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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.

##### MSC:
 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53A25 Differential line geometry
##### Keywords:
maximal surface; mean curvature; neutral Kähler
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