Displaceability of Certain Constant Sectional Curvature Lagrangian Submanifolds
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Results in Mathematics
Displaceability of Certain Constant Sectional Curvature Lagrangian Submanifolds ˙ Nil Ipek S.irikc.i Abstract. We present an alternative proof of a nonexistence result for displaceable constant sectional curvature Lagrangian submanifolds under certain assumptions on the Lagrangian submanifold and on the ambient symplectically aspherical symplectic manifold. The proof utilizes an index relation relating the Maslov index, the Morse index and the Conley– Zehnder index for a periodic orbit of the flow of a specific Hamiltonian function, a result on this orbit’s Conley–Zehnder index and another result on the Morse indices for constant sectional curvature manifolds the utilization of which to prove nondisplaceability is new. Mathematics Subject Classification. Primary 53D12. Keywords. Lagrangian submanifolds, Maslov index, Morse index, Conley– Zehnder index.
1. Introduction 1.1. Results on Displaceability of Lagrangians There are various results in the literature obtained on the displaceability of Lagrangian submanifolds. In a symplectic manifold without boundary that is closed or convex at infinity, if L is a compact Lagrangian submanifold with ω|π2 (M,L) = 0, then ψ(L) ∩ L = ∅ where ψ : L → L is a Hamiltonian symplectomorphism [41, p. 297]. This has been established by Gromov in [28] and a stronger version giving a lower bound for ψ(L) ∩ L whenever ψ(L) and L intersect transversally has been proved by Floer [22,23], [41, p. 297]. There are many other 0123456789().: V,-vol
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related results that has been obtained (e.g. as given in the discussions in the introductions of [30,59]). Chekanov [16] proved the analogue of the Lagrangian intersection theorem of Floer for the case when the exactness condition of the Lagrangian is replaced by an assumption that sets the energy of the symplectomorphism to be less than a certain value. This value is the supremum of the minimum of the set consisting of the minimal area of a J- holomorphic sphere in the manifold and the minimal area of a J-holomorphic disc in the manifold with boundary on the Lagrangian submanifold where the supremum is taken over all almost complex structures on the symplectic manifold such that the symplectic manifold together with the almost complex structure is a tame almost K¨ahler manifold. The nondisplaceability of the zero-section of T ∗ M has been established by Chaperon in [13] for the case of the torus, by Hofer [31], Laudenbach and Sikarov [39], [44, p. 14], [29, p. 202] and Polterovich [47], [6, p. 459]. Lalonde and Polterovich proved that if φ is a bounded compactly supported symplectomorphism of M, L ⊂ M is a closed Lagrangian submanifold admitting a Riemannian metric with non-positive sectional curvature, and the inclusion of L in M induces an injection on fundamental groups, then φ(L) ∩ L = ∅ [38, Theorem 1.4.A]. Frauenfelder and Schlenk proved the nondisplaceability of closed Lagrangian submanifolds which lie outside the boundary of the symplectic manifold and which admit a
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