Some Variational Problems Of Affine Hypersurfaces

The research of affine geometry has a long history of development,and the variational problem is also a key problem in the research of affine geometry.Therefore,this paper focuses on the variational problems of central affine hypersurface,spatial form and mean curvature function on hypersurface,and obtains some meaningful research results.This paper is mainly divided into five chapters:The first chapter mainly introduces affine geometry,central affine geometry and central affine hypersurface,then describes the classification of complete affine spheres and the local classification of affine spheres with constant curvature in affine geometry,finally introduces the development of geometric variational problems from Euclidean space to affine space,and then introduces the variational formulas of arc length and volume,and the variation of the mean curvature function of spatial hypersurfaces,The evolution of the central affine volume variation of nondegenerate hypersurfaces,the geometric variation of submanifolds,the variation of curvature functional on Riemannian manifolds and related problems,and the presentation of the main research purposes of this paper.The second chapter presents some affine geometric quantities needed in this paper,including the basic equation of hypersurface,central affine metric,Levi civita connection and covariant derivative of connection,which is convenient for the follow-up research of this paper.The third chapter mainly studies the non degenerate hypersurface in affine space.Using the representation of central affine geometric quantity under natural parameters,we give a simple and direct proof of the first and second variational formulas of central affine volume.We also further study the variational formulas of techebychev form length square integral and 3 form length square integral.In Chapter 4,we continue to use the parametric representation of Hypersurfaces in Chapter 3 to give the variational formula of higher-order mean curvature function of central affine geometric hypersurfaces.The fifth chapter summarizes the main results of this paper,and points out the key aspects of scientific research on this problem in the future.