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Displaceability of Certain Constant Sectional Curvature Lagrangian Submanifolds

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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.

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Notes

  1. When the map \(\pi _{1}(L)\rightarrow \pi _{1}(M)\) is injective in the long exact sequence of homotopy groups ...\(\pi _{2}(M)\rightarrow \pi _{2}(M,\ L)\rightarrow \pi _{1}(L)\rightarrow \pi _{1}(M)\) ..., the map \(\pi _{2}(M)\rightarrow \pi _{2}(M,\ L)\) is onto. Then, by symplectic asphericity, \(\omega |_{\pi _{2}(M,\ L)}=0\).

  2. Such a function can be obtained [50].

  3. For sufficiently small \(\epsilon _L>0\), F is a Morse function whose critical points outside \(U_{\epsilon }\) agree with those of \(F_0\) and whose critical points in \(U_{\epsilon }\) are precisely the critical points of \(F_L\) on \(L \subset M\).

  4. The set of potentials is dense in \(C^{\infty }(S^1 \times L)\) [35, 58].

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

    Nil İpek Şirikçi

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The original version of this article was revised. In the original publication the author name is incorrectly published as “Nil İpek Şirikçi”. The correct author name should be read as “Nil İpek Şirikçi” and it is available in this correction.

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Şirikçi, N.İ. Displaceability of Certain Constant Sectional Curvature Lagrangian Submanifolds. Results Math 75, 158 (2020). https://doi.org/10.1007/s00025-020-01279-0

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