The operator $(u,v)\mapsto du+J_{\mathcal{C}}dv$ has a kernel which consists of pairs $(u,v)$ such that $u+\sqrt{-1}v$ is holomorphic. In our case this is eliminated since we have the growth control (5.28). Notice that the smallest growth rate of a non-constant holomorphic function on $\mathcal{C}$ is $e^{\frac{1}{2}z^{2}}.$