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A Minimal Maslov Number Condition for Displaceability in Certain Weakly Exact Symplectic Manifolds

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Abstract

We present a proof of a result which gives an upper bound for the minimal Maslov number of a displaceable n-dimensional Lagrangian submanifold in a weakly exact symplectic manifold with minimal Chern number at least n. The proof utilizes a result on the Conley-Zehnder index of a periodic orbit of the flow of a specifically constructed Floer Hamiltonian and an index relation.

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Notes

  1. For a survey of various results obtained on the displaceability of Lagrangian submanifolds, see the introduction of [24].

  2. For example, symplectic tori (\((T^{2n}, \omega )\) for \(n>0\)) satisfy the conditions on the symplectic manifold in Theorem 1.

  3. Here, we have used the following formula for gluing \(A \in \pi _2(M)\) to a spanning disk w of x(t): \(\mu _{cz}(x,w\# A) = \mu _{cz}(x,w) + 2 c_1(A)\). [15]

  4. If \(\delta ([\eta , w_{\eta }]) = [y, {w_y}]\), then \(\delta ([\eta , {w_{\eta }} \#^k A]) = [y, {w_y} \#^k A]\) for \(A \in \pi _2(M)\). Hence if \([y,{w_y}]\) is in the image of \(\delta \), so is \([y, {w_y} \#^k A]\).

  5. This index relation has been used earlier in [26] by Viterbo. Its versions are used also in [8, 13, 15, 23,24,25, 27].

  6. If \(\widehat{H_L}\) denotes the Hamiltonian in in Proposition 5.1 in [14], the Hamiltonian \(H_L\) in Proposition 1 is \(-\widehat{H_L}\). In this proposition in [14], the metric on L is a metric whose energy functional is Morse-Bott, hence \(B_r\) is a submanifold.

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  1. Department of Economics, Middle East Technical University, 06800, Ankara, Turkey

    Nil İpek Şirikçi

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Şirikçi, N.İ. A Minimal Maslov Number Condition for Displaceability in Certain Weakly Exact Symplectic Manifolds. Results Math 77, 172 (2022). https://doi.org/10.1007/s00025-022-01714-4

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