Instability of the geodesic flow for the energy functional

Domenico Perrone

Vol. 249 (2011), No. 2, 431–446

DOI: 10.2140/pjm.2011.249.431

Abstract

Let (Sⁿ(r),g₀) be the canonical sphere of radius r. Denote by G_s the Sasaki metric on the unit tangent bundle T₁Sⁿ(r) induced from g₀ and by G_s the Sasaki metric on T₁ T₁Sⁿ(r) induced from G_s. We resolve here, for n ≥ 7, a question raised by Boeckx, González–Dávila, and Vanhecke: namely, we prove that the geodesic flow

is an unstable harmonic vector field for any r > 0 and n ≥ 7. In particular, in the case r = 1, ξ is an unstable harmonic map. We show that these results are invariant under a four-parameter deformation of the Sasaki metric G_s.

Keywords

geodesic flow, canonical sphere, stability, energy functional, harmonic maps, natural Riemannian metrics

Mathematical Subject Classification 2000

Primary: 53C43, 53D25

Milestones

Received: 9 February 2010

Accepted: 8 November 2010

Published: 1 February 2011