FiniteEnergyPlane.isProperEmbedding_proj_of_isFastPlane

Statement. Let $H:\mathbb{R}^4\to\mathbb{R}$ be a Hamiltonian with strictly star-shaped level $S=\{H=1\}$, let $J$ be a compatible structure on $\xi$, let $P$ be a Reeb orbit of $H$ with image $\mathcal{P}$, and let $u:\mathbb{C}\to\mathbb{R}^4$ be a map such that

• $u$ is a fast plane for $(H,J)$;
• $u$ has a nondegenerate end at infinity at $P$, i.e. $\mathrm{NE}(u,P)$;
• $\mathrm{pr}(u(z))\notin\mathcal{P}$ for every $z\in\mathbb{C}$.

Then the map $\mathrm{pr}\circ u:\mathbb{C}\to\mathbb{R}^4$ satisfies all three of:

1. $\mathrm{pr}\circ u$ is injective;
2. for every $z\in\mathbb{C}$, the real-linear map $D(\mathrm{pr}\circ u)(z):\mathbb{C}\to\mathbb{R}^4$ is injective;
3. for every compact $K\subset\mathbb{R}^4$ with $K\cap\mathcal{P}=\emptyset$, the preimage $(\mathrm{pr}\circ u)^{-1}(K)\subset\mathbb{C}$ is bounded.

The compact set $K$ in item 3 is any compact subset of $\mathbb{R}^4$ disjoint from $\mathcal{P}$; it is not required to lie in $S$ (the preimage only depends on $K\cap\mathrm{pr}(u(\mathbb{C}))$). $P$ is not assumed prime, and no Conley–Zehnder bound on $P$ and no embeddedness of $u$ is assumed. The prime orbit in the definition of "fast plane" is existentially quantified and is not stated to equal $P$. The conclusion does not assert continuity of the inverse or any statement about the closure of the image beyond item 3.

Meaning of every symbol used above

Ambient space and linear structures. $\mathbb{R}^4$ has coordinates $y=(y_0,y_1,y_2,y_3)$. The fixed linear maps and forms are

$$j_0(v)=(-v_1,\;v_0,\;-v_3,\;v_2),\qquad \omega_0(u,v)=\langle j_0u,v\rangle=u_0v_1-u_1v_0+u_2v_3-u_3v_2,\qquad \lambda_x(v)=\tfrac12\,\omega_0(x,v),$$

where $\langle\cdot,\cdot\rangle$ is the Euclidean dot product.

The Hamiltonian $H$ (hypothesis "strictly star-shaped level"). $H:\mathbb{R}^4\to\mathbb{R}$ is $C^\infty$; for every $y\neq 0$ there is exactly one $r>0$ with $H(ry)=1$; and $dH_x(x)>0$ at every $x$ with $H(x)=1$. Write $S=\{H=1\}$. Nothing is assumed about $H(0)$. Further notation:

• $\nabla H(x)$ is the vector of partial derivatives, $X_H(x)=j_0\nabla H(x)$;
• the Reeb field is $R(x)=\lambda_x(X_H(x))^{-1}\,X_H(x)$, with the convention $0^{-1}=0$ (so $R(x)=0$ wherever $\lambda_x(X_H(x))=0$);
• $\xi_x=\{v\in\mathbb{R}^4:\ \lambda_x(v)=0 \text{ and } dH_x(v)=0\}$.

The structure $J$ (hypothesis "compatible on $\xi$"). $J:\mathbb{R}^4\to\mathrm{End}_{\mathbb{R}}(\mathbb{R}^4)$ is $C^\infty$, and for every $x\in S$ and all $v,w\in\xi_x$:

$$J_xv\in\xi_x,\qquad J_xJ_xv=-v,\qquad v\neq0\Rightarrow \omega_0(v,J_xv)>0,\qquad \omega_0(J_xv,J_xw)=\omega_0(v,w).$$

No condition is imposed on $J_x$ outside $\xi_x$, nor for $x\notin S$.

Radial coordinates. For $y\in\mathbb{R}^4$ let $\sigma(y)$ be a (chosen) $r>0$ with $H(ry)=1$ if such an $r$ exists, and $\sigma(y)=1$ otherwise; for $y\neq0$ this $r$ is the unique one from the star-shapedness hypothesis. Put

$$\mathrm{pr}(y)=\sigma(y)\,y,\qquad a(y)=-\tfrac12\log\sigma(y).$$

For $y\neq0$, $\mathrm{pr}(y)$ is the point of $S$ on the ray through $y$ and $y=e^{2a(y)}\,\mathrm{pr}(y)$; always $\mathrm{pr}(0)=0$.

The Reeb orbit $P$. $P$ consists of a map $x_P:\mathbb{R}\to\mathbb{R}^4$ and a number $T_P>0$ such that $x_P(\tau)\in S$, $x_P'(\tau)=R(x_P(\tau))$ and $x_P(\tau+T_P)=x_P(\tau)$ for all $\tau$. The period $T_P$ is not required to be the minimal period. Its image is $\mathcal{P}=x_P(\mathbb{R})\subset\mathbb{R}^4$. A Reeb orbit $Q$ is called prime if $x_Q(\tau)\neq x_Q(0)$ for all $0<\tau<T_Q$.

The almost complex structure $\tilde J$ on $\mathbb{R}^4$. For $y\in\mathbb{R}^4$ let $x=\mathrm{pr}(y)$, $r=\sigma(y)$, and define the linear forms $\alpha(v)=\dfrac{r\,dH_x(v)}{2\,dH_x(x)}$ and $\beta(v)=r\,\lambda_x(v)$ (division by $0$ gives $0$). Then

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

Plane coordinates. For $f:\mathbb{C}\to F$ and $z=s+it$, $\partial_sf(z)=Df(z)\,1$ and $\partial_tf(z)=Df(z)\,i$, where $Df(z)$ is the real derivative (taken to be $0$ where $f$ is not differentiable). For a map $u:\mathbb{C}\to\mathbb{R}^4$ write $a_u=a\circ u$ and $w_u=\mathrm{pr}\circ u$.

Finite-energy curve on $\mathbb{C}$. $u$ is $C^\infty$ on $\mathbb{C}$; $u(z)\neq0$ for every $z$; at every $z$, $u$ is differentiable and $\partial_su(z)+\tilde J(u(z))\,\partial_tu(z)=0$; and $E(u)<\infty$, where

$$E(u)=\sup_{\varphi}\int_{\mathbb{C}}\max\!\big(0,\,e_\varphi(z)\big)\,dz\in[0,\infty],$$ $$e_\varphi=\varphi'(a_u)\big(\partial_sa_u\,\lambda_{w_u}(\partial_tw_u)-\partial_ta_u\,\lambda_{w_u}(\partial_sw_u)\big)+\varphi(a_u)\,\omega_0(\partial_sw_u,\partial_tw_u),$$

the supremum ranging over all $C^\infty$, nondecreasing $\varphi:\mathbb{R}\to[0,1]$ (the integral is the Lebesgue integral of the nonnegative part).

Nondegenerate end at infinity at an orbit $Q$ (written $\mathrm{NE}(u,Q)$). With $c_\infty(s,t)=e^{2\pi s}e^{2\pi it}$, there exist $d,c\in\mathbb{R}$ and $b>0$ such that, as $s\to+\infty$, uniformly in $t\in\mathbb{R}$,

$$a\big(u(c_\infty(s,t))\big)-T_Q\,s-d\;\to\;0,\qquad e^{bs}\,\Big\|\mathrm{pr}\big(u(c_\infty(s,t))\big)-x_Q(T_Qt+c)\Big\|\;\to\;0 .$$

("Uniformly" means: for every $\varepsilon>0$ there is $s_0$ with the quantity at most $\varepsilon$ in absolute value/norm for all $s\ge s_0$ and all $t$.)

The function $\pi\partial_s$. For $x\in\mathbb{R}^4$ define $e_1(x)=|x|^{-1}\big(q_2(x)-\tfrac{dH_x(q_2(x))}{dH_x(x)}x\big)$ and $e_2(x)=|x|^{-1}\big(q_1(x)-\tfrac{dH_x(q_1(x))}{dH_x(x)}x\big)$ with $q_1(x)=(-x_2,x_3,x_0,-x_1)$, $q_2(x)=(x_3,x_2,-x_1,-x_0)$ and $|x|$ the Euclidean norm (again $0^{-1}=0$), and the complex coordinate $\zeta_x(v)=\omega_0(v,e_2(x))+i\,\omega_0(e_1(x),v)\in\mathbb{C}$. Then

$$\pi\partial_s u(z)=\zeta_{w_u(z)}\Big(\partial_sw_u(z)-\lambda_{w_u(z)}\big(\partial_sw_u(z)\big)\,R\big(w_u(z)\big)\Big)\in\mathbb{C}.$$

Winding of a loop. A map $f:\mathbb{R}\to\mathbb{C}$ has winding $k\in\mathbb{Z}$ if $f$ is continuous, $1$-periodic, nowhere zero, and there is a continuous $\theta:\mathbb{R}\to\mathbb{R}$ with $f(t)=|f(t)|e^{i\theta(t)}$ for all $t$ and $\theta(1)-\theta(0)=2\pi k$.

Fast plane. $u:\mathbb{C}\to\mathbb{R}^4$ is a fast plane if all three hold:

1. $u$ is a finite-energy curve on $\mathbb{C}$;
2. there exists some prime Reeb orbit $Q$ (not necessarily equal to $P$) with $\mathrm{NE}(u,Q)$;
3. the zero set $Z=\{z\in\mathbb{C}:\pi\partial_su(z)=0\}$ is finite, and there are integers $k_z$ ($z\in Z$) with $\sum_{z\in Z}k_z=0$ such that for each $z\in Z$ there is $r>0$ with $\pi\partial_su\neq0$ on the closed disc $\overline{B}(z,r)$ minus $z$, and for every $\rho\in(0,r)$ the loop $t\mapsto\pi\partial_su(z+\rho e^{2\pi it})$ has winding $k_z$. (If $Z$ is empty this condition holds automatically.)