Read-back: FiniteEnergyPlane.isEmbeddedPlane_of_tendstoCInftyLocOn

Statement. Let $H : \mathbb{R}^4 \to \mathbb{R}$ be a function whose level set $S = \{H = 1\}$ is strictly star-shaped (in the sense (1) below), let $J$ be a compatible complex structure on the contact planes (in the sense (2) below), let $(u_k)_{k \in \mathbb{N}}$ be a sequence of maps $u_k : \mathbb{C} \to \mathbb{R}^4$ and $v : \mathbb{C} \to \mathbb{R}^4$ a map. Assume:

• every $u_k$ is a fast plane for $(H, J)$ (definition (6) below);
• every $u_k$ is an embedded plane (definition (7) below);
• $v$ is a fast plane for $(H, J)$;
• $u_k \to v$ in $C^\infty_{\mathrm{loc}}$ on all of $\mathbb{C}$ (definition (8) below).

Then $v$ is an embedded plane. No other hypotheses appear; in particular nothing is assumed about the asymptotic orbits of the $u_k$ or $v$ being the same, about energies, or about the Reeb orbits beyond what is contained in "fast plane".

Notation used throughout. $\mathbb{R}^4$ is the space of real 4-tuples $x = (x_0, x_1, x_2, x_3)$, normed by the sup norm $\|x\| = \max_i |x_i|$. Put

$$j_0(v) = (-v_1,\, v_0,\, -v_3,\, v_2), \qquad \omega_0(u, v) = \langle j_0 u, v\rangle = u_0 v_1 - u_1 v_0 + u_2 v_3 - u_3 v_2, \qquad \lambda_x(v) = \tfrac12\, \omega_0(x, v).$$

$\nabla H(x)$ is the vector of partial derivatives $\big(dH_x(e_i)\big)_i$, $X_H(x) = j_0 \nabla H(x)$, and the Reeb field is

$$R(x) = \lambda_x\big(X_H(x)\big)^{-1}\, X_H(x),$$

with the convention $0^{-1} = 0$. Let $\xi_x = \{ w : \lambda_x(w) = 0 \text{ and } dH_x(w) = 0\}$. For $f$ defined on $\mathbb{C} = \{s + it\}$ write $\partial_s f(z) = Df(z)\cdot 1$ and $\partial_t f(z) = Df(z)\cdot i$, where $Df(z)$ is the real Fréchet derivative (equal to $0$ by convention where $f$ is not differentiable).

(1) Strictly star-shaped level. $H$ is $C^\infty$; for every $u \ne 0$ there is exactly one $r > 0$ with $H(ru) = 1$; and $dH_x(x) > 0$ for every $x$ with $H(x) = 1$.

(2) Compatible structure. $J$ is a $C^\infty$ map from $\mathbb{R}^4$ to real linear maps $\mathbb{R}^4 \to \mathbb{R}^4$ such that, for every $x$ with $H(x) = 1$ and all $v, w \in \xi_x$: $J(x)v \in \xi_x$; $J(x)J(x)v = -v$; $\omega_0(v, J(x)v) > 0$ if $v \ne 0$; and $\omega_0(J(x)v, J(x)w) = \omega_0(v, w)$. Nothing is required of $J(x)$ off $S$ or outside $\xi_x$.

(3) Radial coordinates. For $y \in \mathbb{R}^4$ let $\rho(y)$ be some chosen $r > 0$ with $H(ry) = 1$ if such an $r$ exists, and $\rho(y) = 1$ otherwise (for $y \ne 0$ it is the unique one by (1); for $y = 0$ it is an arbitrary positive number if $H(0) = 1$, and $1$ otherwise). Put

$$\pi(y) = \rho(y)\, y, \qquad a(y) = -\tfrac12 \log \rho(y).$$

The almost complex structure $\tilde J(y)$ on $\mathbb{R}^4$ is: with $x = \pi(y)$, $r = \rho(y)$, $\alpha(w) = \frac{r}{2\, dH_x(x)}\, dH_x(w)$, $\beta(w) = r\,\lambda_x(w)$,

$$\tilde J(y) w = \alpha(w)\, r^{-1} R(x) - 2\beta(w)\, y + J(x)\Big( w - 2\alpha(w)\, y - \beta(w)\, r^{-1} R(x) \Big).$$

(4) Reeb orbits. A Reeb orbit $P$ consists of a map $x_P : \mathbb{R} \to \mathbb{R}^4$ and a number $T_P > 0$ with $H(x_P(\tau)) = 1$, $x_P'(\tau) = R(x_P(\tau))$ and $x_P(\tau + T_P) = x_P(\tau)$ for all $\tau$ ($T_P$ is a period, not necessarily the minimal one). $P$ is prime if $x_P(\tau) \ne x_P(0)$ for all $0 < \tau < T_P$.

(5) Nondegenerate end at $\infty$. For $u : \mathbb{C} \to \mathbb{R}^4$ and a Reeb orbit $P$, write $e(s, t) = e^{2\pi s} e^{2\pi i t}$. "$u$ has a nondegenerate end at $\infty$ asymptotic to $P$" means: there are real $d, c$ and $b > 0$ with

$$\sup_{t \in \mathbb{R}} \big| a(u(e(s,t))) - T_P\, s - d \big| \to 0, \qquad \sup_{t \in \mathbb{R}} e^{bs}\, \big\| \pi(u(e(s,t))) - x_P(T_P t + c) \big\| \to 0 \qquad (s \to +\infty).$$

(6) Fast plane. $u : \mathbb{C} \to \mathbb{R}^4$ is a fast plane for $(H, J)$ if all of the following hold:

• $u$ is $C^\infty$ on $\mathbb{C}$, $u(z) \ne 0$ for all $z$, and $u$ is $\tilde J$-holomorphic on $\mathbb{C}$: at every $z$, $u$ is real-differentiable and $\partial_s u(z) + \tilde J(u(z))\, \partial_t u(z) = 0$;
• the energy is finite: with $\mathcal{A}$ the set of $C^\infty$, nondecreasing $\varphi : \mathbb{R} \to [0, 1]$, and with $\mathbf a = a \circ u$, $w = \pi \circ u$,

$$E(u) = \sup_{\varphi \in \mathcal{A}} \int_{\mathbb{C}} \max\Big(0,\; \varphi'(\mathbf a)\big(\partial_s \mathbf a\, \lambda_w(\partial_t w) - \partial_t \mathbf a\, \lambda_w(\partial_s w)\big) + \varphi(\mathbf a)\, \omega_0(\partial_s w, \partial_t w)\Big)\, dz < \infty$$

(Lebesgue measure, value in $[0, \infty]$);

• there exists a prime Reeb orbit $P$ such that $u$ has a nondegenerate end at $\infty$ asymptotic to $P$ (5);
• the "winding of $\pi_\xi \partial_s w$" condition: let $\Pi(z) = \zeta_{w(z)}\big(\partial_s w(z) - \lambda_{w(z)}(\partial_s w(z))\, R(w(z))\big) \in \mathbb{C}$, where $\zeta_x(v) = \omega_0(v, F_2(x)) + i\, \omega_0(F_1(x), v)$ with

$$F_1(x) = |x|^{-1}\Big(Q_2 x - \tfrac{dH_x(Q_2 x)}{dH_x(x)}\, x\Big), \quad F_2(x) = |x|^{-1}\Big(Q_1 x - \tfrac{dH_x(Q_1 x)}{dH_x(x)}\, x\Big),$$

$Q_1 x = (-x_2, x_3, x_0, -x_1)$, $Q_2 x = (x_3, x_2, -x_1, -x_0)$, $|x|$ the Euclidean norm. Then: the zero set $Z$ of $\Pi$ in $\mathbb{C}$ is finite, and there are integers $k_z$ ($z \in Z$) with $\sum_{z \in Z} k_z = 0$ such that each zero $z$ is isolated with index $k_z$, meaning: there is $r > 0$ with $\Pi \ne 0$ on $\overline{B}(z, r) \setminus \{z\}$, and for every $\rho \in (0, r)$ the loop $t \mapsto \Pi(z + \rho e^{2\pi i t})$ is continuous, 1-periodic, nowhere zero, and has a continuous argument $\theta$ ($\Pi = |\Pi| e^{i\theta}$) with $\theta(1) - \theta(0) = 2\pi k_z$. (If $\Pi$ has no zeros this holds with $Z = \emptyset$; the condition only requires the indices to sum to $0$.)

(7) Embedded plane. $u : \mathbb{C} \to \mathbb{R}^4$ is an embedded plane if $u$ is injective and for every $z \in \mathbb{C}$ the real derivative $Du(z) : \mathbb{C} \cong \mathbb{R}^2 \to \mathbb{R}^4$ is injective. (Since the derivative is taken to be $0$ where $u$ is not differentiable, this includes differentiability at every point.)

(8) $C^\infty_{\mathrm{loc}}$ convergence on $\mathbb{C}$. For every compact $K \subseteq \mathbb{C}$ and every $m \in \mathbb{N}$, the $m$-th iterated real Fréchet derivatives $D^m u_k$ converge to $D^m v$ uniformly on $K$ as $k \to \infty$ (for $m = 0$: $u_k \to v$ uniformly on $K$).