On Some Problems On Affine Hypersurfaces

In affine geometry, once the affine hyperspheres are classified, a further interestingquestion emerged is to study the complete hypersurfaces with constant affine meancurvature (abbreviated by CAMC). Many results on this direction are achieved recently([LSZ-1], [LSZ-2], [LJ-1], [LJ-2], [LSC], [TW-1], [TW-2]).Although one can locally construct many hyperbolic affine hypersuffaces withCAMC, the classification of complete hyperbolic affine hypersurfaces with CAMC iscompletely open. Few results on this problem are known. For example, no any examplebut affine hyperspheres is known so far.In chapter one, by solving a fully nonlinear fourth order PDE that degeneratesat boundary, we construct a large class of affine hypersurfaces with CAMC; further-more, we develop a method for gradient estimates and therefore be able to prove theEuclidean completeness of these surfaces. To summary, we prove that: for a givenbounded convex domainΩ(?)Rⁿ with smooth boundary and a functionφ∈C^∞((?)),there is an Euclidean complete hypersurface with negative CAMC. Furthermore, whenn=2, these hypersurfaces are also affine complete.In chapter two, we study the following fourth order PDE: sum from i,j=1 to n U^ijω_ij=-L,ω=(det(f_ij))^a onΩ(0.1)where a is a nonzero constant, f is a smooth convex function on the smooth boundedconvex domainΩ(?)Rⁿ, L∈C^∞((?)) and U^ij is the cofactor of f_ij in the Hessian matrix of f. As we know●when a=-(n+1)/(n+2) (0.1) is the equation for the hypersurface with prescribed affinemean curvature L;●when a=-1, (0.1) is the famous Abreu equation that is related to the complex toruswith constant scalar curvature in K(?)hler geometry.One may ask if there is certain geometric meaning when a takes other values. It turnsout that (0.1) is the equation for hypersurface with prescribed relative affine meancurvature in relative affine geometry. Using the method given in chapter one, we areable to solve the equation for -1＜a＜0, therefore we obtain the following result:for a given smooth bounded domainΩ(?)Rⁿ and a functionφ∈C^∞((?)), there is anEuclidean complete hypersurface with constant relative affine mean curvature L＜0.When a＞0, we prove that: if M is an Euclidean complete hypersurface with constantrelative affine mean curvature L＜0 and is given by a graph of a convex function f,then the Legendre transformation domain of f is Rⁿ.In chapter three and four, we give affirmative answers to the conjectures proposedin [LSC]and [LSZ-2]. In [LSC], Li-Simon-Chen studied the locally strongly convexhypersurfaces of hyperbolic type with constant affine Gauss-Kronecker curvature S_n=1. They proved that given a bounded convex domainΩ(?)/Rⁿ with smooth boundaryand a functionφ∈C^∞((?)Ω), there is a convex hypersurface M with constant affineGauss-Kronecker curvature that is both Euclidean complete and W-complete. WhenΩis a standard ball, they proved that M is also affine complete. In [LSZ-2], Li-Simon-Zhao further solved the problem for a prescribed Gauss-Kronecker curvature that maynot be constant. The following problem remains open and was conjectured by them: Mis affine complete for any domainΩ. This is solved in chapter three and four for twocases respectively.