FiniteEnergyPlane.exists_bubble

Statement. Let $n\in\mathbb N$, let $H:V\to\mathbb R$ satisfy the strictly star-shaped level hypothesis, and let $J:V\to\mathcal L(V,V)$ be a compatible $\xi$-structure for $H$ (all terms are defined below). Let:

• $\Gamma\subset\mathbb C$ be a finite set (possibly empty);
• $(U_k)_{k\in\mathbb N}$ be open subsets of $\mathbb C$ with $U_k\subseteq U_{k+1}$, $U_k\subseteq\mathbb C\setminus\Gamma$ for every $k$, and $\bigcup_kU_k=\mathbb C\setminus\Gamma$;
• $u_k:\mathbb C\to V$ be, for every $k$, a finite-energy curve on $U_k$;
• $C\in\mathbb R$ with $E(u_k;U_k)\le\max(C,0)$ for every $k$;
• $z_k\in U_k$ for every $k$, with

$$g_k(z_k)\longrightarrow+\infty,\qquad g_k(z)=\big\|D(a\circ u_k)(z)\big\|+\big\|D(\pi\circ u_k)(z)\big\|$$

(operator norms of the derivatives of $z\mapsto a(u_k(z))\in\mathbb R$ and $z\mapsto\pi(u_k(z))\in V$, with the modulus on $\mathbb C$ and the sup norm on $V$; a non-existent derivative counts as $0$);

• some $\delta>0$ with $|z_k-\gamma|\ge\delta$ for all $k$ and all $\gamma\in\Gamma$ (vacuous when $\Gamma=\varnothing$);
• either the sequence $(z_k)$ is bounded, or there are $m\in\mathbb N$ and $R\in\mathbb R$ with $\{w\in\mathbb C:|w|>R\}\subseteq U_m$.

Conclusion. There exist a strictly increasing $\phi:\mathbb N\to\mathbb N$, points $z'_j\in\mathbb C$, radii $r_j\in\mathbb R$ and a Reeb orbit $P=(x,T)$ such that

• $|z_{\phi(j)}-z'_j|\to0$ as $j\to\infty$;
• $r_j>0$ for all $j$, and $r_j\to0$;
• the open disk $B(z'_j,r_j)$ is contained in $U_{\phi(j)}$ for every $j$;
• $T\le C$;

$$T\le\limsup_{j\to\infty}E_{d\lambda}\big(u_{\phi(j)};B(z'_j,r_j)\big)$$

(the $\limsup$ is taken in $[0,\infty]$).

Since a Reeb orbit has $T>0$, the conclusion $T\le C$ entails $C>0$; for $C\le0$ the statement therefore asserts that the hypotheses cannot all hold. The orbit $P$ is linked to the maps $u_k$ only through the last inequality; no convergence of the maps $u_k$ (or of rescalings of them) is asserted.

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.)