预定仿射曲率的Monge-Ampère方程

仿射微分几何中,完备双曲型仿射球的分类可以归结到一个Monge-Ampère方程的边值问题,它等价于一个实变量情形下的Fefferman方程.这些方程凸解的边界估计,我们仍然知之甚少.预定仿射Gauss-Kronecker曲率的四阶方程可以分解成两个联立的Monge-Ampère方程,给定边值解这个联立方程,存在一个预定仿射Gauss-Kronecker曲率的完备超曲面,但这个超曲面的一些整体性质仍待研究.本项目利用方程和仿射几何的方法研究这些方程解的边界行为,试图建立这些方程的解关于距离函数的精确估计,得到对应超曲面的一些曲率性质.我们也考虑凸区域上由特征函数和核函数定义的两个射影不变度量,用仿射不变量来描述这些度量的标量曲率的渐近展开式.最后我们研究一类Monge-Ampère方程的Bernstein性质.

In affine differential geometry, the classification of complete hyperbolic affine hyperspheres is deduced to the boundary value problem of a Monge-Ampère equation, it is equivalent to a real analogue of Fefferman equation. It is still less known to the boundary estimates of the convex solutions of these equations. The fourth order equation of prescribed affine Gauss-Kronecker curvature can be written as two linked Monge-Ampère equations. By solving of these equation with given boundary values, there exists a complete hypersurface with prescibed affine.Gauss-Kronecker curvature, but some global properties of the constructed hypersurface should be more studied. In this project, we use the methods of equations and affine geometry to study the boundary behaviors of the solutions of these equations, we try to obtain the exact estimates of the solutions with respect to the distance function, hence give some curvature properties of the corresponding hypersurfaces. On a convex domain, we also consider two projectively invariant metrics defined by the characteristic function and the kernel function, and try to use the affine invariants to describe the asymptotic expansions of the scalar curvatures. At last we study the Bernstein property of a class of Monge-Ampère equations.