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 higher dimensions by Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces by Helein, as well as on the structure of a singular set of minimizing harmonic maps and stationary harmonic maps by Simon and Lin.The second part of the book contains a systematic coverage of heat flow of harmonic maps that includes Eells-Sampson's theorem on global smooth solutions, Struwe's almost regular solutions in dimension two, Sacks-Uhlenbeck's blow-up analysis in dimension two, Chen-Struwe's existence theorem on partially smooth solutions, and blow-up analysis in higher dimensions by Lin and Wang. The book can be used as a textbook for the topic course of advanced graduate students and for researchers who are interested in geometric partial differential equations and geometric analysis.
Author(s): Fanghua Lin, Changyou Wang
Publisher: World Scientific Publishing Company
Year: 2008
Language: English
Pages: 280
3.3 Stationary harmonic maps in higher dimensions......Page 6
Preface......Page 8
Organization of the book......Page 10
Acknowledgements......Page 12
1.1 Dirichlet principle of harmonic maps......Page 14
1.2 Intrinsic view of harmonic maps......Page 15
1.3 Extrinsic view of harmonic maps......Page 16
1.4 A few facts about harmonic maps......Page 17
1.5 Bochner identity for harmonic maps......Page 18
1.6 Second variational formula of harmonic maps......Page 20
2.1 Minimizing harmonic maps in dimension two......Page 22
2.2 Minimizing harmonic maps in higher dimensions......Page 28
2.3 Federer's dimension reduction principle......Page 40
2.4 Boundary regularity for minimizing harmonic maps......Page 44
2.6 Integrability of Jacobi fields and its applications......Page 53
3.1 Weakly harmonic maps into regular balls......Page 62
3.4 Stable-stationary harmonic maps into spheres......Page 90
4.1 Preliminary analysis......Page 100
4.2 Rectifiability of defect measures......Page 107
4.3 Strong convergence and interior gradient estimates......Page 114
4.4 Boundary gradient estimates......Page 119
5.1 Motivation......Page 122
5.2 Existence of short time smooth solutions......Page 123
5.3 Existence of global smooth solutions under RN < 0......Page 126
5.4 An extension of Eells-Sampson's theorem......Page 131
6 Bubbling analysis in dimension two......Page 138
6.1 Minimal immersion of spheres......Page 139
6.2 Almost smooth heat ows in dimension two......Page 147
6.3 Finite time singularity in dimension two......Page 154
6.4 Bubbling phenomena for 2-D heat ows......Page 157
6.5 Approximate harmonic maps in dimension two......Page 166
7.1 Monotonicity formula and a priori estimates......Page 174
7.2 Global smooth solutions and weak compactness......Page 178
7.3 Finite time singularity in dimensions at least three......Page 186
7.4 Nonuniqueness of heat flow of harmonic maps......Page 187
7.5 Global weak heat flows into spheres......Page 189
7.6 Global weak heat flows into general manifolds......Page 192
8 Blow up analysis on heat ows......Page 202
8.1 Obstruction to strong convergence......Page 203
8.2 Basic estimates......Page 205
8.3 Stratification of the concentration set......Page 210
8.4 Blow up analysis in dimension two......Page 217
8.5 Blow up analysis in dimensions n > 3......Page 220
9 Dynamics of defect measures in heat flows......Page 232
9.1 Generalized varifolds and rectifiability......Page 233
9.2 Generalized varifold flows and Brakke's motion......Page 244
9.3 Energy quantization of the defect measure......Page 252
9.4 Further remarks......Page 262
Bibliography......Page 264
Index......Page 278
