The second variation of area of minimal surfaces in four-manifolds
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.
scientific article; zbMATH DE number 537252
scientific article