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The second variation of area of minimal surfaces in four-manifolds. (English) Zbl 0788.58016

We study the second variation of area of an oriented minimal surface in an oriented Riemannian 4-manifold \(N\) whose curvature operator satisfies \({1 \over 6} S-W_ + \geq 0\) where \(S\) is the scalar curvature and \(W_ + :\Lambda^ 2_ + \to \Lambda^ 2_ +\) is the part of the Weyl curvature operator which maps self-dual two-forms \((\Lambda^ 2_ +)\) into themselves. Special attention is given to the case when \(N\) is Kähler and sharp results are obtained when \(N\) is hyper-Kähler. In particular, the relation between stability, holomorphicity and the complex and anti-complex tangent points of the minimal surface is investigated.

MSC:

58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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References:

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