The aim of this book is to study harmonic maps, minimal and parallel mean curvature immersions in the presence of symmetry. In several instances, the latter permits reduction of the original elliptic variational problem to the qualitative study of certain ordinary differential equations: the authors' primary objective is to provide representative examples to illustrate these reduction methods and their associated analysis with geometric and topological applications.
The material covered by the book displays a solid interplay involving geometry, analysis and topology: in particular, it includes a basic presentation of 1-cohomogeneous equivariant differential geometry and of the theory of harmonic maps between spheres.
| 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 |
Presents a study of harmonic maps, minimal and parallel mean curvature immersions in the presence of symmetry. This book covers the material which displays an interplay involving geometry, analysis and topology. It includes a basic presentation of 1-cohomogeneous equivariant differential geometry and of the theory of harmonic maps between spheres.
