Overview

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 IntroductionResults and ConditionsConjecture 2 and Sketch of the Proof of Theorem 1: OutlineThe Difference of TopologyOpen ProblemsPreliminary Estimates and Expansions, the Principal TermsPreliminary EstimatesProof of the Morse Lemma at Infinity when the Concentrations are ComparableProof of the Morse Lemma at Infinity Contact Form Geometry: General IntroductionOn the Dynamics of a Contact Structure Along a Vector Field of Its KernelAppendix 1: The Normal Form of (α, υ) Near an Attractive Periodic Orbit of υCompactnessTransmutationsOn the Morse Index of a Functional Arising in Contact Form Geometryand other chaptersReadership: Researchers seeking new and fresh directions in the field of conformal geometry.

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9781281120656

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