Cauchy–Riemann equations of a J̃-holomorphic curve in the symplectization coordinates along a cylindrical end

Let $S = H^{-1}(1) \subset \mathbb{R}^{2n+2}$ be a strictly star-shaped level, $\lambda_0 = \tfrac12 \sum_k (q_k\,dp_k - p_k\,dq_k)$ the Liouville form, and $J$ a compatible complex structure on the contact planes $\xi$. Let $\tilde J$ be the induced almost complex structure on $\mathbb{R}^{2n+2} \setminus \{0\} \cong \mathbb{R} \times S$, with coordinates $y = e^{2a} w$, $w \in S$.

Let $u : \mathbb{D} \setminus \{0\} \to \mathbb{R}^{2n+2} \setminus \{0\}$ be $\tilde J$-holomorphic, and write $a(s,t)$, $w(s,t)$ for its coordinates in the positive cylindrical coordinates $(s, t) \mapsto e^{-2\pi(s + it)}$ at the puncture $0$.

Then for every $s > 0$ and $t \in \mathbb{R}$, $a$ and $w$ are differentiable at $(s,t)$ and

This expresses the $\mathbb{R}$-component of the Cauchy–Riemann equation $\partial_s u + \tilde J(u)\,\partial_t u = 0$ in the coordinates on an end, the form used to compare $a$ with the period coordinate of a nearby Reeb orbit.

Formalization Note. $u$ is assumed $\tilde J$-holomorphic and nonvanishing on ball 0 1 \ {0}; the end coordinates are endA H u cylEnd and endW H u cylEnd, partial derivatives are fderiv applied to (1, 0) and (0, 1), and $\lambda_0(w)(v)$ is liouville w v.