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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:
 5.8e+21 Harmonic maps, etc.
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##### References:
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