Suppose $ (X,\omega,J)$ is a closed symplectic manifold with a compatible almost complex structure. The fact below follows from McDuff-Salamon’s book on $ J$ -holomorphic curves (specifically, Lemma 4.7.3).

Given $ 0<\mu< 1$ , there exist constants $ 0<C<\infty$ and $ \hbar>0$ such that the following property holds. Given a $ J$ -holomorphic map $ u:(-R-1,R+1)\times S^1\to X$ with energy $ E(u)<\hbar$ defined on an annulus (with $ R>0$ ), we have the exponential decay estimates

(1) $ E(u|_{[-R+T,R-T]\times S^1})\le C^2e^{-2\mu T}E(u)$

(2) $ \sup_{[-R+T,R-T]\times S^1}\|du\|\le Ce^{-\mu T}\sqrt{E(u)}$

for all $ 0\le T\le R$ . Here, we take $ S^1 = \mathbb R/2\pi\mathbb Z$ and use the standard flat metric on the cylinder $ \mathbb R\times S^1$ and the metric on $ X$ to measure the norm $ \|du\|$ .

Now, if $ J$ were integrable, we can actually improve this estimate further in the following way: at the expense of decreasing $ \hbar$ and increasing $ C$ , we can actually take $ \mu=1$ in (1) and (2) above. The idea would be to use (2) to deduce that $ u|_{[-R,R]\times S^1}$ actually maps into a complex coordinate neighborhood on $ X$ where we can use the Fourier expansion of $ u$ along the cylinder $ [-R,R]\times S^1$ to obtain the desired estimate.

**I would like to know: is it possible to improve the estimate to $ \mu=1$ also in the case when $ J$ is not integrable? If so, a proof with some details or a reference would be appreciated. If not, what is the reason and is it possible to come up with a (counter-)example to illustrate this?**