Harmonic maps with symmetry, harmonic morphisms, and deformations of metrics (Research notes in mathematics)

Cover......Page 1 Title: Harmonic Maps with Symmetry, Harmonic Morphisms and Deformations of Metrics......Page 2 ISBN 0-273-08603-0......Page 3 Preface......Page 4 Contents......Page 6 Introduction......Page 9 1. 1 THE LAPLACIAN ON THE SPHERE......Page 15 1. 2 Harmonic maps into spheres......Page 17 1. 3 Joins of spheres and Smith •s construction......Page 19 1. 4 Outline of the solution of Smith's equation......Page 24 1. 5 Hyperbolic space......Page 27 1.6 Polar coordinates on hyperbolic space and an analogous construction to Smith's......Page 30 1. 7 Solving the equation for hyperbolic spaces......Page 32 2.1 Definition of isoparametric function......Page 35 2.2 Properties of isoparametric functions andMU"nzner•s classification theorem......Page 37 2. 3 Examples of isoparametric functions......Page 42 2.4 Generalizing the notion of isoparametric families of hypersurfaces......Page 47 3.1 Derivation of the stress-energy tensor......Page 50 3. 2 Examples......Page 54 3.3 The eigenvalue decomposition of the stress-energy tensor......Page 55 4 Equivariant theory......Page 57 4. 1 Maps which are equivariant with respect to isoparametric functions.......Page 58 4.2 Generalized equivariant maps between Riemannian manifolds......Page 73 5. 1 Maps from Euclidean space to the sphere......Page 77 5. 2 Maps from hyperbolic space to the sphere......Page 84 5. 3 Maps from sphere to sphere.......Page 88 6. 1 Existence of solutions......Page 102 6. 2 Asymptotic estimates......Page 119 6.3. Smoothness of certain equivariant harmonic maps......Page 125 7. 1 General theory......Page 130 7. 2 Examples and non-examples of harmonic morphisms......Page 133 7.3 Maps 0: (M,g) - (N,h) where 0 *h has two distinct non-zero eigenvalues.......Page 141 8. 1 Properties of harmonic polynomial morphisms......Page 143 8.2. Some examples......Page 147 8. 3. Harmonic morphisms defined bt homogeneous polynomials of degree bigger than two.......Page 150 8.4 Harmonic pol omial morphisms and equivariant maps between spheres......Page 154 9. 1 Deformations of the metric for harmonic morphisms......Page 161 9.2 Examples......Page 168 9.3. Deformations of metrics for equivariant maps......Page 170 9. 4 Examples......Page 180 References......Page 185 Index of definitions......Page 188 "The aim of this book is to construct harmonic maps between Riemannian manifolds, and in particular between spheres. These maps have a delightful geometry associated with them - they preserve families of level hypersurfaces of constant mean curvature. New maps between Euclidean spheres are constructed, as well as harmonic maps from hyperbolic space to sphere and from Euclidean space to sphere. The author makes considerable use of the stress-energy tensor, which has not previously been used in the context of harmonic maps...In particular, it is used to solve the rendering problem for certain classes of maps between spheres." - back cover