The analysis of harmonic maps and their heat flows / Fanghua Lin, Changyou Wang.

Format:

Book

Author/Creator:

Lin, Fanghua.

Contributor:

Wang, Changyou, 1967-

Language:

English

Subjects (All):

Harmonic maps--Textbooks.

Harmonic maps.

Heat equation--Textbooks.

Heat equation.

Riemannian manifolds--Textbooks.

Riemannian manifolds.

Physical Description:

1 online resource (280 p.)

Edition:

1st ed.

Place of Publication:

Hackensack, NJ : World Scientific, c2008.

Language Note:

English

Summary:

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 harmon

Contents:

Contents; 3.2 Weakly harmonic maps in dimension two; 3.3 Stationary harmonic maps in higher dimensions; 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 principle2.4 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; 3.4 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 solutions5.3 Existence of global smooth solutions under RN < 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 maps7.5 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 > 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

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 251-264) and index.

ISBN:

9786611938086

9781281938084

1281938084

9789812779533

9812779531

OCLC:

308160132