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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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