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Schrödinger flow of maps into symplectic manifolds. (English) Zbl 0918.53017
Let $$(N,\omega)$$ be a symplectic manifold with an almost complex structure $$J$$. Then $$h=\omega\circ J$$ is a Riemannian metric. Let $$(M,g)$$ be a Riemannian manifold and consider a map $$u:M\to N$$. The Schrödinger flow is the flow on $$N$$ induced by the Hamiltonian vector field $$V_E$$ of the energy functional $$E(u)=\int_M e(u) dv_g$$ where, in local coordinates, $$e(u)=1/2 g^{\alpha\beta}h_{ij} \partial_\alpha u^i \partial_\beta u^j$$. It is shown that the flow of a ferromagnetic chain can be represented as Schrödinger flow of maps into $$S^2$$ and that there exists a unique local smooth solution for the initial value problem of the 1-dimensional Schrödinger flow of maps into a Kähler manifold. In the case that the targets are Kähler manifolds with constant curvature, it is proved that the 1-dimensional Schrödinger flow admits a unique global smooth solution.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 37D99 Dynamical systems with hyperbolic behavior 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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##### References:
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