Abstract

Consider the following equation

sum_{i,j=1}^n U^{ij} w_{ij} = - L, w =[det((partial^2 u)/(partial xi_i partial xi_j))]^a,

where L is some given C^infty function, u(xi) is a smooth and strictly convex function defined in a convex domain in R^n, (U^{ij}) denotes the cofactor matrix of the Hessian matrix (partial^2 u)/(partial xi_i partial xi_j) and a <> 0 is a constant. When a = -(n+1)/(n+2) and L=0, the above equation is the equation for affine maximal hypersurfaces. When a = -1 it is called the Abreu equation, which appears in the study of the differential geometry of toric varieties, where L is the scalar curvature of the Kahler metric. In this talk, we will discuss some recent development on the study of the relevant differential equations in the differential geometry.