On the biharmonic and harmonic indices of the Hopf map

Details On the biharmonic and harmonic indices of the Hopf map On the biharmonic and harmonic indices of the Hopf map

2004-02-18

arxiv.org/abs/math/0402295v1

Mathematics

Differential Geometry

17 pages

Scientific paper

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $\psi:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $\phi:\s^3\to \s^3$. We show $\phi$ to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.

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