Read-back: FiniteEnergyPlane.tendsto_aCoord_atTop_of_small_area

Ambient notation. Fix $n \in \mathbb{N}$ (any value, including $n = 0$) and write $V_n = \mathbb{R}^{2(n+1)}$ with coordinates $y_0, \dots, y_{2n+1}$. Let $j_0$ be the linear map with $(j_0 v)_{2i} = -v_{2i+1}$ and $(j_0 v)_{2i+1} = v_{2i}$, let $\omega_0(u,v) = \sum_k (j_0 u)_k v_k = \sum_i (u_{2i} v_{2i+1} - u_{2i+1} v_{2i})$, and let $\lambda_x(v) = \tfrac12 \omega_0(x, v)$. For $H : V_n \to \mathbb{R}$ put $(\nabla H(x))_k = dH_x(e_k)$, $X_H = j_0 \nabla H$, and define the Reeb field

with the convention $0^{-1} = 0$. Write $\xi_x = \{ v : \lambda_x(v) = 0,\ dH_x(v) = 0 \}$ and $S_H = \{H = 1\}$.

• Hypothesis on $H$ ("strictly star-shaped level"): $H$ is $C^\infty$; for every $u \ne 0$ there is exactly one $r > 0$ with $H(ru) = 1$; and $dH_x(x) > 0$ whenever $H(x) = 1$.
• Hypothesis on $J$ ("compatible $\xi$-structure"): $J : V_n \to \mathrm{End}(V_n)$ is $C^\infty$, and for every $x \in S_H$ and all $v, w \in \xi_x$: $J(x)v \in \xi_x$, $J(x)^2 v = -v$, $\omega_0(v, J(x)v) > 0$ if $v \ne 0$, and $\omega_0(J(x)v, J(x)w) = \omega_0(v,w)$. Nothing is required of $J(x)$ off $S_H$ or outside $\xi_x$.
• Scaling, projection and $a$-coordinate. For $y \in V_n$, $s_H(y)$ is some $r > 0$ with $H(ry) = 1$ if such an $r$ exists (chosen by choice), and $s_H(y) = 1$ otherwise. Put $\pi_H(y) = s_H(y)\, y$ and $a_H(y) = -\tfrac12 \log s_H(y)$. Since $s_H(y) > 0$ always, $y = e^{2a_H(y)} \pi_H(y)$.
• The almost complex structure $\tilde J$ on $V_n$: for $y \in V_n$ let $x = \pi_H(y)$, $r = s_H(y)$, $\alpha(v) = \frac{r\, dH_x(v)}{2\, dH_x(x)}$, $\beta(v) = r \lambda_x(v)$, $w(v) = v - 2\alpha(v)\, y - \beta(v)\, r^{-1} R_H(x)$, and

• Derivatives. For $f : \mathbb{C} \to F$, $\partial_s f(z) = Df(z)[1]$ and $\partial_t f(z) = Df(z)[i]$ (Fréchet derivatives over $\mathbb{R}$; equal to $0$ where $f$ is not differentiable).
• Energy. An admissible $\varphi : \mathbb{R} \to \mathbb{R}$ is $C^\infty$, monotone non-decreasing, with values in $[0,1]$. For $u : \mathbb{C} \to V_n$ write $a = a_H \circ u$, $w = \pi_H \circ u$ and

For $U \subseteq \mathbb{C}$, $E_H(U, u) = \sup_{\varphi \text{ admissible}} \int_U \max(e_\varphi, 0)\, d\mathrm{Leb} \in [0, \infty]$ (negative parts of the density are cut off before integrating). The $d\lambda$-energy is $E^{d\lambda}_H(U,u) = \int_U \max\big(\omega_0(\partial_s w, \partial_t w), 0\big)\, d\mathrm{Leb} \in [0,\infty]$.

• Finite-energy curve on $U$: $u$ is $C^\infty$ on $U$ (in the sense of smoothness on the set $U$), $u(z) \ne 0$ for $z \in U$, at every $z \in U$ the map $u$ is differentiable and $\partial_s u(z) + \tilde J(u(z))\, \partial_t u(z) = 0$, and $E_H(U,u) < \infty$.
• Reeb orbit: a map $x : \mathbb{R} \to V_n$ and a number $T > 0$ with $x(\tau) \in S_H$, $x'(\tau) = R_H(x(\tau))$ and $x(\tau + T) = x(\tau)$ for all $\tau$. $T$ is any period, not necessarily the minimal one, so multiple covers are included.
• Loop action. For $\rho \in \mathbb{R}$ let $c_\rho(t) = \rho\, e^{2\pi i t}$. The loop action of $u$ along $c_\rho$ is

where $\ell'(t)$ is taken as $0$ at points where $\ell$ is not differentiable, and the integral is the Lebesgue interval integral (taken as $0$ if the integrand is not integrable).

Statement. Let $H$ be a strictly star-shaped level function and $J$ a compatible $\xi$-structure as above. Let $T, \gamma, e, N \in \mathbb{R}$ with $\gamma > 0$ and $e > 0$ ($T$ and $N$ are unrestricted in sign). Assume every Reeb orbit $Q$ of $H$ with period $Q.T \le T$ satisfies $\gamma < Q.T$. Let $(r_k)_{k \in \mathbb{N}}$ be positive reals with $r_k \to 0$, and $(v_k)_{k \in \mathbb{N}}$ maps $\mathbb{C} \to V_n$ such that, for every $k$:

1. $v_k$ is a finite-energy curve on all of $\mathbb{C}$;
2. $E_H(\mathbb{C}, v_k) \le \max(T, 0)$;
3. $E^{d\lambda}_H\big(\{ z : |z| > r_k \}, v_k\big) \le \gamma$;
4. for every $\rho > r_k$, $e \le \mathcal{A}(v_k, \rho)$;
5. $a_H(v_k(z)) \ge -N$ for every $z \in \mathbb{C}$.

Then for every compact set $K_c \subseteq \mathbb{C}$ with $0 \notin K_c$ and every $M \in \mathbb{R}$ there is $k_0$ such that for all $k \ge k_0$ and all $z \in K_c$,

Edge cases visible in the statement. $K_c = \emptyset$ is allowed (the conclusion is then trivial). If $T < 0$, condition 2 says $E_H(\mathbb{C}, v_k) = 0$. The hypothesis on Reeb orbits concerns only orbits with period $\le T$; if no such orbit exists it is vacuous, and when it is not vacuous it says no Reeb orbit has period in $(0, \min(T, \gamma)]$. Condition 4 bounds the loop action from below for every radius $\rho > r_k$, with the same $e$ for all $k$. Condition 5 is a uniform lower bound on the $a$-coordinate, with the same $N$ for all $k$ and $z$.