Self-intersections of closed parametrized minimal surfaces in generic Riemannian manifolds

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

This article shows that for a generic choice of Riemannian metric on a compact manifold M of dimension at least five, all prime compact parametrized minimal surfaces within M are imbeddings. Moreover, if M has dimension four, all prime compact parametrized minimal surfaces within M have transversal self-interstions, and at any self-intersection the tangent planes fail to be complex for any choi...

Bumpy Metrics and Closed Parametrized Minimal Surfaces in Riemannian Manifolds

The purpose of this article is to study conformal harmonic maps f : Σ→M , where Σ is a closed Riemann surface and M is a compact Riemannian manifold of dimension at least four. Such maps define parametrized minimal surfaces, possibly with branch points. We show that when the ambient manifold M is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, a...

Bumpy Riemannian Metrics and Closed Parametrized Minimal Surfaces in Riemannian Manifolds∗

This article is concerned with conformal harmonic maps f : Σ → M , where Σ is a closed Riemann surface and M is a compact Riemannian manifold of dimension at least four. We show that when the ambient manifold M is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, and are as Morse nondegenerate as allowed by the group of complex automorphisms of Σ.

Correction for: Bumpy Metrics and Closed Parametrized Minimal Surfaces in Riemannian Manifolds

Bifurcation of Minimal Surfaces in Riemannian Manifolds

We study the bifurcation of closed minimal surfaces in Riemannian manifolds through higher order variations of the area functional and relate it to elementary catastrophes.