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Morse index of harmonic maps of the sphere. (Indice de Morse des applications harmoniques de la sphère.) (French) Zbl 0924.58012
From the introduction (translated from the French): “A harmonic map \(\varphi\) between two Riemannian manifolds is a critical point of the energy functional. It is said to be stable if the second variation \(H_\varphi\) of the energy in \(\varphi\) is semipositive on the space \(\Gamma(\varphi)\) of vector fields along \(\varphi\).
In this article, we are essentially interested in the case in which the source manifold is an \(m\)-dimensional canonical sphere \(S^m\). In fact, it is well-known that if \(m\geq 3\) then every stable harmonic map from \(S^m\) into a Riemannian manifold \((N,h)\) is constant. The question that arises naturally is to determine the least unstable nonconstant harmonic maps from \(S^m\) into \((N,h)\). The instability degree of a harmonic map \(\varphi\) is measured by the index, denoted by \(\text{Ind}_E(\varphi)\) of \(H_\varphi\), i.e., the dimension of the maximal subspaces of \(\Gamma(\varphi)\) where \(H_\varphi\) is negative.
Our first result in this paper is that under all nonconstant harmonic maps, defined on \(S^m\), the identity \(I\) of \(S^m\) has the smallest index. That means, for all nonconstant harmonic \(\varphi: S^m\to (N,h)\) with \(m\geq 3\), one has \[ \text{Ind}_E(\varphi)\geq \text{Ind}_E(I)= m+1. \] In the second part of the paper, we calculate the canonical inversion index \(j_m: S^m\to\mathbb{R} P^m\to\mathbb{C} P^m\) (which is totally real), then we show that under all non(anti-)holomorphic harmonic functions \(\varphi\) of \(S^2\) in \(\mathbb{C} P^d\), the application \(j_2\) possesses the smallest index, i.e., \[ \text{Ind}_E(\varphi)\geq \text{Ind}_E(j_2)= 6\text{''}. \]
Reviewer: Reviewer (Berlin)

MSC:
58E20 Harmonic maps, etc.
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References:
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