Annulus energy between level-separated circles is bounded by the loop actions

Let $S=H^{-1}(1)\subset\mathbb R^{2n+2}$ be a strictly star-shaped level with contact form $\lambda=\lambda_0|_S$ and $d\lambda$-compatible complex structure $J$ on $\xi$, and let $u=(a,w):\dot D=D\setminus\{0\}\to\mathbb R\times S\cong\mathbb R^{2n+2}\setminus\{0\}$ be a finite-energy $\tilde J$-holomorphic curve on the punctured unit disc. For a level $A\in\mathbb R$ let $\varphi_A(s)=\chi(s-A)^2$, where $\chi$ is Mathlib's smooth transition function ($\chi=0$ on $(-\infty,0]$, $\chi=1$ on $[1,\infty)$), so $\varphi_A$ is an admissible function in Hofer's energy. Let $0<\rho_2<\rho_1<1$, and suppose that on each circle $|z|=\rho_i$ either $a\circ u\le A$ everywhere or $a\circ u\ge A+1$ everywhere. Then

where $Q(\rho)=\int_0^1 w(\gamma_\rho(t))^*\lambda$ is the action of the clockwise loop $\gamma_\rho(t)=\rho e^{-2\pi i t}$.

In words: the Hofer-energy integrand at level $A$ over an annulus whose boundary circles avoid the slab $A<a<A+1$ is controlled by the actions of the two boundary loops. This is a step in the zero-charge case of the asymptotic description of finite-energy punctured discs (Hofer; Wendl, Theorem 9.8).

Formalization Note $\mathbb R\times S$ is identified with $\mathbb R^{2n+2}\setminus\{0\}$ by $(a,x)\mapsto e^{2a}x$, and $a$ is aCoord H. The integrand of $u^*d(\varphi(a)\lambda)$ with respect to $ds\wedge dt$ is energyDensity H φ u; the integral is the lower Lebesgue integral of its positive part over ball 0 ρ₁ \ ball 0 ρ₂, and $Q(\rho)$ is loopAction H u (cwLoop 0 ρ).