Suppose $(X,\omega)$ is a closed symplectic manifold. Let $H$ denote a time-dependent Hamiltonian, all of whose critical points are non-degenerate, and fix an $\omega$-compatible time dependent family of almost complex structures.

Let $\mathcal{M}$ denote the set of all *finite energy* Floer flow lines. That is, maps $u:\mathbb{R} \times S^1 \rightarrow X$ that satisfy the Floer equation
$\partial_s u + J(u)(\partial_t u - X_{H}(u))=0$, and satisfy $e(u) \lt \infty$, where $e(u)$ is the energy of $u$, defined by
$e(u) = \int_{\mathbb{R}} \int_{S^1} |\partial_s u |^2 dsdt$.

Then it's well known that if $u$ is any map satisfying the Floer equation, then $u\in \mathcal{M}$ (i.e. $u$ has finite energy) is equivalent to asking for one of the following two properties:

$u \in \mathcal{M}(x,y)$ for two 1-periodic orbits $x,y$ of $H$, that is, $\lim_{s \rightarrow \infty}u(s,t)=x(t)$ and $\lim_{s \rightarrow -\infty}u(s,t)=y(t)$.

$u$

*decays exponentially*- there exist constants $C,\delta$ such that $|\partial_s u(s,t)| \lt Ce^{-\delta |s|}$.

This means that if $u \in \mathcal{M}$ then the *length* $l(u(\cdot,t)) = \int_{\mathbb{R}} |\partial_s u(s,t)|ds$ is finite for each $t \in S^1$.

Fix two 1-periodic orbits $x,y$. Then if $u \in \mathcal{M}(x,y)$, not only is the energy finite, but there is a uniform bound on the energy $e(u)$ for any such $u$ - namely $\mathcal{A}_{H}(x)-\mathcal{A}_H(y)$ - where $\mathcal{A}_H$ is the action functional.

My question is the following: **do there exist uniform bounds (i.e. depending only on $x$ and $y$) on the length $l(u(\cdot,t))$ for every $u \in \mathcal{M}(x,y)$?**