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

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.

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...

Closed minimal surfaces of high Morse index in manifolds of negative curvature

We use Bott’s equivariant Morse theory to show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index. 1 Sources of noncompactness for parametrized minimal surfaces In this article, we apply the theory of parametrized two-dimensional minimal surfaces in a compact n-dimensional Riemannian manifold (M, 〈·, ·〉) to t...

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

Least Area Incompressible Surfaces in 3-Manifolds

Let M be a Riemannian manifold and let F be a closed surface. A map f: F---,M is called least area if the area of f is less than the area of any homotopic map from F to M. Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function. The existence of least area immersions in a ...