Read-back: FiniteEnergyPlane.embedded_fastPlane_fredholm

Statement. Let $H_0 : \mathbb{R}^4 \to \mathbb{R}$ have a strictly star-shaped level (1), and let $H$ be the same function regarded on $V = \mathbb{R}^4$ (the indexing set of 4 coordinates is re-identified; concretely $H(y) = H_0(y)$ coordinate by coordinate). For every compatible structure $J$ (2), every Reeb orbit $P$ of $H$ (4) with Conley–Zehnder index at least $3$ (9), and every map $u : \mathbb{C} \to \mathbb{R}^4$ such that

• $u$ is a fast plane for $(H, J)$ (6),
• $u$ has a nondegenerate end at $\infty$ asymptotic to $P$ (5),
• $u$ is an embedded plane (7),

the following three assertions hold, where $w = \pi \circ u$ and $\pi$ is the radial projection (3):

(A) For all $z \in \mathbb{C}$: $\pi(u(z)) \notin \{x_P(\tau) : \tau \in \mathbb{R}\}$.

(B) The map $w = \pi \circ u : \mathbb{C} \to \mathbb{R}^4$ is injective; for every $z$ the real derivative $Dw(z) : \mathbb{R}^2 \to \mathbb{R}^4$ is injective (so $w$ is differentiable at every $z$, since the derivative is $0$ by convention at non-differentiability points); and for every compact $C \subseteq \mathbb{R}^4$ disjoint from $\{x_P(\tau)\}$, the preimage $w^{-1}(C)$ is a bounded subset of $\mathbb{C}$.

(C) For every natural number $\ell \ge 1$ there exist a real $r > 0$ and a map $f : \mathbb{C} \times \mathbb{C} \to \mathbb{R}^4$, with $D_r = \{\tau \in \mathbb{C} : |\tau| < r\}$ the open complex disc, such that:

1. $f$ is real $C^\ell$ on $\mathbb{C} \times D_r$;
2. $f$ is injective on $\mathbb{C} \times D_r$;
3. at each $p \in \mathbb{C} \times D_r$ the real derivative $Df(p) : \mathbb{R}^4 \to \mathbb{R}^4$ of $f$ (as a map of $\mathbb{C} \times \mathbb{C} \cong \mathbb{R}^4$) is injective;
4. $f(z, 0) = u(z)$ for all $z \in \mathbb{C}$;
5. for every $\tau \in D_r$, the map $f_\tau = f(\cdot, \tau) : \mathbb{C} \to \mathbb{R}^4$ is a fast plane for $(H, J)$, has a nondegenerate end at $\infty$ asymptotic to $P$, and is an embedded plane;
6. for every $\tau_0 \in D_r$ and every sequence $(v_k)_{k \in \mathbb{N}}$ of maps $\mathbb{C} \to \mathbb{R}^4$ such that every $v_k$ is a fast plane for $(H, J)$, has a nondegenerate end at $\infty$ asymptotic to $P$ and is an embedded plane, and such that $v_k \to f_{\tau_0}$ in $C^\infty_{\mathrm{loc}}$ on $\mathbb{C}$ (8), there exist sequences $(\tau_k)$, $(A_k)$, $(B_k)$ of complex numbers with $\tau_k \to \tau_0$, $A_k \to 1$, $B_k \to 0$ such that for all sufficiently large $k$: $\tau_k \in D_r$ and

$$f(A_k z + B_k,\ \tau_k) = v_k(z) \quad \text{for all } z \in \mathbb{C}.$$

The radius $r$ and the map $f$ may depend on $\ell$; the regularity in (C.1) is the finite order $\ell$. The only hypothesis stated on $P$ is (9): nondegeneracy of $P$, conditions on other Reeb orbits, and primeness of $P$ are not hypotheses. "Fast plane" separately contains the existence of some prime Reeb orbit $P'$ as asymptotic limit, not stated to be $P$. The conclusion of (C.6) concerns only all sufficiently large $k$.

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

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

$X_H(x) = j_0 \nabla H(x) = (-\partial_1 H, \partial_0 H, -\partial_3 H, \partial_2 H)(x)$, 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 $g$ on $\mathbb{C} = \{s + it\}$: $\partial_s g(z) = Dg(z)\cdot 1$, $\partial_t g(z) = Dg(z)\cdot i$, with $Dg(z)$ the real Fréchet derivative, taken to be $0$ where $g$ is not differentiable.

(1) Strictly star-shaped level. $H_0$ is $C^\infty$; for every $u \ne 0$ there is exactly one $r > 0$ with $H_0(ru) = 1$; and $d(H_0)_x(x) > 0$ whenever $H_0(x) = 1$.

(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 $x = \pi(y)$, $r = \rho(y)$, $\alpha(w) = \frac{r}{2\, dH_x(x)} dH_x(w)$, $\beta(w) = r \lambda_x(w)$:

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

(4) Reeb orbits. A Reeb orbit $Q$ is a pair $(x_Q, T_Q)$ with $x_Q : \mathbb{R} \to \mathbb{R}^4$, $T_Q > 0$, and $H(x_Q(\tau)) = 1$, $x_Q'(\tau) = R(x_Q(\tau))$, $x_Q(\tau + T_Q) = x_Q(\tau)$ for all $\tau$. $T_Q$ need not be the minimal period. $Q$ is prime if $x_Q(\tau) \ne x_Q(0)$ for $0 < \tau < T_Q$.

(5) Nondegenerate end at $\infty$ asymptotic to $Q$. With $e(s, t) = e^{2\pi s} e^{2\pi i t}$, a map $g : \mathbb{C} \to \mathbb{R}^4$ has this property if there are $d, c \in \mathbb{R}$ and $b > 0$ with

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

The constants $d, c, b$ may differ between maps $g$ (in (C.5) and (C.6) they may differ with $\tau$ and with $k$).

(6) Fast plane. $g : \mathbb{C} \to \mathbb{R}^4$ is a fast plane for $(H, J)$ if: $g$ is $C^\infty$ on $\mathbb{C}$, never $0$, and $\tilde J$-holomorphic ($g$ differentiable and $\partial_s g + \tilde J(g)\, \partial_t g = 0$ at every $z$); its energy

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

is finite ($\varphi$ over $C^\infty$ nondecreasing $\mathbb{R} \to [0, 1]$; here $\mathbf a = a \circ g$, $w = \pi \circ g$); there is some prime Reeb orbit $P'$ such that $g$ 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 $\sum_{z \in Z} k_z = 0$, where each $z$ is an isolated zero (nonzero on $\overline B(z, \varrho) \setminus \{z\}$ for some $\varrho > 0$) and for all $\rho \in (0, \varrho)$ 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. $g$ is injective and $Dg(z) : \mathbb{R}^2 \to \mathbb{R}^4$ is injective for every $z$ (in particular $g$ is differentiable everywhere).

(8) $C^\infty_{\mathrm{loc}}$ convergence on $\mathbb{C}$. For every compact $C \subseteq \mathbb{C}$ and every $m \in \mathbb{N}$, the iterated real derivatives $D^m v_k \to D^m f_{\tau_0}$ uniformly on $C$ as $k \to \infty$. (Here $D^m f_{\tau_0}$ is the iterated derivative of $f_{\tau_0}$ on all of $\mathbb{C}$.)

(9) Conley–Zehnder index at least $3$ of $P$. 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 $\{y(t)\} = \{x_P(\tau)\}$ as sets and

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

and for every $Y : \mathbb{R} \to L(\mathbb{R}^4)$ with $Y(0) = \mathrm{id}$ and $Y'(t) = DX_H(y(t))\, Y(t)$ for all $t$, 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 $\mathrm{CZ}(M)$: 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 $\underline w = \inf W$, $\overline w = \sup W$ (each $0$ by convention if $W$ is empty or unbounded on that side). 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 $\Gamma$ or $Y$ exists the condition holds vacuously.