The analysis of harmonic maps and their heat flows

• Main Authors: Lin, Fanghua, Wang, Changyou
• Format: eBook Book
• Language: English
• Published: New Jersey World Scientific Publishing Co. Pte. Ltd 2008; World Scientific; World Scientific Publishing Company; WORLD SCIENTIFIC
• Edition: 1
• Subjects: Harmonic maps; Heat equation; Pure Mathematics; Riemannian manifolds; SCIENCE; Textbooks; Heat Flow of Harmonic Maps; Harmonic Maps; Structure of Singularity; Regularity; Blow-Up Analysis
• ISBN: 9789812779526; 9812779523; 9789812779533; 9812779531
• DOI: 10.1142/6679

This book provides a broad yet comprehensive introduction to the analysis of harmonic maps and their heat flows. The first part of the book contains many important theorems on the regularity of minimizing harmonic maps by Schoen–Uhlenbeck, stationary harmonic maps between Riemannian manifolds in hig...

Table of Contents:

• The analysis of harmonic maps and their heat flows -- Contents -- Preface -- Organization of the book -- Acknowledgements -- Chapter 1: Introduction to harmonic maps -- Chapter 2: Regularity of minimizing harmonic maps -- Chapter 3: Regularity of stationary harmonic maps -- Chapter 4: Blow up analysis of stationary harmonic maps -- Chapter 5: Heat ows to Riemannian manifolds of NPC -- Chapter 6: Bubbling analysis in dimension two -- Chapter 7: Partially smooth heat ows -- Chapter 8: Blow up analysis on heat ows -- Chapter 9: Dynamics of defect measures in heat flows -- Bibliography -- Index
• Intro -- Contents -- Preface -- Organization of the book -- Acknowledgements -- 1 Introduction to harmonic maps -- 1.1 Dirichlet principle of harmonic maps -- 1.2 Intrinsic view of harmonic maps -- 1.3 Extrinsic view of harmonic maps -- 1.4 A few facts about harmonic maps -- 1.5 Bochner identity for harmonic maps -- 1.6 Second variational formula of harmonic maps -- 2 Regularity of minimizing harmonic maps -- 2.1 Minimizing harmonic maps in dimension two -- 2.2 Minimizing harmonic maps in higher dimensions -- 2.3 Federer's dimension reduction principle -- 2.4 Boundary regularity for minimizing harmonic maps -- 2.5 Uniqueness of minimizing tangent maps -- 2.6 Integrability of Jacobi fields and its applications -- 3 Regularity of stationary harmonic maps -- 3.1 Weakly harmonic maps into regular balls -- 3.2 Weakly harmonic maps in dimension two -- 3.3 Stationary harmonic maps in higher dimensions -- 3.4 Stable-stationary harmonic maps into spheres -- 4 Blow up analysis of stationary harmonic maps -- 4.1 Preliminary analysis -- 4.2 Rectifiability of defect measures -- 4.3 Strong convergence and interior gradient estimates -- 4.4 Boundary gradient estimates -- 5 Heat ows to Riemannian manifolds of NPC -- 5.1 Motivation -- 5.2 Existence of short time smooth solutions -- 5.3 Existence of global smooth solutions under RN &lt -- 0 -- 5.4 An extension of Eells-Sampson's theorem -- 6 Bubbling analysis in dimension two -- 6.1 Minimal immersion of spheres -- 6.2 Almost smooth heat ows in dimension two -- 6.3 Finite time singularity in dimension two -- 6.4 Bubbling phenomena for 2-D heat ows -- 6.5 Approximate harmonic maps in dimension two -- 7 Partially smooth heat ows -- 7.1 Monotonicity formula and a priori estimates -- 7.2 Global smooth solutions and weak compactness -- 7.3 Finite time singularity in dimensions at least three
• 7.4 Nonuniqueness of heat flow of harmonic maps -- 7.5 Global weak heat flows into spheres -- 7.6 Global weak heat flows into general manifolds -- 8 Blow up analysis on heat ows -- 8.1 Obstruction to strong convergence -- 8.2 Basic estimates -- 8.3 Stratification of the concentration set -- 8.4 Blow up analysis in dimension two -- 8.5 Blow up analysis in dimensions n &gt -- 3 -- 9 Dynamics of defect measures in heat flows -- 9.1 Generalized varifolds and rectifiability -- 9.2 Generalized varifold flows and Brakke's motion -- 9.3 Energy quantization of the defect measure -- 9.4 Further remarks -- Bibliography -- Index