On families of Lagrangian submanifolds |
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Authors: | Roberto Paoletti |
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Affiliation: | (1) Dipartimento di Matematica “E. De Giorgi”, Universitá di Lecce, Via per Arnesano, 73100 Lecce, Italy. e-mail: roberto.paoletti@unile.it, IT |
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Abstract: | We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ t of diffeomorphisms of X such that ω t =Φ t *(ω0). If L⊂X is a Lagrangian submanifold for (X,ω0), L t =Φ t -1(L) is thus a Lagrangian submanifold for (X,ω t ). Here we show that if we simply assume that L is compact and ω t | L is exact for every t, a family L t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001 |
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Keywords: | Mathematics Subject Classification (2000): 14A10 53D05 53D50 |
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