Branch Points and Bumpy Metrics for Parametrized Minimal Spheres and Tori

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

Spectral Properties of Bipolar Minimal Surfaces In

Abstract. The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori and on Klein bottles. The present paper is concerned with extremal m...

Honours Projects for 2009

This project concerns an important conjecture which appears to have been solved recently: It concerns minimal surfaces, in particular minimal submanifolds of spheres. It combines PDE and geometry, though the PDE required is not very much. It does involve some basic spectral theory for the Laplacian on a Riemannian manifold. There are many examples known of submanifolds in spheres which are mini...