Curves of zeros of the weighted operator give families of $\bar J$-holomorphic planes

Let $\rho, b > 0$, let $(\bar J, J_\infty, S_\infty)$ be a standard tube structure (IsStdTubeStructure ρ b Jbar Jinf Sinf), let $\delta < 0$ and $\delta_T \in (\max(\delta, -2\pi), 0)$, let $\chi_T, \chi_N, \beta$ satisfy IsSlopeWeight χT π (2π + δT), IsSlopeWeight χN π δ and IsLeftCutoff β, and let $l \in \mathbb{N}$. Then there is $\varepsilon_0 > 0$ with the following property.

Let $r > 0$ and let $\gamma : \mathbb{R} \to W^{1,4} \times \mathbb{C} \times W^{1,4}$ be $C^\infty$ on $(-r, r)$, with $\gamma(0) = 0$, $\gamma'(0) = k$, and, for every $t \in (-r, r)$, $\gamma(t) \in B_{\varepsilon_0}(0)$ and qMap Jbar χT χN β (γ t) = 0. Then there are $r_1 > 0$ and $G : \mathbb{C} \times \mathbb{R} \to \mathbb{C} \times \mathbb{C}$, $C^l$ on $\mathbb{C} \times (-r_1, r_1)$, such that

• $G(z, 0) = (z, 0)$ for all $z$, and $\big[\partial_t G(0,0)\big]_2 = k_2$, the $\mathbb{C}$-component of $k$;
• for each $t \in (-r_1, r_1)$ the map $G_t = (\psi_t, v_t)$ is $C^\infty$, takes values in $\mathbb{C} \times B_\rho(0)$, is $\bar J$-holomorphic on $\mathbb{C}$, and satisfies, for some $C$ and $\beta' > 0$, $|\psi_t(z) - z| \le C(1+|z|)^{1-\beta'}$, $|v_t(z)| \le C(1+|z|)^{\delta/2\pi}$, and for every $z$ there is $\Delta$ with $\big[DG(z,t)(\Delta,1)\big]_1 = 0$ and $\big|\big[DG(z,t)(\Delta,1)\big]_2\big| \le C(1+|z|)^{\delta/2\pi}$;
• for every $\varepsilon > 0$ there is $r' \in (0, r_1]$ such that for $|t| < r'$ and all $z$: $|\psi_t(z) - z| \le \varepsilon(1+|z|)$, $|v_t(z)| \le \varepsilon$, $\|D\psi_t(z) - \mathrm{id}\| \le \varepsilon$ and $(1+|z|)\|Dv_t(z)\| \le \varepsilon$.

The hypothesis that every member $\gamma(t)$ is a zero is essential.

Role. Weak solutions of the $\bar J$-Cauchy–Riemann equation in weighted Sobolev coordinates are turned into an honest family of smooth $\bar J$-holomorphic planes, jointly regular in the parameter, with pointwise weighted bounds.