Morse theory and infinite families of harmonic maps between spheres

Abstract

Existence of an infinite sequence of harmonic maps between spheres of certain dimensions was proven by Bizon and Chmaj. This sequence shares many features of the Bartnik-McKinnon sequence of solutions to the Einstein-Yang-Mills equations as well as sequences of solutions that have arisen in other physical models. We apply Morse theory methods to prove existence of the harmonic map sequence and to prove certain index and convergence properties of this sequence. In addition, we generalize the result of Bizon and Chmaj to produce infinite sequences of harmonic maps not previously known. The key features ``responsible'' for the existence and properties of these sequences are thereby seen to be the presence of a reflection symmetry and the existence of a singular harmonic map of infinite index which is invariant under this symmetry.

arXiv comment17 pages, LaTeX

References in corpus (1)

• Saddle point solutions in Yang-Mills-dilaton theory

Cited by in corpus (4)

• Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere
• Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres
• On the equivariant stability of harmonic self-maps of cohomogeneity one manifolds
• Applications of the theory of evolution equations to general relativity