Nondegeneracy of coverings of minimal tori and Klein bottles in Riemannian manifolds

John Douglas Moore

Vol. 230 (2007), No. 1, 147–166

DOI: 10.2140/pjm.2007.230.147

Abstract

We say that a parametrized minimal torus or Klein bottle in an ambient Riemannian manifold is Morse nondegenerate if it lies on a nondegenerate critical submanifold which is also an orbit for the group of isometries of the flat metric of total area one. We show that for a generic choice of a Riemannian metric on a compact manifold of dimension at least four, unbranched multiple covers of prime minimal tori or Klein bottles are Morse nondegenerate. A similar result holds for harmonic tori and Klein bottles. The proofs require a modification of techniques of Bott for studying iterations of smooth closed geodesics.

Keywords

minimal surfaces, harmonic surfaces, bumpy metrics, generic Riemannian metrics

Mathematical Subject Classification 2000

Primary: 53C40, 58E12

Secondary: 58D15, 58E05

Milestones

Received: 17 August 2005

Revised: 13 February 2007

Accepted: 14 February 2007

Published: 1 March 2007