Biharmonic Submanifolds And Biharmonic Maps In Riemannian Geometry

The book aims to present a comprehensive survey on biharmonic submanifolds and maps from the viewpoint of Riemannian geometry. It provides some basic knowledge and tools used in the study of the subject as well as an overall picture of the development of the subject with most up-to-date important results. Biharmonic submanifolds are submanifolds whose isometric immersions are biharmonic maps, thus biharmonic submanifolds include minimal submanifolds as a subclass. Biharmonic submanifolds also appeared in the study of finite type submanifolds in Euclidean spaces. Biharmonic maps are maps between Riemannian manifolds that are critical points of the bienergy. They are generalizations of harmonic maps and biharmonic functions which have many important applications and interesting links to many areas of mathematics and theoretical physics. Since 2000, biharmonic submanifolds and maps have become a vibrant research field with a growing number of researchers around the world, with many interesting results have been obtained. This book containing basic knowledge, tools for some fundamental problems and a comprehensive survey on the study of biharmonic submanifolds and maps will be greatly beneficial for graduate students and beginning researchers who want to study the subject, as well as researchers who have already been working in the field. Contents: Differentiable ManifoldsRiemannian and Pseudo-Riemannian ManifoldsSubmanifoldsBiharmonic Curves and Surfaces in Pseudo-Euclidean SpacesSome Progress on Chen's Biharmonic ConjectureSome Progress on Generalized Chen's ConjectureBiharmonic Submanifolds in SpheresBiharmonic Submanifolds in Some Other Model SpacesHarmonic Maps and Their GeneralizationsBiharmonic Maps Between Riemannian ManifoldsBiharmonic Conformal MapsSecond Variation of Bienergy, Liouville-type and Unique Continuation Theorem Readership: Graduate students and researchers from the fields of geometry and analysis.Biharmonic Submanifolds;Biharmonic Maps, Riemannian Geometry;Chen's Conjecture on Biharmonic Submanifolds;Generalized Chen's Conjecture on Biharmonic Submanifolds;Biharmonic Submanifolds in Spheres;Harmonic Maps;p-harmonic Maps;Infinity Harmonic Maps;f-harmonic Maps, F-harmonic Maps;Biharmonic Maps between Surfaces;Biharmonic Maps into Spheres;Equivariant Biharmonic Maps;Stable Biharmonic Maps;f-biharmonic Maps;Conformal Biharmonic Maps;Biharmonic Conformal Immersions;Biharmonic Riemannian Submersions;Second Variation of Bienergy;Liouville-type Theorems;Unique Continuation Theorems;Minimal Submanifolds;f-Minimal Hypersurfaces0 Key Features: Written by two experts in the subject, this is the first book that gives acomprehensive survey on the study of biharmonic submanifolds and mapsIncludes detailed proofs of most important results and the relations among various directions in the study of the subjectIt is useful to researchers who have been working on the subject or the related topics as well as to graduate students or new researchers who have an interest in studying the subject and the related topics since the book also provides basic knowledge and tools used in the study of biharmonic maps and submanifolds