Let $(M,J,\omega)$ be a symplectic manifold with a compatible almost complex structure, $D$ be the closed unit disk in $\mathbb{C}$, and $u:(D,i)\to (M,J)$ be a $(J,i)$-holomorphic map.


*Question*: Assume $u|\_{\partial D}$ is constant, does this imply $u$ is a constant map?



Pseudo-holomorphic disk which is constant along boundary
Extend $u$ to get a $C^1$ pseudoholpmorphic map defined on $\mathbb{C}$ by setting $u$ constant outside the unit disc. It's $C^1$ because you know the derivative of $u$ along the unit circle vanishes (by assumption), so the Cauchy-Riemann equations satisfied by $u$ on the disc tell you that $du$ vanishes along the unit circle; clearly $du$ continues to vanish outside the disc, hence it's $C^1$. It's pseudoholomorphic because this is a pointwise condition on derivatives which clearly holds piecewise for this map. Now by unique continuation, $u$ is constant.


