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.

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Correspondence to Rui Pacheco.

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The second author was partially supported by Fundação para a Ciência e Tecnologia through the project UID/MAT/00212/2019.

Communicated by Adrian Constantin.

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Aleman, A., Pacheco, R. & Wood, J.C. Harmonic maps and shift-invariant subspaces. Monatsh Math (2021). https://doi.org/10.1007/s00605-021-01516-w

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Keywords

  • Harmonic maps
  • Riemann surfaces
  • Shift-invariant subspaces

Mathematics Subject Classification

  • Primary 58E20
  • Secondary 47B32
  • 30H15
  • 53C43