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Harmonic morphisms and minimal submanifolds

Oskar Riedler

Abstract

Harmonic morphisms, maps which preserve Laplace's equation, are intimately connected to the topic of minimal submanifolds. In this article we first characterise harmonic morphisms between Riemannian manifolds as the weakly horizontally conformal maps that preserve the equation for minimal submanifolds of co-dimension $2$. We further derive additional reduction properties of harmonic morphisms for minimal submanifolds of other co-dimensions. These theorems are then applied in an example case, yielding a novel family of degree $4$ area-minimising hypercones in $\mathbb R^m$, $m\geq32$.

Harmonic morphisms and minimal submanifolds

Abstract

Harmonic morphisms, maps which preserve Laplace's equation, are intimately connected to the topic of minimal submanifolds. In this article we first characterise harmonic morphisms between Riemannian manifolds as the weakly horizontally conformal maps that preserve the equation for minimal submanifolds of co-dimension . We further derive additional reduction properties of harmonic morphisms for minimal submanifolds of other co-dimensions. These theorems are then applied in an example case, yielding a novel family of degree area-minimising hypercones in , .
Paper Structure (9 sections, 21 theorems, 40 equations, 1 table)

This paper contains 9 sections, 21 theorems, 40 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction theorems
  4. Examples: area-minimising cones of $\mathds R^m$
  5. Umbillic polynomial harmonic morphisms
  6. Minimal submanifolds from quadratic harmonic morphisms
  7. The method of subcalibrations
  8. Proof of \ref{['thm: minimal-Scone']}
  9. Local existence of minimal submanifolds

Key Result

Theorem 1.1

Let $\varphi\colon(M,g)\to(N,h)$ be a submersive horizontally conformal map with $\dim(N)=2$. The following are equivalent:

Theorems & Definitions (55)

  • Theorem 1.1: baird-eells-80
  • Theorem 1.2
  • Theorem 1.3: cf. Corollary 3.4 of baird-gudmundsson-92
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Example 1.7
  • Remark 1.8
  • Definition 2.1
  • Theorem 2.2: fuglede-78ishihara-79
  • ...and 45 more