Bahri A., Xu Y. Recent Progress in Conformal Geometry

Singapore: Imperial College Press. – 2007. – 521 p. (Imperial College Press Advanced Texts in Mathematics – Vol. 1). This book is divided into two parts. The first part is about Sign-Changing Yamabe-type problems. A Morse Lemma at infinity, under reasonable basic conjectures, is proved for such problems. This work is an attempt to define a new area of research for nonlinear analysts.The second part of this book is about Contact Form Geometry via Legendrian curves. We had been searching for a long time for such a result. This work entitled Compactness will be published independently by the first author and dedicated to his long time friend and collaborator Haim Brezis for his sixtieth birthday. Both directions of research i.e. Conformal Geometry, Einstein equations, Yang-Mills equations on one hand, Contact Form Geometry on the other hand, are also studied by other techniques due to hard-chore Geometry and Symplectic Geometry. In fact, Geometers have always been our co-area researchers. We view these areas which we have contributed to define – for Contact Form Geometry – in a different way and with different techniques. This book is a book of collaboration and research. It also defines new goals and presents a new understanding. It is not (yet) a textbook for graduate students. It rather informs our collaborators about a definite progress in the two above mentioned areas.Sign-Changing Yamabe-Type Problems
General Introduction
Results and Conditions
Conjecture 2 and Sketch of the Proof of Theorem
The Difference of Topology
Open Problems
Preliminary Estimates and Expansions, the Principal Terms
Preliminary Estimates
Proof of the Morse Lemma at Infinity When the Concentrations are Comparable
Redirecting the Estimates
Proof of the Morse Lemma at Infinity
Contact Form Geometry
General Introduction
On the Dynamics of a Contact Structure along a Vector Field of its Kernel
The normal form for (α, v) when α does not turn well
The Normal Form of (α, v) Near an Attractive Periodic Orbit of v
Compactness
Transmutations
On the Morse Index of a Functional Arising in Contact Form Geometry
Calculation of ∂2J∞(x∞).u
2.u2
Calculation of ∂2J∞(x∞).u
2.u4
Other Second Order Derivatives