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Harmonic Morphisms and p-Harmonic Functions on the Classical Compact Symmetric Spaces via the Cartan Embedding

Adam Lindström

Abstract

Given a symmetric triple $(G,K,σ)$ of compact type, with $G^σ = K$, the well known Cartan embedding $\hatΦ: G/K \to G$ homothetically embeds the symmetric space $M = G/K$ as a totally geodesic submanifold of $G$. In this thesis we show that $\hatΦ$ and the related $K$-invariant Cartan map $Φ= \hatΦ\circ π$ are harmonic. This yields simple formulae relating the tension field $τ$ and the recently introduced conformality operator $κ$ on the symmetric space $M$ to those on the image $Φ(G)$ of the Cartan map. We use these formulae to construct common eigenfunctions of the tension field and conformality operator on all the classical irreducible compact symmetric spaces. On the complex and quaternionic Grassmannians we, in addition, construct eigenfamilies, which are families of compatible eigenfunctions. In the case of the quaternionic Grassmannians, some of these eigenfamilies are entirely new. It was recently discovered by S. Gudmundsson, A. Sakovich and M. Sobak that such eigenfunctions can be employed to construct proper $p$-harmonic functions, and that eigenfamilies can be applied to construct complex-valued harmonic morphisms. The results of this thesis can thus be used to construct harmonic morphisms and $p$-harmonic functions on the classical compact irreducible symmetric spaces, and in the case of the quaternionic Grassmannians, new examples of such maps.

Harmonic Morphisms and p-Harmonic Functions on the Classical Compact Symmetric Spaces via the Cartan Embedding

Abstract

Given a symmetric triple of compact type, with , the well known Cartan embedding homothetically embeds the symmetric space as a totally geodesic submanifold of . In this thesis we show that and the related -invariant Cartan map are harmonic. This yields simple formulae relating the tension field and the recently introduced conformality operator on the symmetric space to those on the image of the Cartan map. We use these formulae to construct common eigenfunctions of the tension field and conformality operator on all the classical irreducible compact symmetric spaces. On the complex and quaternionic Grassmannians we, in addition, construct eigenfamilies, which are families of compatible eigenfunctions. In the case of the quaternionic Grassmannians, some of these eigenfamilies are entirely new. It was recently discovered by S. Gudmundsson, A. Sakovich and M. Sobak that such eigenfunctions can be employed to construct proper -harmonic functions, and that eigenfamilies can be applied to construct complex-valued harmonic morphisms. The results of this thesis can thus be used to construct harmonic morphisms and -harmonic functions on the classical compact irreducible symmetric spaces, and in the case of the quaternionic Grassmannians, new examples of such maps.
Paper Structure (17 sections, 501 equations)

This paper contains 17 sections, 501 equations.

Table of Contents

  1. Introduction
  2. Lie Groups and Lie Algebras
  3. The Classical Compact Lie Groups
  4. Symmetric Spaces
  5. The Cartan Embedding
  6. Harmonic Morphisms and p-Harmonic Functions
  7. The Methods of Eigenfunctions and Eigenfamilies
  8. The Classical Symmetric Spaces and Their Involutions
  9. The Symmetric Spaces $\SU{n}$, $\SO{n}$ and $\Sp{n}$
  10. The Symmetric Space $\U{m+n}/\U{m}\times \U{n}$
  11. The Symmetric Space $\SO{m+n}/\SO{m}\times \SO{n}$
  12. The Symmetric Space $\Sp{m+n}/\Sp{m}\times\Sp{n}$
  13. The Symmetric Space $\SU{n}/\SO{n}$
  14. The Symmetric Space $\SO{2n}/\U{n}$
  15. The Symmetric Space $\Sp{n}/\U{n}$
  16. ...and 2 more sections

Theorems & Definitions (73)

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