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New Minimal Surfaces of the Sphere $S^4$ and the Hyperbolic Space $H^4$ via Harmonic Morphisms

Sigmundur Gudmundsson, Leonard Nygren Löhndorf

Abstract

In this work we introduce a new method for the construction of minimal submanifolds of codimension two in even dimensional spheres and hyperbolic spaces. This is based on the theory of complex-valued harmonic morphisms. This gives the first explicit examples of such maps defined on the sphere $S^4$ and the hyperbolic space $H^4$.

New Minimal Surfaces of the Sphere $S^4$ and the Hyperbolic Space $H^4$ via Harmonic Morphisms

Abstract

In this work we introduce a new method for the construction of minimal submanifolds of codimension two in even dimensional spheres and hyperbolic spaces. This is based on the theory of complex-valued harmonic morphisms. This gives the first explicit examples of such maps defined on the sphere and the hyperbolic space .

Paper Structure

This paper contains 8 sections, 5 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Harmonic Morphisms on the Euclidean ${\mathbb R}^{n}$
  4. Harmonic Morphisms on the Spheres $S^{2n+2}$
  5. Compact Minimal Surfaces in the Sphere $S^{4}$
  6. Harmonic Morphisms on the Lorentian Spaces ${\mathbb R}^n_1$
  7. Harmonic Morphisms on the Hyperbolic Spaces $H^{2n+2}$
  8. Minimal Surfaces in the Hyperbolic Space $H^{4}$

Key Result

Theorem 2.2

Fug-2 A map $\phi:(M,g)\to (N,h)$ between semi-Riemannian manifolds is a harmonic morphism if and only if it is a horizontally (weakly) conformal harmonic map.

Theorems & Definitions (13)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • ...and 3 more