On The Study Of Biharmonic Hypersurfaces In Conformally Flat Spaces

Biharmonic maps are maps between Riemannian manifolds which are critical pointsof the bi-energy functional. They are generalizations of harmonic maps. Since thederivation of the biharmonic equations appeared in [1], the study of biharmonic mapshas attracted more and more attention of mathematicians in the world and has becomea fascinating area of research. One of the central topics in this area is the study of bi-harmonic submanifolds (i.e., those submanifolds whose inclusion maps are biharmonic).Many results about constructions and classification of submanifolds in Euclidean spaces,spheres, hyperbolic space forms have been obtained. However, the following conjecturesremain open:Chen's conjecture: any biharmonic submanifold in Euclidean space Rm is mini-mal.The generalized Chen's conjecture: any biharmonic submanifold in a space ofnon-positive sectional curvature is minimal.In this thesis, we study biharmonic hypersurfaces in a conformally ?at space. Ourmain work includes the following:First, by using the equation of biharmonic hypersurfaces in Riemannian manifoldsin [2], we deduced the equation of biharmonic hypersurfaces in a conformally ?at space.Next, we give applications of the equation in the following areas:(1)we obtained the equation of constant mean curvature biharmonic hypersurfacesand totally umbilical biharmonic hypersurfaces in a conformally ?at space ;(2)Using the equation in spheres and hyperbolic space forms, we recovered theequation of biharmonic hypersurfaces in constant curvature spaces which R. Caddeo,S. Montaldo, C.Oniciuc obtained by other means. We also recovered the biharmonichyperplane Sm(?) is a proper biharmonic hypersurface in Sm+1 ;(3)Identified a family of conformally ?at metric h = f^-2(z)(dx² +dy² +dz²), whichturn the graphs of a linear function in (R³, h) into a proper biharmonic surface;(4)We obtained the equation of biharmonic hypersurfaces in S^m×R. We alsoobtained S^m-1( ?)×R is a proper biharmonic hypersurface in S^m×R and some otherresults.