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On the biharmonic and harmonic indices of the Hopf map. (English) Zbl 1124.58009

As a generalization of harmonic maps, the biharmonic maps are the critical points of the bienergy functional. Any harmonic map is biharmonic, so the authors are interested in nonharmonic biharmonic maps.
Inspired by [E. Loubeau and C. Oniciuc, Compos. Math. 141, No. 3, 729–745 (2005; Zbl 1075.58014)], the authors prove that a nonconstant map \(\psi:M\to \mathbb{S}^n(\frac{1}{\sqrt{2}})\), \(M\) compact, composed with the inclusion of \(\mathbb{S}^n(\frac{1}{\sqrt{2}})\) in \(\mathbb{S}^{n+1}\) becomes a nonharmonic biharmonic map \(\varphi:M\to \mathbb{S}^{n+1}\) if and only if \(\psi\) is harmonic and has constant energy density. In particular, a harmonic Riemannian submersion \(\psi:M\to \mathbb{S}^n(\frac{1}{\sqrt{2}})\) is modified into a nonharmonic biharmonic subimmersion \(\varphi:M\to \mathbb{S}^{n+1}\), which is proved to be unstable.
When \(\psi:\mathbb{S}^3(\sqrt{2})\to \mathbb{S}^2(\frac{1}{\sqrt{2}})\) is the harmonic Hopf map, the authors first determine the spectrum of the vertical Laplacian, and then recover Urakawa’s determination of the harmonic index and nullity [H. Urakawa, Trans. Am. Math. Soc. 301, 557–589 (1987; Zbl 0621.58010)]. A geometrical description of the kernel of Jacobi operator \(J\) associated to \(\psi\) is given.
Next, the authors study the operator \(I\) associated to the biharmonic map \(\varphi:\mathbb{S}^3(\sqrt{2})\to \mathbb{S}^3\), which gives the Hessian of the bienergy. They prove that the biharmonic index of \(\varphi\) is at least 11, while its nullity is greater than 8. Moreover, a geometrical desription of the kernel of \(I\) and of the space where \(I\) is negative definite is given.

MSC:

58E20 Harmonic maps, etc.
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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