(As a particular case, which however is no longer relevant given the new general formulation of the question: Every biholomorphism $\phi$ of $D^2(r)^o \subset \mathbb{C}$ which has a continuous extension to $D^2(r)$ extends to a biholomorphism of the Riemann sphere. Here we use a Schwarz reflection along $\partial D^2(r)$.)