Description
Book SynopsisPresents a study of harmonic maps, minimal and parallel mean curvature immersions in the presence of symmetry. This book covers the material which displays an interplay involving geometry, analysis and topology. It includes a basic presentation of 1-cohomogeneous equivariant differential geometry and of the theory of harmonic maps between spheres.
Table of ContentsIntroductionPt. IBasic Variational and Geometrical PropertiesCh. IHarmonic maps and minimal immersionsBasic properties of harmonic maps13Minimal immersions20Ch. IIImmersions of parallel mean curvatureParallel mean curvature24Alexandrov's theorem29Ch. IIISurfaces of parallel mean curvatureTheorems of Chern and Ruh-Vilms34Theorems of Almgren-Calabi and Hopf37On the Sinh-Gordon equation40Wente's theorem42Ch. IVReduction techniquesRiemannian submersions48Harmonic morphisms and maps into a circle51Isoparametric maps54Reduction techniques58Pt. IIG-Invariant Minimal and Constant Mean Curvature ImmersionsCh. VFirst examples of reductionsG-equivariant harmonic maps64Rotation hypersurfaces in spheres74Constant mean curvature rotation hypersurfaces in R[superscript n]81Ch. VIMinimal embeddings of hyperspheres in S[superscript 4]Derivation of the equation and main theorem92Existence of solutions starting at the boundary95Analysis of the O.D.E. and proof of the main theorem102Ch. VIIConstant mean curvature immersions of hyperspheres into R[superscript n]Statement of the main theorem111Analytical lemmas114Proof of the main theorem120Pt. IIIHarmonic Maps Between SpheresCh. VIIIPolynomial mapsEigenmaps S[superscript m] [actual symbol not reproducible] S[superscript n]129Orthogonal multiplications and related constructions137Polynomial maps between spheres143Ch. IXExistence of harmonic joinsThe reduction equation151Properties of the reduced energy functional J154Analysis of the O.D.E.157The damping conditions161Examples of harmonic maps167Ch. XThe harmonic Hopf constructionThe existence theorem171Examples of harmonic Hopf constructions179[pi][subscript 3](S[superscript 2] and harmonic morphisms182Appendix 1 Second variations188Appendix 2 Riemannian immersions S[superscript m] [actual symbol not reproducible] S[superscript n]200Appendix 3 Minimal graphs and pendent drops204Appendix 4 Further aspects of pendulum type equations208References213Index224
