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Geometry of harmonic maps

Author: Y L Xin
Publisher: Boston : Birkhäuser, 1996.
Series: Progress in nonlinear differential equations and their applications, v. 23.
Edition/Format:   Print book : EnglishView all editions and formats
Database:WorldCat
Summary:

This monograph examines a fundamental mathematical concept connected to differential geometry - stochastic processes.

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Y L Xin
ISBN: 0817638202 9780817638207 3764338202 9783764338206
OCLC Number: 34150486
Description: x, 241 pages ; 25 cm.
Contents: I. Introduction.- 1.1 Vector Bundles.- 1.1.1 Vector Bundles.- 1.1.2 Connections.- 1.2 Harmonic Maps.- 1.2.1 Energy.- 1.2.2 Tension Field.- 1.2.3 The First Variational Formula.- 1.2.4 Examples of Harmonic Maps.- 1.3 A Bochner Type Formula.- 1.3.1 Hodge-Laplace Operator and Weitzenbock Formula.- 1.3.2 A Bochner Type Formula and Its Applications.- 1.4 Basic Properties of Harmonic Maps.- 1.4.1 Maximum Principle.- 1.4.2 Unique Continuation Theorems.- 1.4.3 Second Variational Formula and Stable Harmonic Maps.- II. Conservation Law.- 2.1 Stress-Energy Tensor and Conservation Law.- 2.2 Monotonicity Formula.- 2.3 Applications of Conservation Law to Liouville type Theorems.- 2.4 Further Generalizations.- III. Harmonic Maps and Gauss Maps.- 3.1 Generalized Gauss Maps.- 3.2 Cone-like Harmonic Maps.- 3.3 Generalized Maximum Principle.- 3.4 Estimates of Image Diameter and its Applications.- 3.5 Gauss Image of a Space-Like Hypersurface in Minkowski Space.- 3.6 Gauss Image of a Space-Like Submanifold in Pseudo-Euclidean Space.- 3.6.1 Geometry of ?IV(2).- 3.6.2 Gauss Map.- 3.6.3 Gauss Image of a Space-like Surface in R24.- IV. Harmonic Maps and Holomorphic Maps.- 4.1 Partial Energies.- 4.2 Harmonicity of Holomorphic Maps.- 4.3 Holomorphicity of Harmonic Maps.- V. Existence, Nonexistence and Regularity.- 5.1 Direct Method of the Calculus of Variations.- 5.2 Regularity Theorems.- 5.3 Nonexistence and Existence.- 5.4 Regularity Results of Harmonic Maps into Positively Curved Manifolds.- VI. Equivariant Harmonic Maps.- 6.1 Riemannian Submersions and Equivariant Harmonic Maps.- 6.2 Reduction Theorems.- 6.3 Equivariant Variational Formulas.- 6.4 On Harmonic Representatives of ?m(Sm).- 6.4.1 ODE of Smith's Construction.- 6.4.2 The Solvability of ODE (6.31) and (6.33).- 6.4.3 Application of Smith's Construction.- 6.4.4 Another Construction of Equivariant Maps.- 6.4.5 The Solvability of ODE (6.63) and (6.65).- 6.4.6 On Harmonic Representatives of Homotopy Group of the Higher Dimensional Sphere.- 6.5 Harmonic Maps via Isoparametric Maps.- 6.6 Harmonic Maps of Projective Spaces.- 6.6.1 Harmonic Maps from QPn - 1 into Sml.- 6.6.2 Harmonic Maps from QPn - 1 into QPm-1.- 6.7 Equivariant Boundary Value Problems.- 6.7.1 The Reduced PDE.- 6.7.2 The Solvability of PDE.- 6.7.3 Construction of Equivariant Maps into CP2.- 6.7.4 Heat Flow.- 6.7.5 Global Existence and Subconvergence.- References.
Series Title: Progress in nonlinear differential equations and their applications, v. 23.
Responsibility: Yuanlong Xin.
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