Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation

$$ u(t,s): S^1 \times \mathbb{R} \to M$$

$$(du+X_H\otimes dt)^{0,1}=0$$

Suppose that at $s=\infty,$ the solution is asymptotic to some constant orbit $p$ of $X_H$. We can then try to extend u to a map $\bar{u}$ from $\mathbb{C} \to M$ by mapping the origin $\mathfrak{o}$ to $p$. This map should be continuous.

Question: Under what conditions on $H$ can we say anything about the regularity of this map $\mathbb{C} \to M$? It seems possible that even though the one form $dt$ doesn't extend to $\mathbb{C}$, the fact that $X_H$ tends to zero may sometimes allow for some smooth extension.

Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation

$$ u(t,s): S^1 \times \mathbb{R} \to M$$

$$(du+X_H\otimes dt)^{0,1}=0$$

Suppose that at $s=\infty,$ the solution is asymptotic to some constant orbit $p$ of $X_H$. We can then try to extend u to a map $\bar{u}$ from $\mathbb{C} \to M$ by mapping the origin $\mathfrak{o}$ to $p$. This map should be continuous.

Question: Under what conditions on $H$ can we say anything about the regularity of this map $\mathbb{C} \to M$? It seems possible that even though the one form $dt$ doesn't extend to $\mathbb{C}$, the fact that $X_H$ tends to zero may sometimes allow for some smooth extension with a critical point at $z=0$.