Equivariant harmonic maps depend real analytically on the representation

Abstract

We prove that when assuming suitable non-degeneracy conditions equivariant harmonic maps into symmetric spaces of non-compact type depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in [EL81] can be applied. Statement of the resultsWe first collect some preliminary definitions and results needed to give a statement of the main theorem. Harmonic mapsIf (M, g) and (N, h) are Riemannian manifolds with M compact then a C 1 map f : (M, g) → (N, h) is called harmonic if it is a critical point of the

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“…Another example is provided by equivariant harmonic maps mapping into symmetric spaces of non-compact type. A result of Sunada ([7]) implies that such harmonic maps are non-degenerate critical points of the energy if and only if they are unique (see [6,Lemma 2.1]).…”