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Arnold’s conjecture and symplectic reduction

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http://www.sciencedirect.com/science/article/pii/0393044096895386
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Publication Date
1996
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Elsevier
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Abstract
Fortune (1985) proved Arnold's conjecture for complex projective spaces, by exploiting the fact that CPn-1 is a symplectic quotient of C-n. In this paper, we show that Fortune's approach is universal in the sense that it is possible to translate Arnold's conjecture on any closed symplectic manifold (Q,Omega) to a critical point problem with symmetry on loops in R(2n) With its Standard symplectic structure.
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UCM subjects
Análisis matemático , Ecuaciones diferenciales
Unesco subjects
1202 Análisis y Análisis Funcional , 1202.07 Ecuaciones en Diferencias
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V.I. Arnold, Commentary on “On a geometrical theorem”, in: Collected Works, Vol. II, ed. H. Poincare (Nauka, Moscow, MR 52 # 5337,1972) pp. 987-989. V.I. Arnold, Fixed points of symplectic diffeomorphisms, in: Mathematical Developments Arisingfrom the Hilbert Problems, Proc. Symp. Pure Math., Amer. Math. Sot. 28 (1976) 66. A. Fleer, Symplectic fixed points and holomorphic spheres, Commun. Math. Phys. 120 (1989) 575-611. B. Fortune, A symplectic fixed point theorem for CP”, Inv. Math. 81 (1985) 29-46. B. Fortune and A. Weinstein, A symplectic fixed point theorem for complex projective spaces, Bull.Amer. Math. Sot. 12 (1985) 128-130. A.B. Givental, A symplectic fixed point theorem for toric manifolds, Progress in Math. Fleer Memorial Volume, 1994, to appear. M.J. Gotay and G.M. Tuynman, R2” is a universal symplectic manifold for reduction, Lett. Math. Phys. 18 (1989) 55-59. L& Hong Van and K. Ono, Symplectic fixed points, the Calabi invariant and Novikov homology,Topology 34 (1995) 155-176. H. Hofer and D. Salamon, Fleer homology and Novikov rings, preprint (1992). A. Ibort and C. Martinez Ontalba, A universal setting for Arnold’s conjecture, C. R. Acad. Sci. Paris t.318, Strie II (1994) 561-566. M. Kummer, On the construction of the reduced phase space of a hamiltonian system with symmetry,Indiana Univ. Math. J 30 (1981) 281-291. Y.-G. Oh, A symplectic fixed point theorem on R*” x Cpk, Math. Z. 203 (1990) 535-552. K. Ono, On the Arnold conjecture for weakly monotone symplectic manifolds, Inv. Math. I 19 (1995)519-537. A. Weinstein, A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys. 2 (1978)417-420.
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https://hdl.handle.net/20.500.14352/57636
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