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Harmonic maps between complex projective spaces. (English) Zbl 0697.58014
Let $${\mathcal H}^{p,q}$$ denote the Hermitian vector space of homogeneous harmonic polynomials in the variables $$z_ 0,...,z_ m,\bar z_ 0,...,\bar z_ m$$ of bidegree (p,q). We call f: $$S^{2m+1}\to S^{2n+1}$$ a polynomial harmonic map of bidegree (p,q) if the components of f in $${\mathbb{C}}^{n+1}$$ belong to $${\mathcal H}^{p,q}$$. We show that the space of all such maps (for m and (p,q) fixed) that project down to harmonic maps $$\bar f:$$ $${\mathbb{C}}P^ m\to {\mathbb{C}}P^ n$$ (modulo the action of the unitary group on the ranges) can be parametrized by a compact convex body lying in a $$U(m+1)$$-submodule of $${\mathcal H}^{p,q}\otimes {\mathcal H}^{q,p}$$. Using the Littlewood-Richardson rule we give an asymptotically best possible lower bound for the dimension of the parameter space. As a byproduct we obtain the exact dimension of the space parametrizing polynomial harmonic maps f: $${\mathbb{C}}P^ m\to S^ n$$.
Reviewer: G.Tóth
##### MSC:
 5.8e+21 Harmonic maps, etc.
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