Generalization Of J(?)rgens-Calabi-Pogorelov Theorem

In this paper, we concern the generalization of J(?)rgens-Calabi-Pogorelov theorem, and consider the following three problems:the rigidity theorems for Lagrangian translat-ing solitons in pseudo-Euclidean space Rn2n; a rigidity theorem for an affine K(?)hler-Ricci flat graph;a rigidity theorem of α-relative parabolic hyperspheres.In Chapter one, let f be a smooth strictly convex solution of defined on Rn, where ai, bi and c are constants, then the graph M▽f={(x,▽f)} of ▽f is a space-like translating solitons for mean curvature flow in pseudo-Euclidean space Rn2n with the indefinite metric ∑ dxidyi. We classify the entire solutions of the PDE above for dimension n= 1 and show every entire classical strictly convex solution (n≥2) must be a quadratic polynomial under a decay condition on the hessian(D2f). The main results are the following theorems: Theorem 1.1.5. Let f be a smooth solution of defined on the whole R1, where a, b and c are constants with ab≠0. Then, in case ab> 0, f(x) must be a quadratic polynomial or where C1> 0 and C2 are constants. In case ab< 0, there does not exist a smooth solution defined on the whole R1.Theorem 1.1.6. Let f be an entire smooth strictly convex solution of above PDE (n≥2). If the hessian (D2f) satisfies that for any e>0 then f(x) must be a quadratic polynomial.Li and Xu [9] proved that any entire strictly convex C∞-solution of the Monge-Ampere equation where do, d1,...,dn are constants, must be a quadratic polynomial. Their result extended a well-known theorem of J(?)rgens-Calabi-Pogorelov. The proof of the theorem is relatively simple in dimension n< 4, but very difficult and quite technical for n≥ 5,and the computations are rather complicated, see [9] or Section 4.5 in [17]. In chapter two, we first observe the relationship between the determinant p (the determinant of the Hessian (D2u)) and the inner product <grad ln　ρ, grad u>. Secondly we can directly estimate the determinant of the Hessian (D2u) on the section Su(O,C), see Lemma 2.3.1. With the help of this Lemma we can make use of different barrier functions to do gradient estimate, and we have a better method to deal with the term <grad ln ρ,grad u>. Therefore we provide a different and relatively simple proof for any dimension here.In Chapter three, let f be a smooth strictly convex solution of defmed on Rn, where α is a nonzero constant, and (a1,a2, …,an+1) is a constant vector in Rn+1. Then the graph hypersurface M={(x,f(x))} in Rn+1 is an a-relative parabolic affine hypersphere in Li-geometry. Li and Xu introduce a-relative geometry, and consider Euclidean complete α-relative parabolic affine hypersphere in [22]. In this chapter, we will further classify Euclidean complete α-relative parabolic affine hyperspheres and show that any smooth strictly convex entire solution of the above PDE with α (?) [n+2/n+1,n+2] must be quadratic polynomial, which generalized the main results in [22].