The Weierstrass representation for pluriminimal submanifolds (Q1772569)

Let \(M\) be a complex manifold of dimension \(m\) and \((X,g)\) a Riemannian manifold. An immersion \(f: M\to X\) is said to be pluriminimal if the restriction to any smooth complex curve in \(M\) is a minimal immersion into \(X\). If \(m= 1\), pluriminimal immersions are minimal. In this paper the case where \((X,g)\) is the Euclidean space is studied. An analogue of the Weierstrass representation for pluriminimal maps is proposed. This formula allows the construction of many examples by explicit calculation or by using techniques of complex geometry to establish existence results. As an application, immersions of \(\mathbb C^2\) in \(\mathbb R^6\) generalizing the example found by Furuhata are constructed. It is also shown that any affine algebraic variety admits a pluriminimal immersion into some Euclidean space.