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Local Schrödinger flow into Kähler manifolds. (English) Zbl 1019.53032
Let $$(M,g)$$ be a Riemannian manifold and $$(N,J,h)$$ a complete Kähler manifold with complex structure $$J$$ and Kähler metric $$h$$. The Schrödinger flow is defined by the initial value problem $$\partial_tu(x,t)=J(u(x,t))\tau(u(x,t))$$, $$u(\cdot,0)=u_0:M\rightarrow N$$, where $$\tau(u)$$ denotes the tension field of $$u$$. It is well known that $$u$$ is a harmonic if and only if $$\tau(u)\equiv 0$$. The authors discuss the short time existence of solutions of the Schrödinger flow from $$(M,g)$$ to $$(N,J,h)$$. It was shown that there exists a unique local smooth solution for the Cauchy problem to the Schrödinger flow for maps from compact Riemannian manifold into a complete Kähler manifold, or from a Euclidean space $$\mathbb{R}^n$$ into a compact Kähler manifold. As a consequence, the authors prove that the Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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##### References:
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