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Area-stationary surfaces in neutral Kähler 4-manifolds. (English) Zbl 1162.53016
Let \((N,g)\) be a Riemannian two-manifold and \(TN\) be its tangent bundle. The authors construct a neutral Kähler structure on \(TN\) and investigate surfaces in \(TN\) that are area-stationary with respect to the introduced Kähler metric. It is firstly shown that holomorphic curves in \(TN\) are area-stationary but area-stationary surfaces are, in general, not holomorphic. If the metric \(g\) is rotationally symmetric, then all area-stationary surfaces that arise as graphs of sections of the bundle \(TN \to N\) and that are rotationally symmetric, can be obtained. If in particular \((N,g)\) is the round two-sphere, then \(TN\) can be identified with the space of oriented affine lines in \(\mathbb R^3\). In this case the authors give a two parameter family of area-stationary tori that are neither holomorphic nor Lagrangian.

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53A25 Differential line geometry
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