The following theories were systematically developed: (1) Harmonic classification theory of Riemannian manifolds, (2) Quasiharmonic classification theory of Riemannian manifolds, (3) Theory of bounded biharmonic functions of Riemannian manifolds, (4) Dirichlet finite biharmonic functions, (5) Bounded Dirichlet finite biharmonic functions, (6) Riesz representation of biharmonic functions, (7) Green's functions of simply supported bodies, (8) Green's functions of clamped bodies. These theories were published in the 83 papers listed in Sections II-IV, and the research monograph described in Section V. (Author).

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.