Harmonic maps and shift-invariant subspaces

Abstract

With the help of operator-theoretic methods, we derive new and powerful criteria for finiteness of the uniton number for a harmonic map from a Riemann surface to the unitary group $${{\,\mathrm{U}\,}}(n)$$ . These use the Grassmannian model where harmonic maps are represented by families of shift-invariant subspaces of $$L^2(S^1,{{\mathbb {C}}}^n)$$ ; we give a new description of that model.