Gradient bound near a puncture of a finite-energy curve

Let $S=H^{-1}(1)\subset\mathbb R^{2n+2}$ be a strictly star-shaped level with contact form $\lambda=\lambda_0|_S$ and $d\lambda$-compatible complex structure $J$ on $\xi$, and let $u=(a,w):\dot D=D\setminus\{0\}\to\mathbb R\times S\cong\mathbb R^{2n+2}\setminus\{0\}$ be a finite-energy $\tilde J$-holomorphic curve on the punctured unit disc. Write $Q(\rho)=\int_0^1 w(\gamma_\rho(t))^*\lambda$ for the action of the clockwise loop $\gamma_\rho(t)=\rho e^{-2\pi i t}$, and call $|z|\,|\nabla u(z)|\le C$ for $0<|z|<r$ the gradient bound near the puncture, where $|\nabla u|=\|Da\|+\|Dw\|$ ($\mathbb R$-invariant). Then there are $r>0$ and $C$ with $|z|\,|\nabla u(z)|\le C$ for all $0<|z|<r$. Otherwise there are $z_k\to0$ with $|z_k|\,|\nabla u(z_k)|\to\infty$; rescaling $u_k(\zeta)=u(|z_k|\zeta)$ on $0<|\zeta|<1/|z_k|$ and bubbling off near $\zeta_k=z_k/|z_k|$ (Hofer's lemma) gives a nonconstant finite-energy plane $v$ whose $d\lambda$-energy on each disc $B_\rho(0)$ is bounded by the liminf of the $d\lambda$-energies of $u$ on discs contained in $B_{2|z_k|}(0)\setminus\{0\}$. These tend to $0$ because $\int_{\dot D}w^*d\lambda\le E(u)<\infty$, so $\int_{\mathbb C}v^*d\lambda=0$, contradicting the positivity of the $d\lambda$-energy of nonconstant finite-energy planes.