Affine Maximal Hyp Ersurfaces And Affine Hyperspheres Generated By A Class Of Radial Function

The thesis mainly studies two properties of the locally strongly convex hy-persurface M generated by a class of radial function. Our main results are as follows:First, when M is an affine maximal hypersurface, we give the equation that f satisfied and reduce the order of the equation. Then we give a characterization of elliptic paraboloid.Then, when M is an affine hypersphere, we give the equation that f satisfied and reduce the order of the equation. Moreover we give the necessary and sufficient condition when M is a improper affine hypersphere.The main method used in this paper is the calculation of graph immersion, we transform the partial differential equation into the ordinary differential equation which f satisfied by the symmetry of f.