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

TL;DR

This paper proves that in generic four-dimensional Riemannian manifolds, all prime compact minimal surfaces typically have transverse self-intersections with non-complex tangent planes, impacting the structure of the manifold's homology.

Contribution

It establishes generic properties of minimal surfaces in 4-manifolds, showing they have transverse self-intersections and non-complex tangent planes, influencing homology class representations.

Findings

Self-intersections are transverse in generic metrics.

Tangent planes at self-intersections are not complex.

Homology classes are generated by embedded minimal surfaces.

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.