Read-back: FiniteEnergyPlane.isSeqCompact_embeddedFastPlaneSet

Statement. Let $H : \mathbb{R}^4 \to \mathbb{R}$ have a strictly star-shaped level set (1), let $J$ be a compatible structure (2), and let $P$ be a Reeb orbit (4). Let $\gamma \in \mathbb{R}$, and $K \subseteq \mathbb{R}^4$. Assume:

• for every Reeb orbit $Q$ (any parametrized closed Reeb orbit with any period, see (4)) with $T_Q \le T_P$: $Q$ is nondegenerate (9) and $Q$ has Conley–Zehnder index at least $3$ in the sense (10);
• $\gamma > 0$;
• $\gamma < T_Q$ for every Reeb orbit $Q$ with $T_Q \le T_P$ (taking $Q = P$, this includes $\gamma < T_P$);
• $\gamma < |T_{Q_1} - T_{Q_2}|$ for all Reeb orbits $Q_1, Q_2$ with $T_{Q_1}, T_{Q_2} \le T_P$ and $T_{Q_1} \ne T_{Q_2}$;
• $K$ is compact, $0 \notin K$, and $\pi(y) \notin \{x_P(\tau) : \tau \in \mathbb{R}\}$ for every $y \in K$.

Then the set $\mathcal{M} = \mathcal{M}(H, J, P, K, \gamma)$ of all maps $u : \mathbb{C} \to \mathbb{R}^4$ such that

• $u$ is a fast plane for $(H, J)$ (6),
• $u$ has a nondegenerate end at $\infty$ asymptotic to $P$ (5),
• $u(0) \in K$,
• $\displaystyle\int_{\{|z| > 1\}} \max\big(0,\, \omega_0(\partial_s w, \partial_t w)\big)\, dz = \gamma$, where $w = \pi \circ u$ (Lebesgue integral with values in $[0, \infty]$; equal to the real number $\gamma > 0$),
• $u$ is an embedded plane (7),

is sequentially compact in $C^\infty_{\mathrm{loc}}$: for every sequence $(u_k)_{k \in \mathbb{N}}$ in $\mathcal{M}$ there are a strictly increasing $\phi : \mathbb{N} \to \mathbb{N}$ and some $v \in \mathcal{M}$ with $u_{\phi(k)} \to v$ in $C^\infty_{\mathrm{loc}}$ on $\mathbb{C}$ (8). (If $\mathcal{M}$ is empty the conclusion holds vacuously.)

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

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

$\nabla H(x) = \big(dH_x(e_i)\big)_i$, $X_H(x) = j_0 \nabla H(x) = (-\partial_1 H, \partial_0 H, -\partial_3 H, \partial_2 H)$, and the Reeb field is $R(x) = \lambda_x(X_H(x))^{-1} X_H(x)$ (with $0^{-1} = 0$). Let $\xi_x = \{ w : \lambda_x(w) = 0,\ dH_x(w) = 0\}$. For $f$ on $\mathbb{C} = \{s + it\}$, $\partial_s f(z) = Df(z)\cdot 1$ and $\partial_t f(z) = Df(z)\cdot i$, with $Df(z)$ the real Fréchet derivative ($0$ by convention where $f$ is not differentiable; the same convention holds for all derivatives below).

(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 $dH_x(x) > 0$ whenever $H(x) = 1$.

(2) Compatible structure. $J$ is a $C^\infty$ map from $\mathbb{R}^4$ to 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 off $\{H = 1\}$ or outside $\xi_x$.

(3) Radial coordinates. $\rho(y)$ is a chosen $r > 0$ with $H(ry) = 1$ if one exists, and $1$ otherwise (unique for $y \ne 0$ by (1)). $\pi(y) = \rho(y)\, y$ and $a(y) = -\tfrac12 \log \rho(y)$. With $x = \pi(y)$, $r = \rho(y)$, $\alpha(w) = \frac{r}{2\, dH_x(x)} dH_x(w)$, $\beta(w) = r \lambda_x(w)$, the structure $\tilde J$ is

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

(4) Reeb orbits. A Reeb orbit $Q$ is a pair $(x_Q, T_Q)$ with $x_Q : \mathbb{R} \to \mathbb{R}^4$, $T_Q > 0$, $H(x_Q(\tau)) = 1$, $x_Q'(\tau) = R(x_Q(\tau))$, $x_Q(\tau + T_Q) = x_Q(\tau)$ for all $\tau$. $T_Q$ is any period (multiple covers and time-shifted parametrizations are separate orbits). $Q$ is prime if $x_Q(\tau) \ne x_Q(0)$ for $0 < \tau < T_Q$. The quantifiers "for every Reeb orbit $Q$ with $T_Q \le T_P$" above range over all such pairs, prime or not.

(5) Nondegenerate end at $\infty$. With $e(s, t) = e^{2\pi s} e^{2\pi i t}$: there are $d, c \in \mathbb{R}$ and $b > 0$ with

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

(6) Fast plane. $u$ is $C^\infty$ on $\mathbb{C}$, never $0$, and $\tilde J$-holomorphic ($u$ differentiable and $\partial_s u + \tilde J(u)\, \partial_t u = 0$ at every $z$); its energy

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

is finite ($\varphi$ over $C^\infty$ nondecreasing $\mathbb{R} \to [0, 1]$, $\mathbf a = a \circ u$, $w = \pi \circ u$); there is some prime Reeb orbit $P'$ (not necessarily $P$) such that $u$ has a nondegenerate end at $\infty$ asymptotic to $P'$; and: with $\Pi(z) = \zeta_{w(z)}\big(\partial_s w(z) - \lambda_{w(z)}(\partial_s w(z)) R(w(z))\big)$, where $\zeta_x(v) = \omega_0(v, F_2(x)) + i\,\omega_0(F_1(x), v)$, $F_1(x) = |x|^{-1}\big(Q_2 x - \frac{dH_x(Q_2x)}{dH_x(x)} x\big)$, $F_2(x) = |x|^{-1}\big(Q_1 x - \frac{dH_x(Q_1x)}{dH_x(x)} x\big)$, $Q_1 x = (-x_2, x_3, x_0, -x_1)$, $Q_2 x = (x_3, x_2, -x_1, -x_0)$, $|x|$ Euclidean: the zero set $Z$ of $\Pi$ is finite and there are integers $k_z$, $z \in Z$, with $\sum_{z \in Z} k_z = 0$, each $z$ being an isolated zero (nonzero on $\overline B(z, r) \setminus \{z\}$ for some $r > 0$) such that for all $\rho \in (0, r)$ the loop $t \mapsto \Pi(z + \rho e^{2\pi i t})$ is continuous, 1-periodic, nowhere zero, with a continuous argument $\theta$ satisfying $\theta(1) - \theta(0) = 2\pi k_z$.

(7) Embedded plane. $u$ is injective and $Du(z) : \mathbb{R}^2 \to \mathbb{R}^4$ is injective for every $z$ (in particular $u$ is differentiable everywhere).

(8) $C^\infty_{\mathrm{loc}}$ convergence. For every compact $C \subseteq \mathbb{C}$ and every $m \in \mathbb{N}$, $D^m u_{\phi(k)} \to D^m v$ uniformly on $C$.

(9) Nondegenerate orbit. For every map $\Phi : \mathbb{R} \to L(\mathbb{R}^4)$ with $\Phi(0) = \mathrm{id}$ and $\Phi'(\tau) = DR(x_Q(\tau))\, \Phi(\tau)$ for all $\tau$, and every $v \in \xi_{x_Q(0)}$: $\Phi(T_Q) v = v$ implies $v = 0$. (Stated for every such $\Phi$; it would hold vacuously if there were none.)

(10) Conley–Zehnder index at least $3$. For every Hamiltonian periodic orbit $\Gamma = (y, T)$ of $H$ (i.e. $T > 0$, $y'(t) = X_H(y(t))$ and $H(y(t)) = 1$ for all $t$, $y(t + T) = y(t)$) such that $\{y(t)\} = \{x_Q(\tau)\}$ as sets and

$$T_Q = \int_0^{T} \lambda_{y(\tau)}\big(y'(\tau)\big)\, d\tau,$$

and for every $Y : \mathbb{R} \to L(\mathbb{R}^4)$ with $Y(0) = \mathrm{id}$ and $Y'(t) = DX_H(y(t))\, Y(t)$, one has $3 \le \mathrm{CZ}(M)$, where $M : \mathbb{R} \to \mathbb{R}^{2\times 2}$ is

$$M(\tau)_{ij} = \Big[\zeta^{\mathbb{R}^2}_{y(T\tau)}\Big( Y(T\tau) f_j - \tfrac{\lambda_{y(T\tau)}(Y(T\tau) f_j)}{\lambda_{y(T\tau)}(X_H(y(T\tau)))}\, X_H(y(T\tau)) \Big)\Big]_i,$$

$f_1 = F_1(y(0))$, $f_2 = F_2(y(0))$, $\zeta^{\mathbb{R}^2}_x(v) = \big(\omega_0(v, F_2(x)),\ \omega_0(F_1(x), v)\big)$. The integer $\mathrm{CZ}(M)$ is defined as follows: let $W$ be the set of numbers $(\theta(1) - s)/(2\pi)$ over all $s \in [0, 2\pi]$ and all $\theta : \mathbb{R} \to \mathbb{R}$ continuous on $[0, 1]$ with $\theta(0) = s$ and $M(t)(\cos s, \sin s) = r_t (\cos\theta(t), \sin\theta(t))$ for some $r_t > 0$, for each $t \in [0, 1]$. Let $\underline w = \inf W$, $\overline w = \sup W$ (with the convention that the infimum/supremum of an empty or unbounded set is $0$). Then $\mathrm{CZ}(M) = 2(\lceil \overline w\rceil - 1)$ if $\underline w \le \lceil \overline w \rceil - 1$, and $\mathrm{CZ}(M) = 2\lceil \overline w\rceil - 1$ otherwise. If no such $\Gamma$ exists the condition holds vacuously.