Read-back: FiniteEnergyPlane.exists_subseq_tendsto_of_fastPlaneSet

Ambient notation. Work in $V = \mathbb{R}^4$ with coordinates $y_0, y_1, y_2, y_3$ (the dimension $2(n+1)$ with $n = 1$). Let $\omega_0(u,v) = u_0 v_1 - u_1 v_0 + u_2 v_3 - u_3 v_2$, $\lambda_x(v) = \tfrac12 \omega_0(x, v)$, $j_0(v) = (-v_1, v_0, -v_3, v_2)$, $(\nabla H(x))_k = dH_x(e_k)$, and $X_H = j_0 \nabla H = (-\partial_1 H, \partial_0 H, -\partial_3 H, \partial_2 H)$. The Reeb field is $R_H(x) = \lambda_x(X_H(x))^{-1} X_H(x)$ (with $0^{-1} = 0$). Write $\xi_x = \{ v : \lambda_x(v) = 0,\ dH_x(v) = 0\}$ and $S_H = \{H = 1\}$.

• Strictly star-shaped level $H$: $H$ is $C^\infty$; every $u \ne 0$ has exactly one $r > 0$ with $H(ru) = 1$; $dH_x(x) > 0$ whenever $H(x) = 1$.
• Compatible $\xi$-structure $J$: $J : V \to \mathrm{End}(V)$ is $C^\infty$, and for every $x \in S_H$ and $v, w \in \xi_x$: $J(x)v \in \xi_x$, $J(x)^2 v = -v$, $\omega_0(v, J(x)v) > 0$ if $v \ne 0$, $\omega_0(J(x)v, J(x)w) = \omega_0(v,w)$. Nothing is required off $S_H$ or outside $\xi_x$.
• Scaling, projection, $a$-coordinate. $s_H(y)$ is some $r > 0$ with $H(ry) = 1$ if one exists (chosen by choice), otherwise $1$; $\pi_H(y) = s_H(y)\, y$; $a_H(y) = -\tfrac12 \log s_H(y)$. As $s_H > 0$ always, $e^{-2a_H(y)} = s_H(y)$, so $e^{-2a_H(y)} y = \pi_H(y)$.
• $\tilde J$: for $y \in V$, with $x = \pi_H(y)$, $r = s_H(y)$, $\alpha(v) = \frac{r\, dH_x(v)}{2 dH_x(x)}$, $\beta(v) = r \lambda_x(v)$, $w(v) = v - 2\alpha(v) y - \beta(v) r^{-1} R_H(x)$: $\ \tilde J(y) v = \alpha(v) r^{-1} R_H(x) - 2\beta(v) y + J(x) w(v)$.
• Derivatives. For $f$ on $\mathbb{C}$: $\partial_s f(z) = Df(z)[1]$, $\partial_t f(z) = Df(z)[i]$ (real Fréchet derivative; $0$ where not differentiable).
• Energies. Admissible $\varphi$: $C^\infty$, non-decreasing, values in $[0,1]$. With $a = a_H \circ u$, $w = \pi_H \circ u$ and $e_\varphi = \varphi'(a)(\partial_s a\, \lambda_w(\partial_t w) - \partial_t a\, \lambda_w(\partial_s w)) + \varphi(a)\, \omega_0(\partial_s w, \partial_t w)$, the energy is $E_H(U,u) = \sup_\varphi \int_U \max(e_\varphi, 0) \in [0,\infty]$, and the $d\lambda$-energy is $E^{d\lambda}_H(U,u) = \int_U \max(\omega_0(\partial_s w, \partial_t w), 0) \in [0,\infty]$ (Lebesgue measure; negative parts of the densities are cut off before integrating).
• Finite-energy curve on $U$: $u : \mathbb{C} \to V$ is $C^\infty$ on $U$, $u(z) \ne 0$ on $U$, at every $z \in U$ it is differentiable with $\partial_s u(z) + \tilde J(u(z))\, \partial_t u(z) = 0$, and $E_H(U,u) < \infty$.
• Reeb orbit $Q$: a map $x_Q : \mathbb{R} \to V$ and a number $T_Q > 0$ with $x_Q(\tau) \in S_H$, $x_Q'(\tau) = R_H(x_Q(\tau))$, $x_Q(\tau + T_Q) = x_Q(\tau)$. $T_Q$ is any period, not necessarily minimal, so multiple covers count as separate orbits. $Q$ is prime if $x_Q(\tau) \ne x_Q(0)$ for $0 < \tau < T_Q$. $Q$ is nondegenerate if for every $\Phi : \mathbb{R} \to \mathrm{End}(V)$ with $\Phi(0) = \mathrm{id}$ and $\Phi'(\tau) = DR_H(x_Q(\tau)) \circ \Phi(\tau)$ for all $\tau$, every $v \in \xi_{x_Q(0)}$ with $\Phi(T_Q)v = v$ is $0$.
• Hamiltonian periodic orbits and the index $\mu$. A Hamiltonian periodic orbit $R$ of a function $G$ on $\mathbb{R}^4$ is $x_R : \mathbb{R} \to \mathbb{R}^4$, $T_R > 0$ with $x_R' = X_G(x_R)$, $G(x_R) = 1$, $x_R(t + T_R) = x_R(t)$. A linearized flow along $x_R$ is $Y : \mathbb{R} \to \mathrm{End}(\mathbb{R}^4)$ with $Y(0) = \mathrm{id}$, $Y'(t) = DX_G(x_R(t)) \circ Y(t)$. With $Q_1 x = (-x_2, x_3, x_0, -x_1)$, $Q_2 x = (x_3, x_2, -x_1, -x_0)$, $|x|$ the Euclidean norm, set $F_1(x) = |x|^{-1}\big(Q_2 x - \tfrac{dG_x(Q_2 x)}{dG_x(x)} x\big)$, $F_2(x) = |x|^{-1}\big(Q_1 x - \tfrac{dG_x(Q_1 x)}{dG_x(x)} x\big)$ (division by $0$ read as multiplication by $0$), the coordinates $c_x(v) = \big(\omega_0(v, F_2(x)),\ \omega_0(F_1(x), v)\big) \in \mathbb{R}^2$, and $p_x(v) = v - \frac{\lambda_x(v)}{\lambda_x(X_G(x))} X_G(x)$. The linearized $\xi$-path is the $2 \times 2$ matrix path

$$\phi_{R,Y}(\tau) = \Big[\, c_{x_R(T_R\tau)}\big(p_{x_R(T_R\tau)}\big(Y(T_R\tau)\, F_j(x_R(0))\big)\big) \,\Big]_{j = 1,2} \quad (\text{column } j).$$

For a $2\times 2$ matrix path $\phi$, an angle lift from $s$ is $\theta : \mathbb{R} \to \mathbb{R}$, continuous on $[0,1]$, with $\theta(0) = s$ and, for each $t \in [0,1]$, $\phi(t)(\cos s, \sin s) = r(\cos\theta(t), \sin\theta(t))$ for some $r > 0$. Let $I(\phi) = \{ (\theta(1) - s)/(2\pi) : s \in [0, 2\pi],\ \theta \text{ an angle lift from } s \}$, $a = \inf I(\phi)$, $b = \sup I(\phi)$ (real infimum/supremum, which take the value $0$ when $I(\phi)$ is empty or unbounded on that side), and

$$\mu(\phi) = \begin{cases} 2(\lceil b \rceil - 1) & \text{if } a \le \lceil b \rceil - 1, \\ 2\lceil b \rceil - 1 & \text{otherwise.} \end{cases}$$

• "CZ index at least $m$" for a Reeb orbit $Q$ of $H$: for every Hamiltonian periodic orbit $R$ of $G = H$ with $x_R(\mathbb{R}) = x_Q(\mathbb{R})$ and $T_Q = \int_0^{T_R} \lambda_{x_R(\tau)}(x_R'(\tau))\, d\tau$, and every linearized flow $Y$ along $x_R$, $m \le \mu(\phi_{R,Y})$. (Vacuous if no such $R$ exists.)
• Cylindrical end at infinity $e_\infty(s,t) = e^{2\pi s} e^{2\pi i t}$, with $W(s,t) = \pi_H(u(e_\infty(s,t)))$, $A(s,t) = a_H(u(e_\infty(s,t)))$. $u$ has a nondegenerate end at $\infty$ asymptotic to $Q$ if there are $d, c \in \mathbb{R}$, $b > 0$ with $\sup_t |A(s,t) - T_Q s - d| \to 0$ and $\sup_t e^{bs} |W(s,t) - x_Q(T_Q t + c)| \to 0$ as $s \to +\infty$.
• Winding and zero index. A loop $f : \mathbb{R} \to \mathbb{C}$ has winding $k \in \mathbb{Z}$ if it is continuous, $1$-periodic, never $0$, and $f(t) = |f(t)| e^{i\theta(t)}$ for some continuous $\theta$ with $\theta(1) - \theta(0) = 2\pi k$. $f : \mathbb{C} \to \mathbb{C}$ has zero index $m$ at $z_0$ if $f(z_0) = 0$ and for some $r > 0$, $f \ne 0$ on $0 < |z - z_0| \le r$ and every loop $t \mapsto f(z_0 + \rho e^{2\pi i t})$, $\rho \in (0, r)$, has winding $m$.
• $\Pi_u$: with $\zeta_x(v) = \omega_0(v, F_2(x)) + i\, \omega_0(F_1(x), v)$ (the frame $F_j$ built from $G = H$) and $\pi^\xi_x(v) = v - \lambda_x(v) R_H(x)$, put $\Pi_u(z) = \zeta_p\big(\pi^\xi_p(\partial_s(\pi_H \circ u)(z))\big)$ with $p = \pi_H(u(z))$.
• Fast plane: $u$ is a finite-energy curve on all of $\mathbb{C}$; there is a prime Reeb orbit $Q$ such that $u$ has a nondegenerate end at $\infty$ asymptotic to $Q$; and the zero set of $\Pi_u$ in $\mathbb{C}$ is a finite set $Z$ for which there are integers $\mathrm{idx}(z)$, each the zero index of $\Pi_u$ at $z$, with $\sum_{z \in Z} \mathrm{idx}(z) = 0$.
• The set $\mathcal{F}(P, K, \gamma)$ (for a Reeb orbit $P$, $K \subseteq V$, $\gamma \in \mathbb{R}$): all $u : \mathbb{C} \to V$ that are fast planes, have a nondegenerate end at $\infty$ asymptotic to $P$, satisfy $u(0) \in K$, and satisfy $E^{d\lambda}_H(\{|z| > 1\}, u) = \max(\gamma, 0)$.
• $C^\infty_{\mathrm{loc}}$ convergence on $O \subseteq \mathbb{C}$: $f_k \to g$ means that for every compact $K' \subseteq O$ and every $m \in \mathbb{N}$, the $m$-th iterated real Fréchet derivatives $D^m f_k$ converge to $D^m g$ uniformly on $K'$ (the iterated derivatives are total functions, defined at every point).
• Sequential $C^\infty_{\mathrm{loc}}$-compactness of a set $\mathcal{A}$ of maps $\mathbb{C} \to V$: for every sequence $(u_k)$ in $\mathcal{A}$ there are a strictly increasing $\varphi : \mathbb{N} \to \mathbb{N}$ and $v \in \mathcal{A}$ with $u_{\varphi(k)} \to v$ in $C^\infty_{\mathrm{loc}}$ on all of $\mathbb{C}$.

Common hypotheses. $H : \mathbb{R}^4 \to \mathbb{R}$ is a strictly star-shaped level function and $J$ is a compatible $\xi$-structure. $P$ is a Reeb orbit of $H$ such that every Reeb orbit $Q$ with $T_Q \le T_P$ (including $P$ itself and all multiple covers with period $\le T_P$) is nondegenerate and has CZ index at least $3$ in the sense above. $\gamma \in \mathbb{R}$ satisfies $\gamma > 0$, $\gamma < T_Q$ for every Reeb orbit $Q$ with $T_Q \le T_P$, and $\gamma < |T_{Q_1} - T_{Q_2}|$ for all Reeb orbits $Q_1, Q_2$ with $T_{Q_1}, T_{Q_2} \le T_P$ and $T_{Q_1} \ne T_{Q_2}$. $K \subseteq \mathbb{R}^4$ is compact with $0 \notin K$ ($K = \emptyset$ is allowed).

Statement. Under the common hypotheses, let $(u_k)_{k \in \mathbb{N}}$ be a sequence with $u_k \in \mathcal{F}(P, K, \gamma)$ for every $k$. Then there exist a strictly increasing $\varphi : \mathbb{N} \to \mathbb{N}$, a finite set $\Gamma \subset \mathbb{C}$ with $|z| \le 1$ for every $z \in \Gamma$, and a map $w : \mathbb{C} \to \mathbb{R}^4$ such that:

1. $w$ is a finite-energy curve on $\mathbb{C} \setminus \Gamma$;
2. $0 < E_H(\mathbb{C} \setminus \Gamma, w) \le T_P$;
3. the rescaled maps

$$\tilde u_k(z) = e^{-2 a_H(u_{\varphi(k)}(2))}\, u_{\varphi(k)}(z) = s_H\big(u_{\varphi(k)}(2)\big)\, u_{\varphi(k)}(z)$$

(rescaling by the constant attached to the value at the point $z = 2 \in \mathbb{C}$) converge to $w$ in $C^\infty_{\mathrm{loc}}$ on $\mathbb{C} \setminus \Gamma$;

and at least one of the following two alternatives holds (inclusive "or"):

(a) for every $M \in \mathbb{R}$, for all sufficiently large $k$ there is $z \in \mathbb{C}$ with $a_H(u_{\varphi(k)}(z)) \le M$; $\Gamma$ has exactly one element; $\Pi_w(z) = 0$ for every $z \notin \Gamma$; and there is $y \in K$ with $\pi_H(y) \in x_P(\mathbb{R})$.

(b) there is $A \in \mathbb{R}$ such that

• for every $\varepsilon > 0$, for all sufficiently large $k$: $a_H(u_{\varphi(k)}(z)) \ge A - \varepsilon$ for all $z \in \mathbb{C}$;
• for every $\varepsilon > 0$, for all sufficiently large $k$: there is $z$ with $a_H(u_{\varphi(k)}(z)) \le A + \varepsilon$;
• $\Gamma = \emptyset$;

and there is $c \in \mathbb{R}$ such that $a_H(u_{\varphi(k)}(2)) \to c$, the unrescaled subsequence $u_{\varphi(k)}$ converges in $C^\infty_{\mathrm{loc}}$ on all of $\mathbb{C}$ to $e^{2c} w$, the map $e^{2c} w$ belongs to $\mathcal{F}(P, K, \gamma)$, and $a_H(e^{2c} w(z)) \ge A$ for every $z \in \mathbb{C}$.

Remarks on the literal content. The energy bound in 2 compares an element of $[0, \infty]$ with the real number $T_P > 0$. In (a), the first clause says $\inf_z a_H(u_{\varphi(k)}(z))$ eventually drops below every bound; nothing is said there about convergence of $u_{\varphi(k)}$ itself. In (a), "$\Pi_w(z) = 0$ off $\Gamma$" is computed from the total-function definition of $\Pi_w$. The common hypotheses on Reeb orbits only concern orbits with period $\le T_P$.