Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds

Abstract

This article shows that for generic choice of Riemannian metric on a smooth manifold $M$ of dimension four, all prime compact parametrized minimal surfaces within $M$ have self-intersections in general position in the following sense: self-intersections are transverse and the two tangent planes at any self-intersection point fail to be complex with respect to any orthogonal complex structure on the ambient manifold $M$. This implies via a result of Sheldon Chang that $H_2(M;{\mathbb Z})$ is generated by homology classes that are represented by imbedded minimal surfaces.