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Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras. (English) Zbl 0796.53063

Let \(\varphi= \mathbb{C}^ n/\Gamma\to G/K\) be a pluriharmonic map of a complex torus into a symmetric space of compact type. The authors introduce a class of pluriharmonic maps of finite type which is a set of solutions of a family of commuting Hamiltonian systems on finite- dimensional subspaces of the algebra of based loops in the Lie algebra of \(G\). The main result is the following Theorem: Let \(\varphi\) be a surjective pluriharmonic map and \(Z\) a holomorphic vector field on \(\mathbb{C}^ n/\Gamma\) such that for some \(x_ 0\), \(\varphi_ *(Z_{x_ 0})\in{\mathfrak p}_{\varphi(x_ 0)}\) is a regular semisimple element. Then \(\varphi\) is of finite type.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E20 Harmonic maps, etc.
58A50 Supermanifolds and graded manifolds
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