Cheng and Yau’s Work on Affine Geometry and Maximal Hypersurfaces

As part of their great burst of activity in the late 1970s, Cheng and Yau proved many geometric results concerning differential structures invariant under affine transformations of R n. Affine differential geometry is the study of those differential properties of hypersurfaces in R n+1 which are invariant under volume-preserving affine transformations. One way to develop this theory is to start with the affine normal, which is an affine-invariant transverse vector field to a convex C 3 hypersurface. A hypersurface is an affine sphere if the lines formed by the affine normals all meet at a point, called the center. A convex affine sphere is called elliptic, parabolic, or hyperbolic according to whether the affine normals point toward the center, are parallel (the center being at infinity), or away from the center, respectively. The global theory of elliptic and parabolic affine spheres is quite tame: Every properly immersed elliptic affine sphere is an ellipsoid, while every properly immersed parabolic affine sphere is a paraboloid. In this generality, both these results follow from Cheng-Yau's paper [8], in which they show that any properly immersed affine sphere must have complete affine metric. Then one may appeal to earlier results of Calabi [3] to classify global elliptic and parabolic affine spheres. The global classification of parabolic affine spheres is an extension of the well-known result of Jörgens, Calabi, and Pogorelov that any entire convex solution to the Monge-Ampère equation det u ij = 1 on R n is a quadratic polynomial. Calabi realized that hyperbolic affine spheres are more varied, by noting that two quite different convex cones contain hyperbolic affine spheres asymptotic to their boundaries. In addition to the hyperboloid, asymptotic to a round cone over an ellipsoid, Calabi also wrote down an affine sphere asymptotic to the boundary of the first orthant in R n+1 , which is a cone over an n-dimensional simplex [3]. Based on these explicit examples in these two extremal cases of convex cones, Calabi conjectured that each proper convex cone admits a unique (up