Length of Floer flow lines

Question

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 $u \in \mathcal{M}(x,y)$?

Answer 1

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.