Harmonic Maps and Minimal Immersions With

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
Basic Variational and Geometrical Properties
Harmonic maps and minimal immersions Basic properties of harmonic maps	p. 13
Minimal immersions	p. 20
Immersions of parallel mean curvature
Parallel mean curvature	p. 24
Alexandrov's theorem	p. 29
Surfaces of parallel mean curvature
Theorems of Chern and Ruh-Vilms	p. 34
Theorems of Almgren-Calabi and Hopf	p. 37
On the Sinh-Gordon equation	p. 40
Wente's theorem	p. 42
Reduction techniques
Riemannian submersions	p. 48
Harmonic morphisms and maps into a circle	p. 51
Isoparametric maps	p. 54
Reduction techniques	p. 58
G-Invariant Minimal and Constant Mean Curvature Immersions
First examples of reductions
G-equivariant harmonic maps	p. 64
Rotation hypersurfaces in spheres	p. 74
Constant mean curvature rotation hypersurfaces in R[superscript n]	p. 81
Minimal embeddings of hyperspheres in S[superscript 4]
Derivation of the equation and main theorem	p. 92
Existence of solutions starting at the boundary	p. 95
Analysis of the O.D.E. and proof of the main theorem	p. 102
Constant mean curvature immersions of hyperspheres into R[superscript n]
Statement of the main theorem	p. 111
Analytical lemmas	p. 114
Proof of the main theorem	p. 120
Harmonic Maps Between Spheres
Polynomial maps
Eigenmaps S[superscript m] [actual symbol not reproducible] S[superscript n]	p. 129
Orthogonal multiplications and related constructions	p. 137
Polynomial maps between spheres	p. 143
Existence of harmonic joins
The reduction equation	p. 151
Properties of the reduced energy functional J	p. 154
Analysis of the O.D.E	p. 157
The damping conditions	p. 161
Examples of harmonic maps	p. 167
The harmonic Hopf construction
The existence theorem	p. 171
Examples of harmonic Hopf constructions	p. 179
[pi][subscript 3](S[superscript 2] and harmonic morphisms	p. 182
Second variations	p. 188
Riemannian immersions S[superscript m] [actual symbol not reproducible] S[superscript n]	p. 200
Minimal graphs and pendent drops	p. 204
Further aspects of pendulum type equations	p. 208
References	p. 213
Index	p. 224
Table of Contents provided by Publisher. All Rights Reserved.

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Harmonic Maps and Minimal Immersions With Symmetries

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