Displaceability questions in the symplectic 2-sphere for level sets of a Morse function

Question

Consider the symplectic $2$-sphere $S^2$ with the canonical symplectic form $\omega$. A subset $A$ is called displaceable if there exists $H:S^2\rightarrow\mathbb{R}$ smooth such that $\Phi_H^{1}(A)\cap A=\emptyset$, where $\Phi_H^1$ is the $1$-time Hamiltonian flow generated by $H$. Otherwise the subset $A$ is called non-displaceable.

In the symplectic $2$-sphere it is easy to see that for example if $A$ divides the sphere into two sets of equal area then since a Hamiltonian diffomorphism preserves area then the set $A$ is non-displaceable.

I am more interested in the reverse question, i.e., suppose $A=h^{-1}(p)$, where $h$ is a Morse function and p can be a regular point or a critical point, but let's assume that $h^{-1}(p)\neq p$. Suppose that the area of $h^{-1}(]-\infty,p[)$ is different than the area of $h^{-1}(]p,\infty[)$, then is the set $A$ is displaceable? I am mostly interested in the case $p$ is an hyperbolic critical point.

From what I am aware for questions on displaceability there is the method of probes by McDuff and extended probes by Abreu & Borman & McDuff, but this applies to toric systems, or at least one needs to have some sort of action-angle coordinates in order to apply these methods. Since I am interested in $A$ to be critical level sets such as the figure eight, these methods are not satisfying in order to study the problem. I am wondering if there is any hope of studying this question and there is some insight I might be missing.

Any help is appreciated.

Answer 1

Consider the following example: $f: S^2 \to \bf{R}$ for the standard unit sphere and $f$ is the projection to the z-axis.

1. Prove that for $L_{\pm \lambda} = f^{-1}(\pm \lambda)$ that $L_\lambda = L_{-\lambda} \cup L_{+\lambda}$ is non-displaceable provided $0 \leq \lambda < 1/2$.

2. Prove that all that really matters here is that $S^2 \setminus L = D_1 \cup D_2 \cup A_3$ as connected components with $area(D_1) = area(D_2)$ and $2\,area(D_1) > area(A_3)$.

3. In particular a figure-8 shape $L \subset S^2$ will be non-displaceable if $S^2 \setminus L = D_1 \cup D_2 \cup A_3$ and with $area(D_1) = area(D_2)$ and $2\,area(D_1) > area(A_3)$. Hence there is a Morse function on $S^2$ that has a connected critical level set L as a figure-8 with the upper and lower level sets having different areas.