which maps the zero section to $N$; in other words, we can write any nearby submanifold to $N$ as the graph of a normal vector field on $N$. We know that $N$ is Lagrangian, so the complex structure $J$ gives an isomorphism between $\nu(N)$ and $TN$ and the metric gives an isomorphism between $TN$ and $T^*N$: $v\mapsto g(Jv,.)=\omega(v,.)=\alpha_v$. Therefore any deformation of $N$ in $T$ is given as the graph of a 1-form. In fact, using the Lagrangian neighbourhood theorem, we can arrange that any $N'\in T$ is the graph of a 1-form $\alpha$, so that if $f_\alpha:N\rightarrow N_{\alpha}$ is the natural diffeomorphism then $$f_{\alpha}^*(\omega)=\d\alpha\quad\text{and}\quad -*f_{\alpha}^*(\operatorname{Im}\Upsilon)=F(\alpha,\nabla\alpha)=\d^*\alpha+Q(\alpha,\nabla\alpha),$$ where the second formula follows from a calculation using the special Lagrangian condition on $N$ and the fact that the ambient structure is Calabi–Yau. Hence, $N_{\alpha}$ is special Lagrangian if and only if $P(\alpha)=(F(\alpha,\nabla\alpha),\d\alpha)=0$. This means that infinitesimal special Lagrangian deformations are given by closed and coclosed 1-forms, which is the kernel of $L_0P$.

Since $\operatorname{Im}\Upsilon=0$ on $N$ we have that $[\operatorname{Im}\Upsilon]=0$ on $N_\alpha$, which means that $f_{\alpha}^*(\operatorname{Im}\Upsilon)$ is exact. Thus $F(\alpha,\nabla\alpha)=-*f_{\alpha}^*(\operatorname{Im}\Upsilon)$ is coexact and so $$P:C^{\infty}(S)\rightarrow \d^*(C^{\infty}(T^*N))\oplus\d(C^{\infty}(T^*N))\subseteq C^{\infty}(\Lambda^0T^*N\oplus\Lambda^2T^*N).$$ If we let $X=C^{1,a}(T^*N)$, $Y=\d^*(C^{1,a}(T^*N))\oplus\d(C^{1,a}(T^*N))$ and $U=C^{1,a}(S)$ we can apply the Implicit Function Theorem if 