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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
58E20 Harmonic maps, etc.
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