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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

##### MSC:
 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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