Hamiltonian non-displaceability of the Gauss images of isoprametric hypersurfaces (A survey)

Abstract

This is a survey of the joint work [13] (Bull Lond Math Soc 48(5), 802–812, 2016) with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.). The Floer homology of Lagrangian intersections is computed in few cases. Here, we take the image L=G(N) of the Gauss map of isoparametric hypersurfaces N in Sn+1, that are minimal Lagrangian submanifolds of the complex hyperquadric Qn(ℂ). We call L Hamiltonian non-displaceable if L∩ φ (L)≠ ∅ holds for any Hamiltonian deformation φ. Hamiltonian non-displaceability is needed to define the Floer homology HF(L), since HF(L) is generated by points in L∩ φ (L). We prove the Hamiltonian non-displaceability of L=G(N) for any isoparametric hypersurfaces N with principal curvatures having plural multiplicities. The main result is stated in Sect. 4.