Read-back: FiniteEnergyPlane.exists_subseq_gradNorm_bounded_off_finite

We work in $\mathbb{R}^4$, with coordinates $0,1,2,3$ and the maximum norm. The theorem takes the following data and hypotheses.

• $H:\mathbb{R}^4\to\mathbb{R}$ has a strictly star-shaped level.
• $J:\mathbb{R}^4\to L(\mathbb{R}^4)$ is a compatible $\xi$-structure.
• $P$ is a Reeb orbit of $H$.
• $K\subseteq\mathbb{R}^4$ is an arbitrary set; it need not be bounded or compact.
• $\gamma>0$ is a real number such that every Reeb orbit $Q$ with $T_Q\le T_P$ satisfies $\gamma<T_Q$. In particular this applies to $Q=P$.
• $(u_k)_{k\in\mathbb{N}}$ is a sequence with every $u_k\in\mathcal{F}(H,J,P,K,\gamma)$.

If $\mathcal{F}(H,J,P,K,\gamma)$ is empty, no such sequence exists and the statement is vacuous.

Conclusion. There exist a strictly increasing $\varphi:\mathbb{N}\to\mathbb{N}$ and a finite set $\Gamma\subset\mathbb{C}$ with the following properties.

1. $|z|\le1$ for every $z\in\Gamma$.
2. $\#\Gamma\cdot\gamma\le T_P$.
3. For every compact $K_c\subseteq\mathbb{C}\setminus\Gamma$ there is $M\in\mathbb{R}$ with

$$|\nabla u_{\varphi(k)}|(z)\le M\quad\text{for all }k\in\mathbb{N}\text{ and all }z\in K_c.$$

4. For every $z\in\Gamma$ there is a sequence $\zeta_k\to z$ in $\mathbb{C}$ with $|\nabla u_{\varphi(k)}|(\zeta_k)\to+\infty$.

The gradient size used in 3 and 4 is

$$|\nabla u|(z)=\big\|D(a\circ u)(z)\big\|+\big\|D(\mathrm{proj}\circ u)(z)\big\|,$$

a sum of operator norms of Fréchet derivatives (each $0$ where the map is not differentiable), computed with the modulus on $\mathbb{C}$ and the maximum norm on $\mathbb{R}^4$. $\Gamma=\varnothing$ is allowed.

Definitions used

Forms.

• $j_0(v)=(-v_1,v_0,-v_3,v_2)$.
• $\omega_0(u,v)=u_0v_1-u_1v_0+u_2v_3-u_3v_2$.
• $\lambda_x(v)=\tfrac12\omega_0(x,v)$.
• $X_H=j_0\nabla H$, with $\nabla H(x)_k=DH(x)e_k$.
• $R_H(x)=\lambda_x(X_H(x))^{-1}X_H(x)$, with $0^{-1}=0$.
• $\xi_x=\{v:\lambda_x(v)=0,\ DH(x)v=0\}$.

Strictly star-shaped level. $H$ is $C^\infty$; every $u\ne0$ has a unique $r>0$ with $H(ru)=1$; and $DH(x)x>0$ on $\{H=1\}$.

Compatible $\xi$-structure. $J$ is $C^\infty$, and for $H(x)=1$ and $v,w\in\xi_x$:

• $J(x)v\in\xi_x$;
• $J(x)^2v=-v$;
• $\omega_0(v,J(x)v)>0$ for $v\ne0$;
• $\omega_0(J(x)v,J(x)w)=\omega_0(v,w)$.

Reeb orbit. $Q=(x_Q,T_Q)$ with $T_Q>0$, $H(x_Q)=1$, $x_Q'=R_H(x_Q)$, and $x_Q(\tau+T_Q)=x_Q(\tau)$. The period $T_Q$ is not necessarily minimal. $Q$ is prime if $x_Q(\tau)\ne x_Q(0)$ for $0<\tau<T_Q$.

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 $a(y)=-\tfrac12\log r(y)$.

$\tilde J$. With $x=\mathrm{proj}(y)$, $r=r(y)$, $\alpha(v)=\frac{r}{2DH(x)x}DH(x)v$ and $\beta(v)=r\lambda_x(v)$:

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

Energies. Write $a_u=a\circ u$ and $w_u=\mathrm{proj}\circ u$, with $\partial_s f=Df(z)1$ and $\partial_t f=Df(z)i$.

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

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

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

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

Exponential end at $\infty$. $\mathrm{NdEnd}_\infty(u,P)$ means there are $d,c\in\mathbb{R}$ and $b>0$ with, as $s\to+\infty$ and uniformly in $t\in\mathbb{R}$,

$$a(u(e^{2\pi s}e^{2\pi it}))-T_Ps-d\to0,\qquad e^{bs}\big(\mathrm{proj}(u(e^{2\pi s}e^{2\pi it}))-x_P(T_Pt+c)\big)\to0.$$

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 $\theta$ has $f=|f|e^{i\theta}$;
• $\theta(1)-\theta(0)=2\pi k$.

$\xi$-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 $|\cdot|_2$ is the Euclidean norm.

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

• $u$ is a finite-energy curve on all of $\mathbb{C}$;
• $\mathrm{NdEnd}_\infty(u,P')$ holds for some prime Reeb orbit $P'$;
• the zero set $Z$ of $\pi\partial_s(z)=\xi_{\mathbb{C}}\big(w_u(z),\partial_sw_u(z)-\lambda_{w_u(z)}(\partial_sw_u(z))R_H(w_u(z))\big)$ is finite, with local indices summing to $0$.

A local index of $z\in Z$ is an integer $k$ such that, for some $r>0$, $\pi\partial_s$ has no other zero in $\bar B(z,r)$ and $t\mapsto\pi\partial_s(z+\rho e^{2\pi it})$ has winding number $k$ for all $\rho\in(0,r)$.

The set $\mathcal{F}(H,J,P,K,\gamma)$. It consists of the fast planes $u$ that also satisfy:

• $\mathrm{NdEnd}_\infty(u,P)$ for the given $P$;
• $u(0)\in K$;
• $E_{d\lambda}(\{|z|>1\},u)=\gamma$.