Conformal, Riemannian, and Lagrangian Geometry

by Chang, Sun-Yung A.; Yang, Paul C.; Grove, Karsten; Wolfson, Jon G.; Freire, Alexandre

• ISBN13: 9780821832103
• ISBN10: 0821832107
• Format: Paperback
• Copyright: 2002-07-01
• Publisher: Amer Mathematical Society

Summary

Recent developments in topology and analysis have led to the creation of new lines of investigation in differential geometry. The 2000 Barrett Lectures present the background, context and main techniques of three such lines by means of surveys by leading researchers. The first chapter (by Alice Chang and Paul Yang) introduces new classes of conformal geometric invariants, and then applies powerful techniques in nonlinear differential equations to derive results on compactificationsof manifolds and on Yamabe-type variational problems for these invariants. This is followed by Karsten Grove's lectures, which focus on the use of isometric group actions and metric geometry techniques to understand new examples and classification results in Riemannian geometry, especially inconnection with positive curvature. The chapter written by Jon Wolfson introduces the emerging field of Lagrangian variational problems, which blends in novel ways the structures of symplectic geometry and the techniques of the modern calculus of variations. The lectures provide an up-do-date overview and an introduction to the research literature in each of their areas. The book is a very enjoyable read, which should prove useful to graduate students and researchers in differential geometryand geometric analysis.

Table of Contents

Preface	ix
Partial Differential Equations Related to the Gauss-Bonnet-Chern Integrand on 4-manifolds
Sun-Yung A. Chang
Paul C. Yang
Introduction	1	(2)
Finiteness of conformally flat structures	3	(5)
Background on σ2	8	(8)
Deforming σ2 to a positive function	16	(6)
Deforming σ2 to a constant	22	(10)
Bibliography	29	(2)
Geometry of, and via, Symmetries
Karsten Grove
Introduction	31	(1)
Geometry of isometry groups	32	(3)
Structure and classification program	35	(4)
Constructions and examples	39	(4)
Emergence of isometries	43	(4)
Open problems	47	(8)
Bibliography	51	(4)
Lagrangian Cycles and Volume
Jon G. Wolfson
Introduction	55	(1)
The variational problem	55	(9)
Existence	64	(10)
Regularity: cones and monotonicity	74
Bibliography	85