Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces

Abstract

Abstract. In this article we study the Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

On Floer Homology of the Gauss Images of Isoparametric Hypersurfaces

“…In the case when g = 3, that is, N n is a Cartan hypersurface, we proved Lemma 2 ( [11]). The Gauss image of L n = G (N n ) of each isoparametric hypersurface with g = 3 is a Z 2 -homology sphere ( i.e.…”

“…The Gauss images of Cartan hypersurfaces provide new examples of Lagrangian Z 2 -homology spheres embedded in compact Hermitian symmetric spaces. This result is quite essential for the proof of main theorem [11] in the case when g = 3. When g = 3 and m = m 1 = m 2 = 2, 4 or 8, by Lemma 1 we have ν = 1.…”

“…Theorem 5 (IMMO [11]). In the case when g = 4 or g = 6 except for the remaining three cases as below, the lifted Floer homology HF N (L) is non-zero: Since the Floer homology is based on the Mores homology, it is quite natural to study the following problems: Problem 4.…”

“…Recently in our recent joint paper with Hiroshi Iriyeh, Hui Ma and Reiko Miyaoka ( [11]) in order to study their Hamiltonian non-displaceability we used the Floer homology and the lifted Floer homology for monotone Lagrangian submanifolds. In this note we will explain the construction of the Floer homology and the spectral sequences for monotone Lagrangian submanifolds, and also their lifted Floer homology (Floer, Y.-G. Oh, Biran, Damian), and their applications to the Gauss images of isoparametric hypersurfaces.…”

Lagrangian submanifolds of the complex hyperbolic quadric

“…Minimal Lagrangian immersions into Q n were studied for example in [8], [9], [3] and [7], by identifying them with Gauss maps of isoparametric hypersurfaces of the unit sphere S n+1 (1). The relation between the geometric invariants of (not necessarily minimal) Lagrangian submanifolds of Q n and (not neccesarily isoparametric) hypersurfaces of S n+1 (1) was stated in full generality in [17].…”