Harmonic Maps and Minimal Immersions with Symmetries (AM-130), Volume 130

Introduction

Pt. I Basic Variational and Geometrical Properties

Ch. I Harmonic maps and minimal immersions

Basic properties of harmonic maps 13

Minimal immersions 20

Ch. II Immersions of parallel mean curvature

Parallel mean curvature 24

Alexandrov's theorem 29

Ch. III Surfaces of parallel mean curvature

Theorems of Chern and Ruh-Vilms 34

Theorems of Almgren-Calabi and Hopf 37

On the Sinh-Gordon equation 40

Wente's theorem 42

Ch. IV Reduction techniques

Riemannian submersions 48

Harmonic morphisms and maps into a circle 51

Isoparametric maps 54

Reduction techniques 58

Pt. II G-Invariant Minimal and Constant Mean Curvature Immersions

Ch. V First examples of reductions

G-equivariant harmonic maps 64

Rotation hypersurfaces in spheres 74

Constant mean curvature rotation hypersurfaces in R[superscript n] 81

Ch. VI Minimal embeddings of hyperspheres in S[superscript 4]

Derivation of the equation and main theorem 92

Existence of solutions starting at the boundary 95

Analysis of the O.D.E. and proof of the main theorem 102

Ch. VII Constant mean curvature immersions of hyperspheres into R[superscript n]

Statement of the main theorem 111

Analytical lemmas 114

Proof of the main theorem 120

Pt. III Harmonic Maps Between Spheres

Ch. VIII Polynomial maps

Eigenmaps S[superscript m] [actual symbol not reproducible] S[superscript n] 129

Orthogonal multiplications and related constructions 137

Polynomial maps between spheres 143

Ch. IX Existence of harmonic joins

The reduction equation 151

Properties of the reduced energy functional J 154

Analysis of the O.D.E. 157

The damping conditions 161

Examples of harmonic maps 167

Ch. X The harmonic Hopf construction

The existence theorem 171

Examples of harmonic Hopf constructions 179

[pi][subscript 3](S[superscript 2] and harmonic morphisms 182

Appendix 1 Second variations 188

Appendix 2 Riemannian immersions S[superscript m] [actual symbol not reproducible] S[superscript n] 200

Appendix 3 Minimal graphs and pendent drops 204

Appendix 4 Further aspects of pendulum type equations 208

References 213

Index 224