# Displaceability of submanifolds

My question is motivated by the following question.

How transitive are the actions of symplectomorphism groups ?

A subset $X$ of a symplectic manifold $M^{2n}$ is called $\it displaceable$ if there is a Hamiltonian (or, symplectic if you want weaker condition) diffeomorphism $f\colon M \rightarrow M$ such that $f(X) \cap X = \phi$. Now assume $X$ is a $k$-dimensional closed submanifold of $M$.

When $k=0$, $X$ is displaceable as explained in the above question. When $k=n$, some $X$ is not displaceable since we know many nondisplaceable Lagrangians (such as great circle in $S^2$). Then when $k<n$, can we expect that $X$ is displaceable?

If so, can we expect more? Suppose two diffeomorphic submanifolds $X$ and $Y$ of dimension $k<n$ represent the same homology class. Is there a symplectomorphism sending $X$ to $Y$? Are there any topological obstructions for such symplectomorphism?

• Here is one vaguely related result I'm aware of: if $X$ and $Y$ are two symplectic submanifolds of a closed symplectic manifold $(M, \omega)$ which are smoothly isotopic, then there exists a Hamiltonian diffeomorphism of $M$ mapping $X$ to $Y$. This is Proposition 4 in S.Haller, C.Vizman: Non-linear Grassmannians as coadjoint orbits, arXiv link. May 20, 2014 at 20:20

There is a result of Basak Gurel which says that a nowhere coisotropic submanifold of a symplectic manifold is displaceable (by a Hamiltonian diffeomorphism) as soon as it's infinitesimally displaceable (meaning its normal bundle admits a nowhere vanishing section). A submanifold $X$ of a symplectic manifold $(M,\omega)$ is nowhere coisotropic if for every $x \in X$ we have $(T_xX)^\omega \not\subset T_xX$. This trivially holds for submanifolds of dimension $<n$. Therefore a $k$-dimensional submanifold $X$ with $k < n$ will be displaceable as soon as its normal bundle admits a nonvanishing section (I can't immediately see whether this always holds for such a submanifold). Link: http://arxiv.org/abs/math/0702091