James Eells

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

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

• Format: PDF

Produktbeschreibung

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.

Produktdetails

• Verlag: Princeton University Press
• Seitenzahl: 240
• Erscheinungstermin: 2. März 2016
• Englisch
• ISBN-13: 9781400882502
• Artikelnr.: 45474441

Autorenporträt

James Eells is Professor of Mathematics at the University of Warwick. Andrea Ratto is Professor Mathematics at the Universite de Bretagne Occidentale in Brest.

Inhaltsangabe

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