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…mehr
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.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
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
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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