FiniteEnergyPlane.exists_shiftA_comp_affine_of_range_eq

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 $\mathcal{P}$ satisfying the Conley–Zehnder bound $\mathrm{CZ}(P)\ge3$, and let $u,v:\mathbb{C}\to\mathbb{R}^4$ be two maps such that, for each $f\in\{u,v\}$,

• $f$ is a fast plane for $(H,J)$;
• $f$ has a nondegenerate end at infinity at the same orbit $P$, i.e. $\mathrm{NE}(f,P)$;
• $f$ is embedded;

and such that $\mathrm{pr}(u(\mathbb{C}))=\mathrm{pr}(v(\mathbb{C}))$ as subsets of $\mathbb{R}^4$. Then there exist $c\in\mathbb{R}$ and $A,B\in\mathbb{C}$ with $A\neq0$ such that for every $z\in\mathbb{C}$,

$$u(Az+B)=e^{2c}\,v(z).$$

Here $e^{2c}\,v(z)$ is ordinary scalar multiplication of the vector $v(z)\in\mathbb{R}^4$. For $y\neq0$ one has $\mathrm{pr}(e^{2c}y)=\mathrm{pr}(y)$ and $a(e^{2c}y)=a(y)+c$. The numbers $c,A,B$ are only asserted to exist, not to be unique. $P$ is not assumed prime. The prime orbit in the definition of "fast plane" is existentially quantified separately for $u$ and for $v$ and is not stated to equal $P$. The Conley–Zehnder bound is a statement about $P$ only, and it may hold vacuously as described below.

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.)

Embedded plane. $u:\mathbb{C}\to\mathbb{R}^4$ is called embedded if $u$ is injective and, for every $z\in\mathbb{C}$, the real-linear map $Du(z):\mathbb{C}\cong\mathbb{R}^2\to\mathbb{R}^4$ is injective. (This is a condition on $u$ itself, not on $\mathrm{pr}\circ u$.)

Conley–Zehnder bound $\mathrm{CZ}(P)\ge 3$. This hypothesis says: $3\le\mu(\gamma,Y)$ for all of the following data:

• every $\gamma:\mathbb{R}\to\mathbb{R}^4$ and $T_\gamma>0$ with $\gamma'(t)=X_H(\gamma(t))$, $H(\gamma(t))=1$ and $\gamma(t+T_\gamma)=\gamma(t)$ for all $t$ ($T_\gamma$ not necessarily minimal),
• such that $\gamma(\mathbb{R})=\mathcal{P}$ and $T_P=\displaystyle\int_0^{T_\gamma}\lambda_{\gamma(\tau)}\big(\gamma'(\tau)\big)\,d\tau$,
• and every $Y:\mathbb{R}\to\mathrm{End}(\mathbb{R}^4)$ with $Y(0)=\mathrm{id}$ and $Y'(t)=DX_H(\gamma(t))\circ Y(t)$ for all $t$.

If no such $\gamma$ exists, the hypothesis holds vacuously. The quantity $\mu(\gamma,Y)\in\mathbb{Z}$ is computed from the $2\times2$ real matrix path

$$\Phi(\tau)_{ij}=\Big[\zeta^{\mathbb{R}^2}_{\gamma(T_\gamma\tau)}\Big(\Pi_{\gamma(T_\gamma\tau)}\,Y(T_\gamma\tau)\,f_j\Big)\Big]_i,\qquad f_1=e_1(\gamma(0)),\ f_2=e_2(\gamma(0)),$$

where $\zeta^{\mathbb{R}^2}_x(v)=\big(\omega_0(v,e_2(x)),\,\omega_0(e_1(x),v)\big)$ and $\Pi_x v=v-\dfrac{\lambda_x(v)}{\lambda_x(X_H(x))}X_H(x)$ (division by $0$ gives $0$). Let $W\subset\mathbb{R}$ be the set of numbers $(\theta(1)-s)/(2\pi)$ over all $s\in[0,2\pi]$ and all $\theta$ continuous on $[0,1]$ with $\theta(0)=s$ such that for each $t\in[0,1]$, $\Phi(t)(\cos s,\sin s)^{\mathsf T}$ is a positive multiple of $(\cos\theta(t),\sin\theta(t))^{\mathsf T}$. With $\alpha=\inf W$, $\beta=\sup W$ (both taken to be $0$ if $W$ is empty or unbounded on that side),

$$\mu(\gamma,Y)=\begin{cases}2(\lceil\beta\rceil-1)&\text{if }\alpha\le\lceil\beta\rceil-1,\\ 2\lceil\beta\rceil-1&\text{otherwise.}\end{cases}$$