Geometry Of Polyharmonic Hypersurfaces In Space Forms

The geometry of polyharmonic maps is an important problem in differential geome-try.In this paper,we study the geometry of polyharmonic hypersurfaces in space forms.The main results are the following four parts:1.Through making careful analysis of Gauss and Codazzi equations,we prove that four dimensional biharmonic hypersurfaces in non-flat space forms have constant mean curvature.2.We prove that a proper CMC triharmonic hypersurface with at most three distinct principal curvatures in a space form has constant scalar curvature,and give the complete classification of the complete proper CMC triharmonic hypersurfaces in 4--dimensional sphere.3.We prove that a proper CMC triharmonic hypersurface in a space form with four distinct principal curvatures has constant scalar curvature if the multiplicity of the zero principal curvature is at most one.4.We give an optimal upper bound of the mean curvature for a non-totally umbilical proper CMC k--harmonic hypersurface with constant scalar curvature in a sphere.