Theorem FiniteEnergyPlane.exists_holomorphic_eqOn_of_accPt_intersection

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), and $u,v:\mathbb C\to V_1$ such that:

• $u$ 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 $w\in W$;
• $z_0\in U$ and $w_0\in W$;
• $(z_k)_{k\in\mathbb N}$, $(w_k)_{k\in\mathbb N}$ are sequences in $\mathbb C$ with $z_k\to z_0$ and $w_k\to w_0$;
• for every $k$, $(z_k,w_k)\ne(z_0,w_0)$ as pairs (so one of the two coordinates may equal its limit, for some or all $k$);
• for every $k$, $u(z_k)=v(w_k)$.

The sequences are not required to lie in $U$, $W$. The equality $u(z_0)=v(w_0)$ is not stated as a hypothesis.

Conclusion. There exist $\varepsilon>0$ and $\varphi:\mathbb C\to\mathbb C$ such that:

• $\varphi$ is complex-differentiable on the open disc $B(z_0,\varepsilon)$;
• $\varphi(z_0)=w_0$;
• the complex derivative satisfies $\varphi'(z_0)\ne0$;
• $\varphi(B(z_0,\varepsilon))\subseteq W$;
• $u(z)=v(\varphi(z))$ for every $z\in B(z_0,\varepsilon)$.

The disc $B(z_0,\varepsilon)$ is not asserted to lie in $U$, while the identity $u=v\circ\varphi$ is asserted on the whole disc. $\varphi$ is not asserted to be injective on the disc, and $\varphi'\ne0$ is asserted only at $z_0$. Nothing is said about $\varphi$ outside the disc.

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 $U$.)