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Harmonic Maps and Minimal Immersions with Symmetries (AM-130), Volume 130 : Methods of Ordinary Differential Equations Applied to Elliptic Variational Problems. (AM-130) - James Eells

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

Methods of Ordinary Differential Equations Applied to Elliptic Variational Problems. (AM-130)

Paperback Published: 11th April 1993
ISBN: 9780691102498
Number Of Pages: 240

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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.

Basic Variational and Geometrical Properties
Harmonic maps and minimal immersions Basic properties of harmonic mapsp. 13
Minimal immersionsp. 20
Immersions of parallel mean curvature
Parallel mean curvaturep. 24
Alexandrov's theoremp. 29
Surfaces of parallel mean curvature
Theorems of Chern and Ruh-Vilmsp. 34
Theorems of Almgren-Calabi and Hopfp. 37
On the Sinh-Gordon equationp. 40
Wente's theoremp. 42
Reduction techniques
Riemannian submersionsp. 48
Harmonic morphisms and maps into a circlep. 51
Isoparametric mapsp. 54
Reduction techniquesp. 58
G-Invariant Minimal and Constant Mean Curvature Immersions
First examples of reductions
G-equivariant harmonic mapsp. 64
Rotation hypersurfaces in spheresp. 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 theoremp. 92
Existence of solutions starting at the boundaryp. 95
Analysis of the O.D.E. and proof of the main theoremp. 102
Constant mean curvature immersions of hyperspheres into R[superscript n]
Statement of the main theoremp. 111
Analytical lemmasp. 114
Proof of the main theoremp. 120
Harmonic Maps Between Spheres
Polynomial maps
Eigenmaps S[superscript m] [actual symbol not reproducible] S[superscript n]p. 129
Orthogonal multiplications and related constructionsp. 137
Polynomial maps between spheresp. 143
Existence of harmonic joins
The reduction equationp. 151
Properties of the reduced energy functional Jp. 154
Analysis of the O.D.Ep. 157
The damping conditionsp. 161
Examples of harmonic mapsp. 167
The harmonic Hopf construction
The existence theoremp. 171
Examples of harmonic Hopf constructionsp. 179
[pi][subscript 3](S[superscript 2] and harmonic morphismsp. 182
Second variationsp. 188
Riemannian immersions S[superscript m] [actual symbol not reproducible] S[superscript n]p. 200
Minimal graphs and pendent dropsp. 204
Further aspects of pendulum type equationsp. 208
Referencesp. 213
Indexp. 224
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691102498
ISBN-10: 069110249X
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 240
Published: 11th April 1993
Country of Publication: US
Dimensions (cm): 23.52 x 16.31  x 1.42
Weight (kg): 0.34