Read-back: definition module FiniteEnergyPlane_Compactness

This module introduces four definitions: the predicate $Q.\mathrm{IsCZAtLeast}(m)$ on Reeb orbits, the sets $\mathcal{F}(H,J,P,K,\gamma)$ (fastPlaneSet) and $\mathcal{F}_L(H,J,P,K,\gamma)$ (fastPlaneSetL), and a sequential-compactness predicate for sets of maps $\mathbb{C}\to\mathbb{R}^4$ (IsSeqCompactCInftyLoc). All four work in the four-dimensional case. The ingredients they use are expanded under "Notions used" below.

1. $Q.\mathrm{IsCZAtLeast}(m)$

Let $H : \mathbb{R}^4 \to \mathbb{R}$ be any function, $Q = (x_Q, T_Q)$ a Reeb orbit of $H$ (see below), and $m\in\mathbb{Z}$. Let $\tilde H = H\circ \iota$, where $\iota:\mathbb{R}^4\to\mathbb{R}^4$ is the identity re-indexing of coordinates. $Q.\mathrm{IsCZAtLeast}(m)$ means the following. For every periodic orbit $(y, T_y)$ of the Hamiltonian system of $\tilde H$ that satisfies the two conditions

• (image) $\{x_Q(\tau) : \tau\in\mathbb{R}\} = \{y(\tau) : \tau\in\mathbb{R}\}$, and
• (action) $T_Q = \displaystyle\int_0^{T_y} \lambda_{y(\tau)}\big(y'(\tau)\big)\,d\tau$,

and for every linearized flow $Y$ along $y$, one has

$$m \le \mu_{\mathrm{CZ}}\big(\Psi_{\tilde H, y, Y}\big).$$

Here a periodic orbit of $\tilde H$ is a map $y:\mathbb{R}\to\mathbb{R}^4$ together with a number $T_y>0$ such that $y'(t) = X_{\tilde H}(y(t))$ and $\tilde H(y(t)) = 1$ for all $t$, and $y(t+T_y)=y(t)$. The vector field is $X_{\tilde H}(x) = (-\partial_1\tilde H, \partial_0\tilde H, -\partial_3\tilde H, \partial_2\tilde H)(x)$. $T_y$ is any positive period and need not be minimal. A linearized flow along $y$ is any $Y:\mathbb{R}\to L(\mathbb{R}^4)$ with $Y(0)=\mathrm{id}$ and $Y'(t) = DX_{\tilde H}(y(t))\circ Y(t)$ for all $t$. The index $\mu_{\mathrm{CZ}}$ and the path $\Psi$ are defined under "Notions used". If no pair $(y,T_y)$ satisfies the image and action conditions, or no such $Y$ exists, the predicate holds vacuously for every $m$.

2. $\mathcal{F}(H,J,P,K,\gamma)$ (fastPlaneSet)

The inputs are $H:\mathbb{R}^4\to\mathbb{R}$, $J:\mathbb{R}^4\to L(\mathbb{R}^4)$, a Reeb orbit $P$ of $H$, an arbitrary subset $K\subseteq\mathbb{R}^4$, and $\gamma\in\mathbb{R}$. The set consists of all maps $u:\mathbb{C}\to\mathbb{R}^4$ such that:

1. $u$ is a fast plane for $(H,J)$ (defined below);
2. $u$ has a positive exponentially convergent end at $\infty$ asymptotic to this particular $P$. That is, $\mathrm{NdEnd}(u, e_\infty, P, +1)$ holds with $e_\infty(s,t) = e^{2\pi s}e^{2\pi i t}$;
3. $u(0)\in K$;
4. $E_{d\lambda}(\{z : |z|>1\}, u) = \max(\gamma, 0)$ (this is the real number $\gamma$ cast into $[0,\infty]$, so for $\gamma\le 0$ the condition reads $E_{d\lambda}=0$).

Clause 1 has its own existential orbit, which is not required to equal $P$. Clause 2 is the separate requirement for $P$.

3. $\mathcal{F}_L(H,J,P,K,\gamma)$ (fastPlaneSetL)

This is the set of $u\in\mathcal{F}(H,J,P,K,\gamma)$ with $a(u(z)) \ge -L$ for every $z\in\mathbb{C}$. Here $a$ is the coordinate defined below. $L$ is an arbitrary real number.

4. IsSeqCompactCInftyLoc$(A)$

Let $A$ be a set of maps $\mathbb{C}\to\mathbb{R}^4$. The predicate means: for every sequence $(u_k)_{k\in\mathbb{N}}$ with all $u_k\in A$, there exist a strictly increasing $\varphi:\mathbb{N}\to\mathbb{N}$ and some $v\in A$ with the following property. For every compact $K\subseteq\mathbb{C}$ and every $m\in\mathbb{N}$, the $m$-th iterated Fréchet derivatives $D^m u_{\varphi(k)}$ converge uniformly on $K$ to $D^m v$ as $k\to\infty$. Iterated derivatives are taken to be $0$ wherever they do not exist. The limit $v$ must lie in $A$ itself.

Notions used

Space and forms. $\mathbb{R}^4$ has coordinates $0,1,2,3$. Throughout the module the norm on it is the maximum norm $\max_k|v_k|$.

• $j_0(v) = (-v_1, v_0, -v_3, v_2)$.
• $\omega_0(u,v) = u_0v_1-u_1v_0+u_2v_3-u_3v_2$.
• $\lambda_x(v) = \tfrac12\omega_0(x,v)$.
• $\nabla H(x)_k = DH(x)e_k$, where $D$ is the Fréchet derivative, taken to be $0$ where it does not exist.
• $X_H = j_0\nabla H$.
• The Reeb field is $R(x) = \lambda_x(X_H(x))^{-1}X_H(x)$, with the convention $0^{-1}=0$.
• $S=\{H=1\}$.
• $\xi_x = \{v : \lambda_x(v)=0,\ DH(x)v=0\}$.

Reeb orbit. A Reeb orbit $P=(x_P,T_P)$ of $H$ consists of $x_P:\mathbb{R}\to\mathbb{R}^4$ and $T_P>0$ with:

• $x_P(\tau)\in S$ and $x_P'(\tau)=R(x_P(\tau))$ for all $\tau$;
• $x_P(\tau+T_P)=x_P(\tau)$.

$T_P$ need not be the minimal period. $P$ is prime if $x_P(\tau)\ne x_P(0)$ for $0<\tau<T_P$.

Radial coordinates. $r(y)$ is a chosen $r>0$ with $H(ry)=1$ if one exists, and $r(y)=1$ otherwise. Then $\mathrm{proj}(y) = r(y)\,y$ and $a(y) = -\tfrac12\log r(y)$.

Induced almost complex structure. For $y\in\mathbb{R}^4$ put $x=\mathrm{proj}(y)$, $r=r(y)$, $\alpha(v) = \frac{r}{2DH(x)x}DH(x)v$ and $\beta(v) = r\lambda_x(v)$. Then

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

Derivatives and energies. For $f$ defined on $\mathbb{C}$, $\partial_sf(z) = Df(z)1$ and $\partial_tf(z)=Df(z)i$ (both $0$ where $f$ is not differentiable). For $u:\mathbb{C}\to\mathbb{R}^4$ write $a_u = a\circ u$ and $w_u=\mathrm{proj}\circ u$.

• $E_{d\lambda}(U,u) = \int_U \max\big(\omega_0(\partial_sw_u,\partial_tw_u),0\big)\,dz \in[0,\infty]$.
• $E(U,u)$ is the supremum, over all $C^\infty$ nondecreasing $\phi:\mathbb{R}\to[0,1]$, of

$$\int_U \max\Big(\phi'(a_u)\big(\partial_sa_u\,\lambda_{w_u}(\partial_tw_u) - \partial_ta_u\,\lambda_{w_u}(\partial_sw_u)\big) + \phi(a_u)\,\omega_0(\partial_sw_u,\partial_tw_u),\,0\Big)\,dz.$$

Finite-energy curve on $U$. This means all of the following:

• $u$ is $C^\infty$ on $U$ and $u(z)\neq0$ for $z\in U$;
• at every $z\in U$, $u$ is differentiable and $\partial_su(z)+\tilde J(u(z))\partial_tu(z)=0$;
• $E(U,u)<\infty$.

Exponential end. For $e:\mathbb{R}^2\to\mathbb{C}$, $\epsilon\in\mathbb{R}$ and a Reeb orbit $P$, $\mathrm{NdEnd}(u,e,P,\epsilon)$ means that there are $d,c\in\mathbb{R}$ and $b>0$ such that, as $s\to+\infty$ and uniformly in $t\in\mathbb{R}$,

$$a(u(e(s,t))) - \epsilon T_Ps - d \to 0, \qquad e^{bs}\big(\mathrm{proj}(u(e(s,t))) - x_P(\epsilon T_Pt+c)\big)\to 0.$$

No nondegeneracy of $P$ is part of this condition.

Fast plane. $u$ is a fast plane if all three of the following hold:

• $u$ is a finite-energy curve on all of $\mathbb{C}$;
• there is a prime Reeb orbit $P'$ with $\mathrm{NdEnd}(u,e_\infty,P',+1)$;
• the condition below on $\pi\partial_s$ holds.

For the third condition, define $\pi\partial_s(z) = \xi_{\mathbb{C}}\big(w_u(z),\, \partial_sw_u(z) - \lambda_{w_u(z)}(\partial_sw_u(z))R(w_u(z))\big)\in\mathbb{C}$. The condition says that its zero set $Z=\{z : \pi\partial_s(z)=0\}$ is finite, and that there are integers $\mathrm{idx}(z)$, $z\in Z$, with $\sum_{z\in Z}\mathrm{idx}(z)=0$. Each $\mathrm{idx}(z)$ must be a local index: $\pi\partial_s$ has no other zero in some closed disc $\bar B(z,r)$, $r>0$, and for every $\rho\in(0,r)$ the loop $t\mapsto\pi\partial_s(z+\rho e^{2\pi it})$ has winding number $\mathrm{idx}(z)$.

Winding number. A loop $f:\mathbb{R}\to\mathbb{C}$ has winding number $k$ if all of the following hold:

• $f$ is continuous, $1$-periodic and nowhere zero;
• some continuous $\theta:\mathbb{R}\to\mathbb{R}$ has $f(t)=|f(t)|e^{i\theta(t)}$ for all $t$;
• $\theta(1)-\theta(0)=2\pi k$.

$\xi$-coordinates. $\xi_{\mathbb{C}}(x,v) = \omega_0(v, e_2(x)) + i\,\omega_0(e_1(x),v)$, where

$$e_1(x) = |x|_2^{-1}\Big(q_2(x) - \tfrac{DH(x)q_2(x)}{DH(x)x}x\Big),\qquad e_2(x) = |x|_2^{-1}\Big(q_1(x) - \tfrac{DH(x)q_1(x)}{DH(x)x}x\Big).$$

Here $q_1(x)=(-x_2,x_3,x_0,-x_1)$, $q_2(x)=(x_3,x_2,-x_1,-x_0)$, and $|x|_2$ is the Euclidean norm.

Conley–Zehnder index of a path. Let $\Psi:\mathbb{R}\to\mathbb{R}^{2\times2}$ be a path of matrices.

• Its winding set $W$ is the set of numbers $(\theta(1)-s)/2\pi$ obtained as follows. Choose $s\in[0,2\pi]$ and a $\theta$ continuous on $[0,1]$ with $\theta(0)=s$ such that, for every $t\in[0,1]$, $\Psi(t)(\cos s,\sin s)$ is a positive multiple of $(\cos\theta(t),\sin\theta(t))$.
• With $a=\inf W$ and $b=\sup W$ (real infimum and supremum, which take junk values if $W$ is empty or unbounded), $\mu_{\mathrm{CZ}}(\Psi)=2(\lceil b\rceil-1)$ if $a\le\lceil b\rceil-1$, and $2\lceil b\rceil-1$ otherwise.

The path $\Psi_{\tilde H,y,Y}$. For an orbit $(y,T_y)$ and a linearized flow $Y$, the $(i,j)$ entry of $\Psi_{\tilde H,y,Y}(\tau)$ is the $i$-th real component of $\xi_{\mathbb{C}}$ at $y(T_y\tau)$ (computed with $\tilde H$) applied to

$$v - \frac{\lambda_{y(T_y\tau)}(v)}{\lambda_{y(T_y\tau)}(X_{\tilde H}(y(T_y\tau)))}X_{\tilde H}(y(T_y\tau)),\qquad v = Y(T_y\tau)\,e_{j+1}(y(0)).$$

The frame vectors here are $e_1(y(0))$ and $e_2(y(0))$, for $j=0,1$.