Stability and singularities of harmonic maps into 3-spheres

The behavior of harmonic maps from a domain in R-4 into S-3 is discussed. In this paper, we show that if a is a strictly stable stationary harmonic map : B-4 --> S-3, such that the singular set Sing(u) of a consists of {0}, then the degree deg(u, 0) of a at 0 is zero; and that if a is a weakly stable harmonic map B-4 --> S-3 with Sing(u) = {0}, then deg(u, 0) = 0 or +/-1. Furthermore, the characterization of homogeneous stable stationary harmonic maps from B-4 into S-3 is given. (C) 2003 Elsevier Ltd. All rights reserved.