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

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

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...

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...

ACTION OF SEMISIMPLE ISOMERY GROUPS ON SOME RIEMANNIAN MANIFOLDS OF NONPOSITIVE CURVATURE

A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...