The Notes About Affine Complete And Euclidean Complete Of Affine Hypersurfaces

In this paper, we study the relation between the affine (metric) completeness and the Euclidean (metric) completeness on equiaffine hypersurfaces. The main result is as follows:Let Ω(?)R be a convex domain and f:Ωâ†'R is a smooth and strongly convex function. Let M be the graph hypersurface of f. Assume that the Pick invariant j(B) of the locally strongly convex affine hypersurface M with respect to the Blaschke metric G is bounded from above. Then the Riemannian manifold (M, G) is complete if and only if M is Euclidean complete, i. e. M is complete with respect to its induced metric from the Euclidean space Wa+1.Our proof of the above result is a detailed clarification to one claim made by Professor An-Min Li.From the above result, and combining with S.Y. Cheng and S.T. Yau’s proof to the first part of the Calabi conjecture on the classification of affine complete, hyperbolic affine hyperspheres (under the additional condition of Euclidean completeness), and also T. Sasaki’s proof of the Calabi conjecture’s second part, we obtain immediately a new and complete proof of the Calabi conjecture.