The $\xi$-part of $\partial_s u$ vanishes exactly where the $d\lambda$-density vanishes

Let $S=H^{-1}(1)\subset\mathbb{R}^4$ be a strictly star-shaped level with Liouville form $\lambda_0$, Reeb field $R$ and contact structure $\xi=\ker\lambda_0|_{TS}$, let $J$ be a $d\lambda_0$-compatible complex structure on $\xi$, and $\tilde J$ the induced almost complex structure on $\mathbb{R}^4\setminus\{0\}\cong\mathbb{R}\times S$. Let $U\subset\mathbb{C}$ be open and $u:U\to\mathbb{R}^4\setminus\{0\}$ be $\tilde J$-holomorphic, with $w=\mathrm{pr}\circ u:U\to S$. Then for every $z\in U$,

where $\pi:TS\to\xi$ is the projection along $R$, and $\pi\,\partial_s w(z)$ is read in Hryniewicz's global frame of $\xi$.

So the zeros of $\pi\partial_s u$ are exactly the points where the integrand $u^*d\lambda$ vanishes, and elsewhere this integrand is positive. This lets statements about the $d\lambda$-density be transferred to the section $\pi\partial_s u$ and back.

Formalization Note $\mathbb{R}^4$ is V 1, $\tilde J$ is Jtilde H J, $w=$ proj H ∘ u; $\pi\partial_s w$ in the global frame is piDs H u z (a complex number, xiCoordC), and $\omega_0(\partial_s w,\partial_t w)$ is dlambdaDensity H u z. The hypotheses only ask that $u$ avoid $0$ and be $\tilde J$-holomorphic on the open set $U$ (no energy bound).