The question arose while reading the big book of McDuff & Salamon. Here $\Sigma$ is Riemann surface and M is compact symplectic manifold. Let $u^n(n\in \mathbb N), u : \Sigma \rightarrow M$ be J-holomorphic curve and $J^n(n\in \mathbb N), J$ be almost complex structure on M.
A) What does it mean by saying that $u^n, J^n$ "$C^{\infty}$ converges" to u, J(respectively)?
B) And what does it mean by saying that $u^n$ converges to $u$ "uniformly with all derivative(on compact subset)?"
for A), I would appreciate for explaining other equivalent definition if there exists more than one. for B), I can imagine interpreting first derivative of $u$ as a operator norm $||du||$(given metric on M), but for higher derivative I don't know how to deal with it. does this need connection on the manifold? Thank you for reading.