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Proof of the Arnold conjecture for surfaces and generalizations to certain Kähler manifolds. (English) Zbl 0607.58016
Given a compact symplectic manifold (P,\(\omega)\) and a smooth function H: S’\(\times P\to {\mathbb{R}}: (t,p)\mapsto H_ t(p)\), \(S'={\mathbb{R}}/{\mathbb{Z}}\), a periodic family of (”exact Hamiltonian”) vector fields \(X_ t\) is given by \(\omega (\cdot,X_ t)=dH_ t(\cdot)\). The author presents a proof for the Arnold conjecture in the case of surfaces which is stated in the following way: ”On a compact surface \((F_ g,\omega)\) of genus \(g\geq 1\) with volume form \(\omega\) every exact Hamiltonian vector field of period 1 possesses at least 3 solutions of period 1. If all the 1-periodic solutions are nondegenerate, then there exist at least \(2(g+1)\) of them.” Some generalizations to certain Kähler manifolds are also presented.
Reviewer: W.Oliva

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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