FiniteEnergyPlane.punctured_disk_asymptotics

Statement. Let $n\in\mathbb N$, let $H:V\to\mathbb R$ satisfy the strictly star-shaped level hypothesis, let $J:V\to\mathcal L(V,V)$ be a compatible $\xi$-structure for $H$, and let $u:\mathbb C\to V$ be a finite-energy curve on the punctured open unit disk $D^*=\{z\in\mathbb C: |z|<1\}\setminus\{0\}$ (all terms are defined below). Then at least one of the following holds (the "or" is inclusive):

1. $u$ has a removable puncture at $0$ (radius $1$);
2. there is a real number $Q\neq0$ such that all four of the following hold:
   • the loop actions converge as the radius decreases to $0$:

$$\mathcal A(\rho)\longrightarrow Q\qquad(\rho\to0,\ \rho>0);$$

• if $Q>0$, then $a(u(z))\to+\infty$ as $z\to0$ in $\mathbb C$ with $z\neq0$;
• if $Q<0$, then $a(u(z))\to-\infty$ as $z\to0$ in $\mathbb C$ with $z\ne0$;
• for every sequence of radii $\rho_k>0$ with $\rho_k\to0$, there exist a strictly increasing $\phi:\mathbb N\to\mathbb N$, a Reeb orbit $P=(x,T)$ with $T=|Q|$, and a constant $c\in\mathbb R$ such that

$$\ell_{\rho_{\phi(k)}}\longrightarrow\big(t\mapsto x(Qt+c)\big)\quad\text{in }C^\infty\text{ on }\mathbb R\qquad(k\to\infty).$$

The subsequence $\phi$, the orbit $P$ and the shift $c$ may depend on the chosen sequence $(\rho_k)$; only $Q$ is fixed in advance. The statement does not say which alternative occurs, and does not exclude both.

Definitions used.

Ambient space and conventions. $n\in\mathbb N$ is arbitrary (including $n=0$). $V=\mathbb R^{2n+2}$, with coordinates $v_0,\dots,v_{2n+1}$. The norm $\|\cdot\|$ on $V$ is the sup norm $\max_k |v_k|$; on $\mathbb C$ it is the modulus; on linear maps it is the corresponding operator norm. Every derivative below is a Fréchet derivative, with the convention that a derivative that does not exist at a point is taken to be $0$ there. $\mathcal L$ is Lebesgue measure on $\mathbb C=\mathbb R^2$.

• $j_0:V\to V$ is $(j_0v)_{2i}=-v_{2i+1}$, $(j_0v)_{2i+1}=v_{2i}$; $\langle u,v\rangle=\sum_k u_kv_k$; $\omega_0(u,v)=\langle j_0u,v\rangle$; $\lambda_x(v)=\tfrac12\,\omega_0(x,v)$.
• For $H:V\to\mathbb R$: $\nabla H(x)$ has components $dH_x(e_k)$; $X_H(x)=j_0\nabla H(x)$; $S=\{x: H(x)=1\}$;

$$R(x)=\lambda_x\big(X_H(x)\big)^{-1}X_H(x),$$

with the convention $0^{-1}=0$ (so $R(x)=0$ wherever $\lambda_x(X_H(x))=0$); $\xi_x=\{v\in V:\ \lambda_x(v)=0,\ dH_x(v)=0\}$.

• Hypothesis on $H$ ("strictly star-shaped level"): $H$ is $C^\infty$; for every $y\neq0$ there is exactly one $r>0$ with $H(ry)=1$; and $dH_x(x)>0$ for every $x\in S$.
• Hypothesis on $J$ ("compatible $\xi$-structure"): $J:V\to\mathcal L(V,V)$ is $C^\infty$, and for every $x\in S$ and all $v,w\in\xi_x$: $J(x)v\in\xi_x$, $J(x)^2v=-v$, $\omega_0(v,J(x)v)>0$ if $v\ne0$, and $\omega_0(J(x)v,J(x)w)=\omega_0(v,w)$. Nothing is required of $J(x)$ off $\xi_x$ or for $x\notin S$.
• Radial coordinates. $s(y)$ is a chosen $r>0$ with $H(ry)=1$ if such an $r$ exists, and $s(y)=1$ otherwise (for $y\ne0$ this $r$ is the unique one above; for $y=0$ it is an arbitrary positive number if $H(0)=1$, and $1$ otherwise). The projection is $\pi(y)=s(y)\,y$ and the $\mathbb R$-coordinate is $a(y)=-\tfrac12\log s(y)$.
• The almost complex structure $\tilde J$. For $y\in V$ put $x=\pi(y)$, $r=s(y)$, $\alpha(v)=\dfrac{r}{2\,dH_x(x)}\,dH_x(v)$, $\beta(v)=r\,\lambda_x(v)$, $w(v)=v-2\alpha(v)\,y-\beta(v)\,r^{-1}R(x)$, and

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

• For $f:\mathbb C\to F$: $\partial_sf(z)=Df(z)[1]$, $\partial_tf(z)=Df(z)[i]$.
• Finite-energy curve on a set $U\subseteq\mathbb C$ (no openness or measurability of $U$ is required by the definition): $u:\mathbb C\to V$ is $C^\infty$ on $U$ (in the sense of smoothness within the set $U$), $u(z)\neq0$ for $z\in U$, at every $z\in U$ $u$ is differentiable and $\partial_su(z)+\tilde J(u(z))\,\partial_tu(z)=0$, and $E(u;U)<\infty$, where:
  • $\varphi:\mathbb R\to\mathbb R$ is admissible if it is $C^\infty$, monotone non-decreasing, and $0\le\varphi\le1$;
  • with $a=a\circ u$ and $w=\pi\circ u$ (functions on $\mathbb C$), the density is

$$e_\varphi(z)=\varphi'(a)\Big(\partial_sa\;\lambda_{w}(\partial_tw)-\partial_ta\;\lambda_{w}(\partial_sw)\Big)+\varphi(a)\,\omega_0(\partial_sw,\partial_tw);$$

• $E(u;U)=\sup_{\varphi\text{ admissible}}\int_U\max(e_\varphi,0)\,d\mathcal L\in[0,\infty]$ (a lower Lebesgue integral of the positive part; negative parts of $e_\varphi$ are discarded, no integrability is assumed).
• $E_{d\lambda}(u;U)=\int_U\max\big(\omega_0(\partial_s w,\partial_t w),0\big)\,d\mathcal L\in[0,\infty]$, with $w=\pi\circ u$ (again the positive part only).
• Reeb orbit $P=(x,T)$: a map $x:\mathbb R\to V$ and a number $T>0$ with $x(\tau)\in S$, $x'(\tau)=R(x(\tau))$ and $x(\tau+T)=x(\tau)$ for all $\tau$. ($T$ is a period, not necessarily the minimal one.)
• Loops around the puncture. For $\rho\in\mathbb R$, $\gamma_\rho(t)=\rho\,e^{-2\pi i t}$ ($t\in\mathbb R$; for $\rho>0$ the circle of radius $\rho$ about $0$, traversed clockwise once per unit of $t$). The loop in $S$ is $\ell_\rho(t)=\pi\big(u(\gamma_\rho(t))\big)$, and its action is

$$\mathcal A(\rho)=\int_0^1\lambda_{\ell_\rho(t)}\big(\ell_\rho'(t)\big)\,dt$$

(an interval integral, which by convention is $0$ if the integrand is not integrable).

• $C^\infty$ convergence on $\mathbb R$. For $f_k,g:\mathbb R\to V$, "$f_k\to g$ in $C^\infty$" means: for every $m\ge0$, the $m$-th derivatives $f_k^{(m)}$ converge to $g^{(m)}$ uniformly on all of $\mathbb R$.
• Removable puncture. "$u$ has a removable puncture at $0$ (radius $1$)" means: there is $v:\mathbb C\to V$ which is $C^\infty$ on the open unit disk $D=\{|z|<1\}$, satisfies $v(z)\ne0$ on $D$, is differentiable with $\partial_sv+\tilde J(v)\,\partial_tv=0$ at every point of $D$, and agrees with $u$ on $D\setminus\{0\}$. (No energy condition is imposed on $v$.)