Theorem FiniteEnergyPlane.exists_intersection_of_tendstoCInftyLocOn

Data and hypotheses. Let $H:V_1\to\mathbb R$ be a strictly star-shaped level and $J:V_1\to\mathrm{End}_{\mathbb R}(V_1)$ a compatible $\xi$-structure for $H$ (both defined below). Let $U,W\subseteq\mathbb C$ be open sets (no connectedness assumed). Let $u,v:\mathbb C\to V_1$ and sequences $u_k,v_k:\mathbb C\to V_1$ ($k\in\mathbb N$) satisfy:

• $u$ is a finite-energy curve on $U$, and $v$ is a finite-energy curve on $W$;
• $Du(z)$ is injective for every $z\in U$, and $Dv(w)$ is injective for every $w\in W$;
• each $u_k$ is a finite-energy curve on $U$, and each $v_k$ is a finite-energy curve on $W$ (no injectivity of $Du_k$, $Dv_k$ is assumed);
• $u_k\to u$ in $C^\infty_{\mathrm{loc}}$ on $U$, and $v_k\to v$ in $C^\infty_{\mathrm{loc}}$ on $W$;
• $z_0\in U$, $w_0\in W$, and $u(z_0)=v(w_0)$;
• there is $\varepsilon>0$ such that for all $z\in B(z_0,\varepsilon)$ and all $w\in B(w_0,\varepsilon)$: if $u(z)=v(w)$ then $z=z_0$ and $w=w_0$.

In the last hypothesis the discs are not required to lie in $U$, $W$, so it also constrains the values of $u,v$ at any points of those discs outside $U$, $W$. It cannot be satisfied when $u=v$ and $z_0=w_0$: then $u(z)=v(z)$ for every $z$ in the disc, which would force every such $z$ to equal $z_0$.

Conclusion. There exist sequences $(z_k)_{k\in\mathbb N}$, $(w_k)_{k\in\mathbb N}$ in $\mathbb C$ such that

$$z_k\to z_0,\qquad w_k\to w_0,$$

and there is $k_0$ with

$$u_k(z_k)=v_k(w_k)\quad\text{for all }k\ge k_0.$$

The conclusion does not state $z_k\in U$ or $w_k\in W$, and the equality is asserted only for all sufficiently large $k$. It does not exclude $(z_k,w_k)=(z_0,w_0)$. It gives no count or uniqueness of such intersection points.

Setting and notation (unfolded from the imported definitions)

Space. $V_n=\mathbb R^{2(n+1)}$ (functions $\{0,\dots,2n+1\}\to\mathbb R$); all statements here use $V_1=\mathbb R^4$, with coordinates $y=(y_0,y_1,y_2,y_3)$. The norm on $V_1$ is the sup norm $\|y\|=\max_k|y_k|$; on $\mathbb C$ the norm is the modulus, so $B(z_0,\varepsilon)=\{z\in\mathbb C:|z-z_0|<\varepsilon\}$ is a Euclidean open disc. On $\mathbb C\times\mathbb C$ the norm is $\max(|z|,|\tau|)$. "Smooth" ($C^\infty$) means infinitely real-differentiable (not analytic).

Linear structure. $j_0(y)=(-y_1,\,y_0,\,-y_3,\,y_2)$; $\omega_0(u,v)=\langle j_0u,v\rangle=u_0v_1-u_1v_0+u_2v_3-u_3v_2$; the Liouville form $\lambda_x(v)=\tfrac12\,\omega_0(x,v)$.

Hamiltonian data. For $H:V_1\to\mathbb R$: $\nabla H(x)_k=dH_x(e_k)$; $X_H(x)=j_0\nabla H(x)$; the Reeb field $R(x)=\lambda_x(X_H(x))^{-1}\,X_H(x)$, with the convention $0^{-1}=0$ (so $R(x)=0$ when $\lambda_x(X_H(x))=0$). The energy surface is $\{H=1\}$.

Hypothesis "$H$ is a strictly star-shaped level" ($hH$): $H$ is $C^\infty$ on all of $V_1$; for every $u\ne0$ there is a unique $r>0$ with $H(ru)=1$; and $dH_x(x)>0$ for every $x$ with $H(x)=1$.

Contact plane. $v\in\xi_x$ means $\lambda_x(v)=0$ and $dH_x(v)=0$.

Hypothesis "$J$ is a compatible $\xi$-structure" ($hJ$): $J:V_1\to\mathrm{End}_{\mathbb R}(V_1)$ is $C^\infty$, and for every $x$ with $H(x)=1$ and all $v,w\in\xi_x$: $J(x)v\in\xi_x$, $J(x)J(x)v=-v$, $v\ne0\Rightarrow\omega_0(v,J(x)v)>0$, and $\omega_0(J(x)v,J(x)w)=\omega_0(v,w)$. Nothing is required of $J(x)$ off $\xi_x$, nor for $x\notin\{H=1\}$, beyond smoothness.

Radial coordinates. $\sigma(y)$ ("scaleToS") is a chosen $r>0$ with $H(ry)=1$ if such an $r$ exists, and the junk value $1$ otherwise (e.g. at $y=0$ when $H(0)\ne1$). Then $\mathrm{proj}(y)=\sigma(y)\,y$ and $a(y)=-\tfrac12\log\sigma(y)$.

Almost complex structure $\tilde J$ on $V_1$. For $y\in V_1$ put $x=\mathrm{proj}(y)$, $r=\sigma(y)$, and the real-linear functionals and map

$$\alpha(h)=\frac{r}{2\,dH_x(x)}\,dH_x(h),\qquad \beta(h)=r\,\lambda_x(h),\qquad w(h)=h-\alpha(h)\,2y-\beta(h)\,r^{-1}R(x).$$

Then

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

(Division by $0$ again yields $0$.)

Partial derivatives. For $f:\mathbb C\to F$ (real vector space), $\partial_sf(z)=Df(z)(1)$, $\partial_tf(z)=Df(z)(i)$, where $Df(z)$ is the real Fréchet derivative (equal to $0$ by convention where $f$ is not differentiable).

$\tilde J$-holomorphic on $U$: for each $z\in U$, $u$ is real-differentiable at $z$ and $\partial_su(z)+\tilde J(u(z))\,\partial_tu(z)=0$.

Energy. An admissible $\varphi:\mathbb R\to\mathbb R$ is $C^\infty$, monotone non-decreasing, with $0\le\varphi\le1$. With $a_u=a\circ u$ and $w_u=\mathrm{proj}\circ u$, the energy density is

$$e_\varphi(z)=\varphi'(a_u)\big(\partial_sa_u\,\lambda_{w_u}(\partial_tw_u)-\partial_ta_u\,\lambda_{w_u}(\partial_sw_u)\big)+\varphi(a_u)\,\omega_0(\partial_sw_u,\partial_tw_u)$$

(evaluated at $z$), and

$$E(U,u)=\sup_{\varphi\text{ admissible}}\int_U \max\big(e_\varphi(z),0\big)\,dz\in[0,\infty],$$

i.e. only the positive part of the density is integrated (Lebesgue measure on $\mathbb C$). The $d\lambda$-density is $\omega_0(\partial_sw_u,\partial_tw_u)$ and the $d\lambda$-energy is $E_{d\lambda}(U,u)=\int_U\max\big(\omega_0(\partial_sw_u,\partial_tw_u),0\big)\,dz\in[0,\infty]$.

Finite-energy curve on $U$ ($U\subseteq\mathbb C$): $u:\mathbb C\to V_1$ is $C^\infty$ on $U$, $u(z)\ne0$ for all $z\in U$, $u$ is $\tilde J$-holomorphic on $U$, and $E(U,u)<\infty$. (Nothing is required of $u$ outside $U$.)

$C^\infty_{\mathrm{loc}}$ convergence on $O$. $f_k\to g$ in $C^\infty_{\mathrm{loc}}$ on $O$ means: for every compact $K\subseteq O$ and every $m\ge0$, the $m$-th iterated real Fréchet derivatives $D^mf_k$ converge to $D^mg$ uniformly on $K$. These are derivatives of the functions on the whole plane (equal to $0$ by convention where they do not exist), evaluated at points of $K$.