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Minimal Submanifolds via Complex-Valued Eigenfunctions

Sigmundur Gudmundsson, Thomas Jack Munn

Abstract

In this work we introduce a new method for manufacturing minimal submanifolds in Riemannian geometry. For this we employ the so called complex-valued eigenfunctions. This is particularly interesting in the cases when the Riemannian ambient manifold is compact.

Minimal Submanifolds via Complex-Valued Eigenfunctions

Abstract

In this work we introduce a new method for manufacturing minimal submanifolds in Riemannian geometry. For this we employ the so called complex-valued eigenfunctions. This is particularly interesting in the cases when the Riemannian ambient manifold is compact.
Paper Structure (10 sections, 15 theorems, 118 equations, 1 table)

This paper contains 10 sections, 15 theorems, 118 equations, 1 table.

Table of Contents

  1. Introduction
  2. Eigenfunctions and Eigenfamilies
  3. Horizontal Conformality up to First Order
  4. Minimality of the fibre over the origin
  5. Simple Compact Examples
  6. The Riemannian Lie group $\mathbf{GL}_{n}({\mathbb C})$
  7. The compact special orthogonal group $\mathbf{SO}(n)$
  8. The compact unitary group $\text{\bf U}(n)$
  9. The compact quaternionic unitary group $\text{\bf Sp}(n)$
  10. Acknowledgements

Key Result

Theorem 1.1

Let $\phi:(M^m,g)\to (N,h)$ be a smooth conformal map between Riemannian manifolds. If $m=2$ then $\phi$ is harmonic if and only if the image is minimal in $(N,h)$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 3.1
  • ...and 24 more