A NON-DISPLACEABLE LAGRANGIAN TORUS IN T ∗ S 2

• Main Authors: Peter Albers, Urs Frauenfelder
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Published: 2007
• Subjects: Theorem. The Floer homology of the torus L is; North Pole; South Pole; South pole
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.235.6624; http://arxiv.org/pdf/math/0608356v2.pdf

Abstract. We show that the Lagrangian torus in the cotangent bundles of the 2-sphere obtained by applying the geodesic flow to the unit circle in a fibre is not displaceable by computing its Lagrangian Floer homology. The computation is based on a symmetry argument. Following a construction of Leonid Polterovich we consider the following Lagrangian torus L in T ∗ S 2. Letϕt be the geodesic flow on the cotangent bundle of the standard round S 2 where we identified TS 2 with T ∗ S 2 via the round metric. Fix the unit circle C ⊂ T ∗ N S 2 in the cotangent fiber over the north-pole N ∈ S 2. We consider the map φ: S 1 × S 1 − → T ∗ S 2 (t, v)↦→ϕt(v) where we identify isometrically the circle C⊂T ∗ N S 2 with S 1. The mapφdefines a Lagrangian embedding, we set L:=φ(T 2)⊂T ∗ S 2. Obviously,π2(T ∗ S 2, L)�π2(T ∗ S 2)⊕π1(L)=Z⊕Z⊕Z, where the secondZis generated by the unit disk in T ∗ N S 2 bounded by the loop C. The third disk corresponds to the loop t↦→ϕt(v0) in π1(L). The disk bounded by C lies in a Lagrangian submanifold namely T ∗ N S 2 and thus has vanishing Maslov index and symplectic area. The disk corresponding to the loop t↦→ϕt(v0) has Maslov index 2. We obtain a monotone Lagrangian torus L with minimal Maslov number NL = 2. In particular, its Lagrangian Floer homology is well-defined by standard means since T ∗ S 2 is a convex exact symplectic manifold and L has minimal Maslov number NL ≥ 2, see [Oh95]. Apart from the fibers over the north-pole and the south-pole the torus L intersects each fiber of T ∗ S 2 exactly twice. At north and south-pole it intersects the fiber in a circle. The following question of Leonid Polterovich was posed to us by Felix Schlenk: (1) Is L displaceable? (2) If not, what is HF(L, L)?