FiniteEnergyPlane.eq_trivialCylinder_comp_polynomial

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 $\Gamma \subset \mathbb{C}$ be a finite set (possibly empty) and $u : \mathbb{C} \to V$ a map such that:

1. $u$ is a finite-energy curve on $\mathbb{C} \setminus \Gamma$ (as defined below, with respect to $H$ and $J$);
2. $u$ is non-constant on $\mathbb{C} \setminus \Gamma$: there are $z_1, z_2 \notin \Gamma$ with $u(z_1) \neq u(z_2)$;
3. for every $z_0 \in \Gamma$, $a(u(z)) \to -\infty$ as $z \to z_0$ with $z \neq z_0$;
4. $E_{d\lambda}(\mathbb{C} \setminus \Gamma, u) = 0$.

Then there exist a complex polynomial $p$ of degree $\deg p > 0$ whose zero set is exactly $\{z \in \mathbb{C} : p(z) = 0\} = \Gamma$ (no condition on the multiplicities of the zeros), and a Reeb orbit $P$ (with no condition on its period $T_P$), such that

Nothing is asserted about the values $u(z)$ for $z \in \Gamma$.

Edge cases. Hypothesis 3 only involves values of $u$ at points of $\mathbb{C}\setminus\Gamma$ near $z_0$ (since $\Gamma$ is finite). When $\Gamma = \emptyset$, hypothesis 3 is vacuous, and the conclusion requires a polynomial of positive degree with no complex zero, which does not exist; so for $\Gamma = \emptyset$ the statement asserts that hypotheses 1, 2 and 4 cannot hold simultaneously on all of $\mathbb{C}$.

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 $T_P/\pi > 0$, $\mathrm{Cyl}_P(0) = 0$.