This book presents a new front of research in conformal geometry, on sign-changing Yamabe-type problems and contact form geometry in particular. New ground is broken with the establishment of a Morse lemma at infinity for sign-changing Yamabe-type problems. This family of problems, thought to be out of reach a few years ago, becomes a family of problems which can be studied: the book lays the foundation for a program of research in this direction.

In contact form geometry, a cousin of symplectic geometry, the authors prove a fundamental result of compactness in a variational problem on Legrendrian curves, which allows one to define a homology associated to a contact structure and a vector field of its kernel on a three-dimensional manifold. The homology is invariant under deformation of the contact form, and can be read on a sub-Morse complex of the Morse complex of the variational problem built with the periodic orbits of the Reeb vector-field. This book introduces, therefore, a practical tool in the field, and this homology becomes computable.

Contents:

• Sign-Changing Yamabe-Type Problems:
• General Introduction
• Results and Conditions
• Conjecture 2 and Sketch of the Proof of Theorem 1: Outline
• 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
• 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
• Appendix 1: The Normal Form of (a, u) Near an Attractive Periodic Orbit of u
• Compactness
• Transmutations
• On the Morse Index of a Functional Arising in Contact Form Geometry
• and other chapters

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