Floer cohomology for immersed Lagrangians is introduced by Akahi, Manabu; Joyce, Dominic, Immersed Lagrangian Floer theory, J. Differ. Geom. 86, No. 3, 381-500 (2010) and its one-dimension version (in the absence of "teardrops") is developed in Abouzaid, Mohammed, On the Fukaya categories of higher genus surfaces, Adv. Math. 217, No. 3, 1192-1235 (2008). See also the embedded case described in de Silva, Vin; Robbin, Joel W.; Salamon, Dietmar A., Combinatorial Floer homology. As in the embedded case, non-triviality of Floer cohomology of immersed Lagrangians obstructs Hamiltonian displaceability. For example, consider the following example of an immersion $f: L \to X $ in the two sphere $X$ (thought of as the one-point compactification the plane) dividing the two sphere into areas $A_0,A_1,A_2,A_3$ (smoothing used curve-shortening by A. Carapetis)

an immersed curve

A computation shows that the Floer cohomology admits a weakly bounding cochain and is Floer-non-trivial if $A_2 < \min(A_1,A_3)$ and

$$A_0 = A_1 + A_3 - 3A_2.$$

Indeed in this case using the Morse model there are six generators in the Floer cochains, given by the max and min of the height function and four generators corresponding to the two self-intersection points; the zeroth Fukaya map is non-zero because of the two teardrops but the teardrops can be "killed" by taking a bounding cochain that is the sum of generators coming from the self-intersections with coefficients $q^{A_1 - A_2}$ and $q^{A_3 - A_2}$. On the other hand, Moser's principle Moser, Jürgen, On the volume elements on a manifold, Trans. Am. Math. Soc. 120, 286-294 (1965) implies that $f(L)$ is non-displaceable iff each of the areas is at most the sum of the other three:

$$ A_i \leq A_j + A_k + A_l, \quad i,j,k,l \ \text{distinct}. $$

Indeed, if one of these inequalities fails then there is a symplectomorphism that takes the complement of the interior of one of these regions into its interior, while if they all hold then such a symplectomorphism obviously does not exist. The conditions

$$A_0 + 3A_2 = A_1 + A_3, A_2 < A_1, A_2 < A_3$$

imply the "Moser conditions" (exercise), but are much stronger. So there seems to be a fairly big gap between Floer non-triviality and non-displaceability in this case.

Question: (1) Can anyone do better, i.e. is it possible to detect the non-displaceability of immersed curves in the two-sphere using some other version of Floer theory, or a different weakly bounding cochain?

(2) Suppose one takes the product of two such curves in $S^2 \times S^2$. Here Floer theory still works, but there is no analog of Moser's results although presumably one could use McDuff, Dusa, Displacing Lagrangian toric fibers via probes, Proceedings of Symposia in Pure Mathematics 82, 131-160 (2011). When is the image of an immersion of product form displaceable?


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