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:


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



Length of Floer flow lines
In your symplectically aspherical setting, bounds on length will indeed exist. 


Suppose one has a sequence of solutions $u\_n$ to Floer's equation, of bounded energy, and a sequence of points $t\_n\in S^1$ with lengths $l(u\_n(\cdot,t\_n))\to \infty$. Gromov-Floer compactness tells us that after passing to a subsequence, this sequence converges in the Gromov-Floer topology to a broken trajectory plus bubbles. Each cylinder in the domain of the broken trajectory is associated with a time-translation $\sigma\_k \colon s\mapsto s+s\_k$, and the maps $\sigma\_k^\ast u\_n$ converge to a limiting cylinder $v\_k$ *plus bubbles.* Outside a long but finite cylinder $[-T,T] \times S^1$, $\sigma\_k^\ast u\_n$ is exponentially close to $v\_k$ (meaning $\leq ce^{-as}$, where $a>0$ depends only on the asymptotic limits $x$ and $y$, and $c$ only on $u\_n(s\_k,\cdot)$). Similar decay applies to the first derivative of $u\_n$. The divergence of lengths must therefore occur inside the finite cylinder. 


Inside the finite cylinder but away from finitely many "bad points", $\sigma\_k^\ast u\_n$ converges to $v\_k$, and for any $r$, convergence in $C^r$ is uniform on compact subsets. At the bad points, bubbles form. In the symplectically aspherical case, bubbling is impossible: there are no bad points. Therefore one has uniform $C^1$ bounds on $\sigma\_k^\ast u\_n$ over the finite cylinder, contradicting the divergence of the lengths.


Maybe it would be interesting to write down a bubbling sequence of holomorphic maps $\mathbb{CP}^1\to \mathbb{CP}^1$ and watch what happens to the lengths. 


