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

Book Description

TABLE OF CONTENTS:

	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

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