FiniteEnergyPlane.plane_asymptotics

Statement. Let $n \in \mathbb{N}$, $H : V \to \mathbb{R}$ satisfying the star-shaped level conditions, and $J$ a compatible structure (both defined below). Let $u : \mathbb{C} \to V$ be a finite-energy curve on all of $\mathbb{C}$ (as defined below, with respect to $H$ and $J$) which is non-constant: there are $z_1, z_2 \in \mathbb{C}$ with $u(z_1) \neq u(z_2)$. Write $T := \overline{E}$ for the real number obtained from $E_{d\lambda}(\mathbb{C}, u) \in [0,\infty]$ by sending $+\infty$ to $0$ and keeping finite values. Then all four of the following hold:

1. $0 < T$. (Equivalently: $E_{d\lambda}(\mathbb{C}, u)$ is finite and strictly positive, and $T$ equals it.)
2. $a(u(z)) \to +\infty$ as $|z| \to \infty$ (i.e. along the filter of complements of compact subsets of $\mathbb{C}$).
3. As the real parameter $R \to +\infty$,

where $\gamma_R'$ is the derivative in $t$ (taken as $0$ where $\gamma_R$ is not differentiable) and the integral is the interval integral (taken as $0$ when the integrand is not integrable). 4. For every sequence of reals $(R_j)_{j \in \mathbb{N}}$ with $R_j \to +\infty$, there exist a strictly increasing $\phi : \mathbb{N} \to \mathbb{N}$, a Reeb orbit $P$ with period $T_P = T$, and a real constant $c$ such that the maps $\gamma_{R_{\phi(k)}} : \mathbb{R} \to V$ converge in $C^\infty$ on the whole real line to $t \mapsto x_P(T t + c)$: for every order $m \ge 0$,

where the $m$-th derivatives are iterated one-variable derivatives (with the convention that the derivative of a non-differentiable function is $0$), and the convergence is uniform in $t \in \mathbb{R}$. Here $\gamma_R$ is defined for every real $R$ (including $R \le 0$) by the same formula; only the tail $k \to \infty$ matters. The phase $c$ and the subsequence may depend on the sequence $(R_j)$; the orbit $P$ is not asserted to be independent of the sequence.

Setting and notation (all definitions unfolded)

Throughout, $n \in \mathbb{N}$ is arbitrary (including $n = 0$) and $V = \mathbb{R}^{2(n+1)}$, with coordinates indexed $0, 1, \dots, 2n+1$.

• Standard structure. $j_0 : V \to V$ is $(j_0 v)_k = -v_{k+1}$ for $k$ even and $(j_0 v)_k = v_{k-1}$ for $k$ odd. With the Euclidean dot product $\langle u, v\rangle = \sum_k u_k v_k$, set $\omega_0(u, v) = \langle j_0 u, v\rangle$ and $\lambda_x(v) = \tfrac12\,\omega_0(x, v)$.
• Hamiltonian data. For $H : V \to \mathbb{R}$: $\nabla H(x)$ is the vector of partial derivatives $dH_x(e_k)$; $X_H(x) = j_0 \nabla H(x)$; $S = \{x : H(x) = 1\}$; the "Reeb field" is $R(x) = \lambda_x(X_H(x))^{-1}\, X_H(x)$, where the inverse of $0$ is taken to be $0$. Derivatives are Fréchet derivatives, with the convention that the derivative of a non-differentiable map at a point is $0$.
• Star-shaped level (hypothesis $hH$): $H$ is $C^\infty$; for every $u \neq 0$ there is a unique $r > 0$ with $H(ru) = 1$; and $dH_x(x) > 0$ for every $x$ with $H(x) = 1$.
• Compatible structure (hypothesis $hJ$): $J : V \to \mathrm{End}_{\mathbb{R}}(V)$ is $C^\infty$, and for every $x \in S$ and all $v, w$ in $\xi_x = \{v : \lambda_x(v) = 0,\ dH_x(v) = 0\}$: $J(x)v \in \xi_x$, $J(x)J(x)v = -v$, $\omega_0(v, J(x)v) > 0$ if $v \neq 0$, 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. $\rho(y)$ is a chosen $r > 0$ with $H(ry) = 1$ if one exists, and $\rho(y) = 1$ otherwise (under $hH$, for $y \ne 0$ it is the unique such $r$). Then $\pi_S(y) = \rho(y)\, y$ and $a(y) = -\tfrac12 \log \rho(y)$, so $y = e^{2a(y)}\pi_S(y)$ when $\rho(y)$ is that $r$.
• Almost complex structure $\tilde J$. For $y \in V$ put $x = \pi_S(y)$, $r = \rho(y)$, $\alpha(v) = \dfrac{r}{2\, dH_x(x)}\, dH_x(v)$, $\beta(v) = r\,\lambda_x(v)$, $w(v) = v - 2\alpha(v)\, y - \beta(v)\, r^{-1} R(x)$, and

• Derivatives of maps $f$ on $\mathbb{C} = \mathbb{R}^2 \ni z = s + it$. $\partial_s f(z) = Df(z)(1)$, $\partial_t f(z) = Df(z)(i)$ (each $0$ where $f$ is not differentiable).
• Finite-energy curve on $U \subseteq \mathbb{C}$ for $u : \mathbb{C} \to V$: (1) $u$ is $C^\infty$ on $U$; (2) $u(z) \ne 0$ for $z \in U$; (3) for every $z \in U$, $u$ is differentiable at $z$ and $\partial_s u(z) + \tilde J(u(z))\, \partial_t u(z) = 0$; (4) $E(U, u) < \infty$, where, writing $a_u = a \circ u$ and $w_u = \pi_S \circ u$,

the supremum being over $C^\infty$ nondecreasing $\varphi : \mathbb{R} \to [0,1]$, $[\cdot]^+ = \max(\cdot, 0)$, and the integral a Lebesgue integral of a nonnegative function with respect to area measure (value in $[0,\infty]$).

• $d\lambda$-energy. $E_{d\lambda}(U, u) = \displaystyle\int_U \big[\omega_0(\partial_s w_u, \partial_t w_u)\big]^+\, dz \in [0, \infty]$.
• Reeb orbits. A Reeb orbit $P$ consists of a map $x_P : \mathbb{R} \to V$ and a number $T_P > 0$ such that $x_P(\tau) \in S$ for all $\tau$, $x_P$ is differentiable with $x_P'(\tau) = R(x_P(\tau))$, and $x_P(\tau + T_P) = x_P(\tau)$ for all $\tau$. ($T_P$ is a period, not required to be the minimal one.)
• Trivial cylinder over $P$. For $w \in \mathbb{C}$,

with $\arg w \in (-\pi, \pi]$ the principal argument ($\arg 0 = 0$); since $T_P/\pi > 0$, $\mathrm{Cyl}_P(0) = 0$.