Normalized rescalings at a puncture converge to a curve with zero dλ-energy

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. Assume the gradient bound near the puncture: there are $r>0$ and $C$ with $|z|\,|\nabla u(z)|\le C$ for $0<|z|<r$, where $|\nabla u|=\|Da\|+\|Dw\|$. Let $\rho_k\to0$ with $\rho_k>0$. Then there are a subsequence $\rho_{k_j}$ and a finite-energy $\tilde J$-holomorphic curve $v:\mathbb C\setminus\{0\}\to\mathbb R^{2n+2}\setminus\{0\}$ with $\int_{\mathbb C\setminus\{0\}}v^*d\lambda=0$ such that the normalized rescalings

converge to $v$ in $C^\infty_{\mathrm{loc}}(\mathbb C\setminus\{0\})$.

This is the puncture counterpart of exists_subseq_tendsto_rescaled_plane (rescalings of a plane at infinity); it is the blow-up step in the asymptotic analysis of a finite-energy curve at a nonremovable puncture.

Formalization Note In the coordinates $y=e^{2a}x$ ($x\in S$) of $\mathbb R^{2n+2}\setminus\{0\}$, multiplying by $e^{-2a(u(\rho_k))}$ is the $\mathbb R$-translation by $-a(u(\rho_k))$. $u$ is a function on all of $\mathbb C$; only its values on ball 0 1 \ {0} matter. The rescaled maps are defined on all of $\mathbb C$ and are $\tilde J$-holomorphic on $\{0<|z|<1/\rho_k\}$. $C^\infty_{\mathrm{loc}}$ convergence is JHolomorphic.TendstoCInftyLocOn on {z | z ≠ 0}; the gradient bound is the hypothesis hG with gradNorm.